Tensor Analysis and Curvilinear Coordinates

1 Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: May 19, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers wh en not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-co mparison-chart . The table of contents has live links. Most PDF view ers provide these links as bookmarks on the left. Overview an d Summary ........................................................................................................... .............. 7 1. The Transformation F: invertibility, coordinate lines, and level surfaces.................................. 12 Example 1: Polar coordinates (N=2)............................................................................................ ......13 Example 2: Spherical coordinates (N=3)........................................................................................... 14 Cartesian Space and Qu asi-Cartesian Space...................................................................................... .15 Pictures A,B, C and D.......................................................................................................................... 16 Coordinate Lines................................................................................................................................. 16 Example 1: Polar coordinates, coordinate lines ................................................................................ .17 Example 2: Spherical coordinates, coordinate lines .......................................................................... 18 Level Surfaces..................................................................................................................................... 18 2. Linear Local Transformations associated with F : scalars and two kinds of vectors ................. 20 2.1 Linear Local Transformations............................................................................................... ........ 20 2.2 Scalars........................................................................................................................................... 21 2.3 Contravariant vectors .................................................................................................................... 22 2.4 Covariant vectors .......................................................................................................................... 22 2.5 Bar notation................................................................................................................................... 23 2.6 Origin of the names co ntravariant and covariant .......................................................................... 23 2.7 Other vector types? ........................................................................................................ ............... 24 2.8 Linear transformations ..................................................................................................... ............. 25 2.9 Vectors that are cont ravariant by definition............................................................................... ...26 2.10 Vector Fields............................................................................................................................... 26 2.11 Names and symbols .................................................................................................................... 27 2.12 Definition of the words " scalar", "vector" and "tensor".............................................................. 28 3. Tangent Base Vectors e n and Inverse Tangent Base Vectors u' n.................................................. 30 3.1 Differential Displacements ................................................................................................. .......... 30 3.2 Definition of the e n ; the e n are the columns of S......................................................................... 31 3.3 en as a contrava riant vector ........................................................................................................... 33 3.4 A semantic question: unit vectors ......................................................................................... ....... 33 Example 1: Polar coordinates, tangent base vectors .......................................................................... 34 Example 2: Spherical Coordinates, tangent base vectors................................................................... 35 3.5 The inverse tangent base vectors u' n and inverse coordinate lines................................................ 36 Example 1: Polar coordinates: inverse tange nt base vectors and inverse coordinate lines............... 37 2 4. Notions of length, distance and scalar product in Cartesian Space.............................................. 39 5. The Metric Tensor ........................................................................................................... ................. 41 5.1 The Pictur e D Context ...................................................................................................... ............ 41 5.2 Definition of the metric tensor............................................................................................ .......... 41 5.3 Inverse of the metric tensor............................................................................................... ............ 44 5.4 A metric tensor is symmetric ............................................................................................... ......... 44 5.5 det(g) and g nn of a Cartesian-generated metric tensor are non-negative....................................... 45 5.6 Definition of two ki nds of rank-2 tensors ..................................................................................... 45 5.7 Proof that the metric tensor a nd its inverse are both rank-2 tensors ............................................. 46 5.8 Metric tensor c onverts vector types ........................................................................................ ......49 5.9 Vectors in Cartesian space ................................................................................................. ........... 49 5.10 The covariant dot product A • B and norm |A| ......................................................................... 50 5.11 Metric tensor and tangent base vectors: scale factors and orthogonal coordinates................... 52 5.12 The Jacobian J............................................................................................................ ................. 55 5.13 Some relations between g, R and S in Pictures B and C (Cartesian x-space)............................. 58 Example 1: Polar coordinates: metric tensor and Jacobian............................................................... 60 Example 2: Spherical coordinates: metric tensor and Jacobian ........................................................ 61 5.14 Special Relativity and its Metric Tensor: vectors and spinors................................................... 62 5.15 General Relativity an d its Metric Tensor.................................................................................. ..65 5.16 Continuum Mechanics a nd its Metric Tensors............................................................................ 66 6. Reciprocal Base Vectors E n and Inverse Reciprocal Base Vectors U' n........................................ 73 6.1 Definition of the E n....................................................................................................................... 73 6.2 The e n and En Dot Products and R eciprocity (Duality) ................................................................ 74 6.3 Covariant partners for e n and En.................................................................................................. 78 6.4 Summary of the basic facts about e n and En................................................................................. 80 6.5 Repeat the above for the inverse transformation: definition of the U' n........................................ 80 6.6 Expanding vectors on differe nt sets of basis vectors .................................................................... 81 6.7 Another way to write the E n.......................................................................................................... 84 6.8 Comparison of e ¯n and En.............................................................................................................. 85 6.9 Handedness of coordinate systems: the e n , the sign of det(S), and Parity .................................. 86 7. Translation to the Standard Notation ........................................................................................ .....89 7.1 Outer Products .............................................................................................................................. 89 7.2 Mixed Tensors and Notation Issues .......................................................................................... ....90 7.3 The up/down bell goes off .................................................................................................. .......... 91 7.4 Some Preliminary Translations; rais ing and lowering tens or indices with g............................... 92 7.5 Dealing with the matrices R and S ; various Rules and Theorems ............................................... 96 7.6 Orthogonality Rules, Invers ion Rules, Cancellation Rules......................................................... 101 7.7 About δ and ε.............................................................................................................................. 104 7.8 Covariance and Ma trix Multiplication....................................................................................... .105 7.9 Matrix Inverse, Transpose and Determinant............................................................................... 108 7.10 Tensors of Rank n, direct products, Li e groups, symmetry and Ricci-Levi-Civita .................. 114 7.11 The Contraction Tilt-Reversal Rule........................................................................................ ..117 7.12 The Contraction Neutralization Rule ....................................................................................... .119 7.13 The tangent and reciprocal b ase vectors and expansions on same............................................ 121 7.14 Comment on Covariant versus Contravariant........................................................................... 125 7.15 The Significance of Tensor Analysis ....................................................................................... .126 3 7.16 The Christoffel Business: covariant derivatives ...................................................................... 129 7.17 Expansions of hi gher order tensors........................................................................................... 130 7.18 Collection of Facts about basis vectors e n , u'n and bn. ............................................................. 131 7.19 More on basis vectors and matrix elements of R and S ............................................................ 134 8. Transformation of Differential Length, Area and Volume......................................................... 142 8.1 Overview of Chapter 8...................................................................................................... .......... 142 8.2 The differential N-piped mapping........................................................................................... ....143 8.3 Properties of the finite N-piped spanned by the e n in x-space .................................................... 145 8.4 Back to the differential N-piped mapping: how edges, areas and volume transform ................ 147 (a) The Setup................................................................................................................................. 147 (b) Edge Transformation........................................................................................................ ....... 148 (c) Area Transformation................................................................................................................ 149 (d) Volume Transformation .......................................................................................................... 150 (e) Covariant Magnitudes....................................................................................................... ....... 151 (f) Two Theorems : g'nn g' = cof(g' nn) and |(Πx i≠nei)| = cof(g' nn) ...................................... 152 (g) Cartesian-View Magnitude Ratios .......................................................................................... 154 (h) Nested Cofactor Formulas and STS notation........................................................................... 155 (i) Transformation of arbitrary differential vectors, areas and volume......................................... 158 (j) Concatenation (Composition) of Transformations................................................................... 160 (k) Examples of area magnitude transformation for N = 2,3,4 ..................................................... 161 Example 2: Spherical Coordinates: area patches ........................................................................ 162 8.5 Transformation of Differential Volume applied to Integration................................................... 163 8.6 Interpretations of the Jacobian ............................................................................................ ........ 165 8.7 Volume integration of a tensor field under linear transformations ............................................. 165 9. The Divergence in cu rvilinear coordinates ................................................................................... 167 9.1 Geometric Derivation of the Curvilinear Divergence Formula .................................................. 167 9.2 Various expressions for div B.............................................................................................. ....... 170 9.3 Translation from Picture B to Picture M&S ............................................................................... 172 9.4 Comparison of various authors' notations................................................................................... 173 10. The Gradient in cu rvilinear coordinates.................................................................................... .175 10.1 Expressions for grad f .................................................................................................... ........... 175 10.2 Expressions for grad f • B......................................................................................................... 178 11. The Laplacian in curvilinear coordinates ................................................................................... 180 12. The Curl in curvilinear coordinates ........................................................................................ ....182 12.1 Definition of curl B................................................................................................................... 182 12.2 Computation of the line integral ............................................................................................... 183 12.3 Solving for the curl ................................................................................................................... 185 12.4 Various forms of the curl ................................................................................................. ......... 186 12.5 The curl in orthogonal coordinate systems ............................................................................... 188 12.6 The curl in N > 3 dimensions.............................................................................................. ......189 13. The Vector Laplacian in curvilinear coordinates....................................................................... 192 13.1 Derivation of the Vector Laplacian in general curvilinear coordinates .................................... 192 13.2 The Vector Laplacian in or thogonal curvilinear coordinates.................................................... 195 13.3 The Vector Laplacian in Cartesian coordinates ........................................................................ 196 4 14. Summary of Differential Opera tors in curvilinear coordinates ............................................... 198 14.1 Summary of Conventions and How To..................................................................................... 198 14.2 divergence................................................................................................................ ................. 200 14.3 gradient and gradient dot vector.......................................................................................... ......200 14.4 Laplacian................................................................................................................. .................. 201 14.5 curl ............................................................................................................................................ 202 14.6 vector Laplacian.......................................................................................................... .............. 203 14.7 Example 1: Polar coordinates: a practical curvilinear notation.............................................. 204 15. Covariant derivation of all curvilinear differential operator expressions ............................... 206 15.1 Review of Chapters 9 through 13 ........................................................................................... ..206 15.2 The Covariant Method ...................................................................................................... ........ 207 15.3 divergence (Chapter 9).................................................................................................... .......... 209 15.4 gradient and gradient dot vector (Chapter 10) .......................................................................... 209 15.5 Laplacian (Chapter 11).................................................................................................... .......... 210 15.6 curl (Chapter 12) ....................................................................................................................... 211 15.7 vector Laplacian (Chapter 13)................................................................................................... 211 15.8 Verification that two te nsorizations are the same .................................................................... 214 Appendix A: Reciprocal Base Vectors the Hard Way.................................................................... 217 A.1 Introduction................................................................................................................................ 217 A.2 Definition of E n.......................................................................................................................... 218 A.3 Simpler notation......................................................................................................................... 218 A.4 Generalized Cross Product of N-1 vectors of dimension N ....................................................... 218 A.5 Missing Man Formation...................................................................................................... ....... 220 A.6 Apply this Notation to E ............................................................................................................ 220 A.7 Compute E m • en......................................................................................................................... 221 A.8 Compute E n • Em........................................................................................................................ 222 A.9 Summary of relationship between the tangent and reciprocal base vectors............................... 222 A.10 Another Cross Product Notation and another expression for E ............................................... 223 Appendix B: The Geometry of Pa rallelepipeds in N dimensions................................................... 224 B.1 Overview ................................................................................................................... ................. 224 B.2 Preliminary: Equation of a plane in N dimensions.................................................................... 224 B.3 N-pipeds and their Faces in Various Dimensions ...................................................................... 225 (a) The 1-piped................................................................................................................ .............. 225 (b) The 2-piped ................................................................................................................ ............. 225 (c) The 3-piped................................................................................................................ .............. 227 (d) The N-piped................................................................................................................ ............. 228 B.4 The question of inward versus outward facing normal vectors.................................................. 229 B.5 The Face Area and Volume of N-pipeds in Various Dimensions ............................................. 230 (a) The 2-piped................................................................................................................ .............. 230 (b) The 3-piped ................................................................................................................ ............. 231 (c) The 4-piped................................................................................................................ .............. 234 (d) The N-piped................................................................................................................ ............. 235 B.6 Summary of Main Results of this Appendix .............................................................................. 237 Appendix C: Elliptical Polar Coords, View s of x'-space, Jacobian Integration Rule................. 239 C.1 Elliptical polar coordinates............................................................................................... .......... 239 C.2 Forward coordinate lines................................................................................................... ......... 240 5 C.3 Inverse coordinate lines................................................................................................... ........... 240 C.4 Drawing a contravariant vector V in x-space: the meaning of V' n ........................................... 241 C.5 Drawing a contravariant vector V' in x'-space: two "Views" .................................................... 242 C.6 Drawing the specific contravariant vector dx in x-space and x'-space....................................... 245 C.7 Study of how dx transforms in the mapping between x-space and x'-space .............................. 247 C.8 A Derivation of the Jacobian Integration Rule........................................................................... 249 Appendix D: Tensor Densities and the ε tensor .............................................................................. 252 D.1 Definition of a tensor density............................................................................................. ........ 252 D.2 A few facts about tensor densities......................................................................................... .....253 D.3 Theorem about Totally Antisymmetric Tensors: there is really only one: εabc...................... 256 D.4 The contravariant ε tensor.......................................................................................................... 257 D.5 Some facts about the ε tensor..................................................................................................... 259 D.6 The covariant ε tensor : repeat Section D.4 as if its weight were not known ........................... 261 D.7 Generalized cross products ................................................................................................. ....... 262 D.8 The tensorial nature of curl B............................................................................................. ........ 262 D.9 Tensor E as a weight 0 version of ε : three conventions ........................................................... 263 D.10 Representation of ε, εε and contracted εε as determinants...................................................... 267 D.11 Covariant forms of the previous Section results ...................................................................... 274 D.12 How determinants of rank-2 tensors transform........................................................................ 277 Appendix E: Tensor Expansions: direct product, polyadic and operator notation.................... 281 E.1 Direct Product Notation.................................................................................................... .......... 281 E.2 Tensor Expans ions and Bases................................................................................................ .....282 E.3 Polyadic Notation .......................................................................................................... ............. 288 E.4 Dyadic Products.......................................................................................................................... 289 E.5 Matrix notation for dy adics (Cartesian Space) ........................................................................... 290 E.6 Large and small dots used w ith dyadics (Cartesian Space)........................................................ 291 E.7 Operators and Matrices for Rank-2 tensors: the bra-ket notation (Cartesian Space) ................ 292 E.8 Expansions of tensors on unit tangent base vectors: M and N .................................................. 298 E.9 Application of Section E.8 to Orthogonal Curvilinear Coordinates........................................... 303 E.10 Tensor expansions in a mixed basis ........................................................................................ .309 Appendix F: The Affine Connection Γc ab and Covariant Derivatives .......................................... 311 F.1 Definition and Interpretation of Γ : Γc ab = qc • (∂aqb) = Rc i(∂aRbi)..................................... 311 F.2 Identities of the form ( ∂aRd n) = – Re n Rd m (∂aRem).................................................................... 313 F.3 Identities of the form ( ∂cgab) = – [gan Γ b cn + gbn Γa cn] ......................................................... 314 F.4 Identity: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] ................................................................. 316 F.5 Picture D1 Context ......................................................................................................... ............ 319 F.6 Relations between Γ and Γ '........................................................................................................ 320 F.7 Statement and Proof of the Covariant Derivative Theorem........................................................ 320 F.8 Rules for raising any index on a covariant derivative of a covariant tensor density. ................. 326 F.9 Examples of covariant derivative expressions............................................................................ 328 F.10 The Leibniz rule for the covariant deriva tive of the product of two tensor densities ............... 331 Appendix G: Expansion of ( ∇v) in curvilinear coordinates (v = vector) ...................................... 335 G.1 Continuum Mechanics motivation ............................................................................................. 335 G.2 Expansion of ∇v on ei⊗ej by Method 1: Use the fact that v b;a is a tensor.............................. 336 G.3 Expansion of ∇v on ei⊗ej by Method 2: Use brute force........................................................ 338 6 G.4 Expansion on e i⊗ej and e ^i⊗e^j.................................................................................................. 339 G.5 Orthogonal coordinate systems.............................................................................................. ....340 G.6 Maple evaluation of ( ∇v) in several coordinate systems ........................................................... 341 Appendix H: Expansion of div(T) in cu rvilinear coordinates (T = rank-2 tensor) ...................... 344 H.1 Introduction................................................................................................................................ 344 H.2 Continuum Mechanics motivation ............................................................................................. 344 H.3 Expansion of divT on e n by Method 1: Use fact that Tab ;α is a tensor. .................................... 345 H.4 Expansion of divT on e n by Method 2: Use brute force............................................................ 346 H.5 Adjustment for T expanded on ( e^i⊗e^j) and divT expanded on e^a........................................... 348 H.6 Maple: divT in cylindrical and spherical coordinates ............................................................... 349 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates........................... 351 I.1 Introduction ................................................................................................................................. 351 I.2 Method 1 : a review ....................................................................................................... ............. 352 I.3 Method 1 for spherical coordinates: Maple speaks .................................................................... 354 I.4 Method 1 for spherical coordinates: putting results in traditional form ..................................... 357 I.5 Method 2, Part A ......................................................................................................................... 359 I.6 Method 2, Part B.......................................................................................................................... 360 I.7 Method 2 for spherical coordinates: Maple speaks again........................................................... 362 I.8 Results for Cylindrical Coordinates from both methods ............................................................. 364 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor).......................... 368 J.1 Total time derivative as prototype equation ................................................................................ 368 J.2 Computation of components ( ∇T)'ijk......................................................................................... 369 J.3 Tensor expansions of ∇T on the u n and en base vectors.............................................................. 370 J.4 Tensor expansions of ∇T on the e ^n base vectors......................................................................... 371 J.5 Total time derivative equation written in unit-base-vector curvilinear components................... 372 J.6 Shorthand notations and a c ontinuum mechanics application..................................................... 374 J.7 Maple computation of the ( ∇T)'ijk components for spherical coordinates................................. 376 J.8 Maple computation of the ( ∇T)'ijk components for cylindrical coordinates.............................. 380 J.9 The Lai Method of computing ( ∇T)'ijk for orthogonal coordinates........................................... 381 Appendix K: Deformation Tens ors in Continuum Mechanics ...................................................... 385 K.1 A Preliminary Deformation Flow Picture .................................................................................. 385 K.2 A More Complicated Deformation Flow Picture....................................................................... 391 K.3 Covariant form of a solid constitutive equation involving the deformation tensor.................... 396 K.4 Some fluid constitutive equations .......................................................................................... ....397 K.5 Corotational and other objective time derivatives of the Cauchy stress tensor.......................... 398 References............................................................................................................................................ 406 Overview 7 Overview and Summary In this length y monograph, tensor an alysis (also known as tensor algebra or tensor calculus) is developed starting from Square Zero which is an arb itrary invertible continuous transformation x' = F(x) in N dimensions. The subject was "exposed" by Gregorio Ricci in th e late 1800's under the name "absolute differential calculus". He and his student Tullio Levi-Civita published a masterwork on the subject in 1900 (see References). Christoffel and others had laid the groundwork a few decades earlier. Three somewhat different applications of tensor analysis are treated concurrently. Our primary concern is the subject of curvilinear coordinates in N dimensions. All the basic expressions for the standard differential operators in general curvilinear coordinates are derived from scratch (in several ways). These results are often stated but not so often derived. The second application involves transformations connecting "frames of reference". These transformations could be spatial rotations, Galilean transformations, the Lorentz transformations of special relativity, or the transformations involving the effects of gravity in general relativity. Beyond establishing the tensor analysis formalism, not mu ch is said about this set of applications. The third application deals with material flows in continuum mechanics. The first six Chapters develop the theory of tensor analysis in a simple developmental notation where all indices are subscripts, just as in normal colle ge physics. After providing motivation, Chapter 7 translates this developmental notation into the St andard Notation in use today. Chapter 8 treats transformations of length, area and volume and then the curvilinear differential operator expressions are derived, one per Chapter, with a summary in the penultimate Chapter 14. The final Chapter 15 rederives all the same results using the notion of cova riance and associated covariant derivatives. The information is presented informally as if it were a set of lectures. Little attention is paid to strict mathematical rigor. There is no attempt to be concise: examples are given, tangential remarks are inserted, almost all claims are derived in line, and there is a certain amount of repetition. The material is presented in a planned sequence to minimize the need for forward references, but the sequence is not perfect. The interlocking pieces of tensor analysis do seem to exhibit a certain logical circularity. The reader will find in this document very many worked-out, detailed calculations, derivations and proofs. The intent is to provide the reader with "hands-on experience" in working with all the tensor tools. The only real prerequisites for the reader are a knowledge of calculus of several variables (such as the chain rule and meaning of ∇ 2) and of basic linear algebra (matrices and determinants). Overview 8 The following Table of Contents (TOC) highlights in bol d the main topics which are harder to see in the detailed TOC presented above. In most PDF viewers, th e clickable TOC entries appear also as clickable bookmarks to the left of the text. Our equation references are not clickable, but the constant drumbeat of sequential equation numbers on the right should make them easy to locate. ( Split-screen PDF viewing can be simulated by loading up two copies of the same PDF under different names, a very useful trick.) 1. The Transformation F: invertibility, coordinate lines, and level surfaces 2. Linear Local Transformations associated with F : scalars and two kinds of vectors 3. Tangent Base Vectors e n and Inverse Tangent Base Vectors u'n 4. Notions of length, distance and scalar product in Cartesian Space 5. The Metric Tensor 6. Reciprocal Base Vectors En and Inverse Reciprocal Base Vectors U'n 7. Translation to the Standard Notation 8. Transformation of Differential Length, Area and Volume 9. The Divergence in curvilinear coordinates 10. The Gradient in curvilinear coordinates 11. The Laplacian in curvilinear coordinates 12. The Curl in curvilinear coordinates 13. The Vector Laplacian in curvilinear coordinates 14. Summary of Differential Operators in curvilinear coordinates 15. Covariant derivations of all curvilinear differential operator expressions Appendix A: Reciprocal Base Vectors the Hard Way Appendix B: The Geometry of Parallelepipeds in N dimensions Appendix C: Elliptical Polar Coords , Views of x'-space, Jacobian Integration Rule Appendix D: Tensor Densities and the ε tensor Appendix E: Tensor Expansions : direct product, polyadic and operator notation Appendix F: The Affine Connection Γc ab and Covariant Derivatives Appendix G: Expansion of (∇v) in curvilinear coordinates (v = vector) Appendix H: Expansion of div(T) in curvilinear coordinates (T = rank-2 tensor) Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates Appendix J: Expansion of (∇T) in curvilinear coordinates (T = rank-2 tensor) Appendix K: Deformation Tensors in Continuum Mechanics References Here then is a brief summary of each Chapter and each Appendix. Chapter 1 introduces the notion of the general invertible transformation x' = F(x) as a mapping between x-space and x'-space. The range and domain of this mapping are considered in the familiar examples of polar and spherical coordinates. These same examples are used to illustrate the general ideas of coordinate lines and level surfaces. Certain Pictures are introduced to allow different names for the two inter-mapped spaces, for the function F, and for its associated objects. Chapter 2 introduces the linear transformations R and S=R -1 which approximate the (generally non- linear) x' = F (x) in the local neighborhood of a point x. It is shown that two types of vectors naturally arise in the context of this linearization, called cont ravariant and covariant, and an overbar is used to Overview 9 distinguish a covariant vector. Vector fields are de fined and their transformations stated. The idea of scalars and vectors as tensors of rank 0 and rank 1 is presented. Chapter 3 defines the tangent base vectors en(x) which are tangent to the x'-coordinate lines in x- space. In the example of polar coordinates it is shown that er = r^ and eθ = r θ^. The vectors en exist in x- space and form there a complete basis which in general is non-orthogonal. The tangent base vectors u' n(x') of the inverse transformation x = F-1(x') are also defined. Chapter 4 gives a brief review of the notions of norm, metric and scalar product in Cartesian Space. Chapter 5 addresses the metric tensor, called g ¯ in x-space and g ¯' in x'-space. The metric tensor is first defined as a matrix object g ¯, and then g ≡ g¯-1. A definition is given for two kinds of (pure) rank-2 tensors (both matrices), and it is then shown that g ¯ transforms as a covariant rank-2 tensor while g is a contravariant rank-2 tensor. It is demonstrated how g ¯ applied to a contravariant vector V produces a vector that is covariant V¯= g¯ V, and conversely g V¯ = V . In Cartesian space g = 1, so the two types of vectors coincide. The role of the metric tensor in the covariant vector dot product is stated, and the metric tensor is related to the tangent base vectors of Ch apter 3. The Jacobian J and associated functions are defined, though the significance of J is deferred to Chapter 8. The last Sections briefly discuss the connection between tensor algebra and special relativ ity (with a mention of spinor algebra), general relativity, and contin uum mechanics. Chapter 6 introduces the reciprocal (dual) base vectors En which are later called en in the Standard Notation. Of special interest are the covariant dot products among the e n and En. It is shown how an arbitrary vector can be expanded onto different basis sets. It is found that when a contravariant vector in x-space is expanded on the tangent base vectors e n, the vector components in that expansion are in fact those of the contravariant vector in x'-space, V' i = RijVj. This fact proves useful in later Chapters which express differential operators in x-space in terms of curvilinear coordinates and objects of x'-space. The reciprocal base vectors U' n of the inverse transformation are also discussed. Chapter 7 motivates and then makes the transition from the developmental notation to the Standard Notation where contravariant indices are up and cova riant ones are down. Although such a transition might seem completely trivial, many confusing issues do arise. Once a matrix can have up and down indices, matrix multiplication and other matrix operations become hazy: a matrix becomes four different matrices. The matrices R and S act like tensors, but are not tensors, and in fact are not even located in a well-defined space. Section 7.15 discusses the significance of tensor analysis with respect to physics in terms of covariant equations , and Section 7.16 broaches the topic of the covariant derivative of a vector field with its associated Christoffel symbols. Secti on 7.17 describes how to expand tensors of any rank in various bases and notations, and finally Section 7.18 summarizes information about the basis vectors. The focus then fully shifts to curvilinear coordinates as an application of tensor analysis. The final Chapters are all written in the Standard Notation. Chapter 8 shows how differential length, area and volume transform under x' = F (x). It considers the inverse mapping of a differential orthogonal N-piped (N dimensional parallelepiped) in x'-space to a skewed one in x-space. It is shown how the scale factors h' n = g'nn describe that ratio of N-piped edges, while the Jacobian J = det(g'nn) describes the ratio of N-piped vol umes. The relationship between the vector areas of the N-pipeds is more complicated, and it is found that the ratio of vector area magnitudes is cof(g'nn) . Heavy use is made of the results of Appendices A and B, as outlined below. Chapters 9 through 13 use the information of Chapter 8 and earlier material to derive expressions for all the standard differential operators expressed in general non-orthogonal curvilinear coordinates: divergence, gradient, Laplacian, curl, and vector Lapl acian. The last two operators are treated only in N=3 Overview 10 dimensions where the curl has a vector representation, but then the curl is generalized to N dimensions. For each differential operator, simplified results for orthogonal coordinates are also stated. Chapter 14 gathers all the differential operator expressions into a set of tables, and revisits the polar coordinates example one last time to illustrate a re asonably clean and practical curvilinear notation. Chapter 15 rederives the general results of Chapters 9 through 13 using the ideas of covariance and covariant differentiation. These derivations are elegantl y brief, but lean heavily on the idea of tensor densities (Appendix D) and on the implications of covariance of tensor objects involving covariant derivatives (Appendix F). About half of our content is c ontained in a set of Appendices. Appendix A develops an alternative expression for the reciprocal base vector E n as a generalized cross product of the tangent base vectors en, applicable when x-space is Cartesian. This alternate E n is shown to match the En defined in Chapter 6, and the covariant dot products involving En and en are verified. Appendix B presents the geometry of a parallepiped in N dimensions (called an N-piped). Using the alternate expression for En developed in Appendix A, it is shown that the vector area of the nth pair of faces on an N-piped spanned by the en is given by ± An, where An = |det(S)| En , revealing a geometric significance of the reciprocal base vect ors. Scaled by differentials so d An = |det(S)| En(Πi≠n dx'i), this equation is then used in Chapter 9 wh ere the divergence of a vector field is defined as the total flux of that field flowing out through all the faces of the skew ed differential N-piped in x-space divided by its volume. This same d An appears in Chapter 8 with regard to the transformation of N-piped face vector areas. Appendix C presents a case study of an N=2 non-orthogonal coordinate system, elliptical polar coordinates. Both the forward and inverse coordinate lines are displayed. The meaning of the curvilinear (x'-space) component V' n of a contravariant vector is explored in the context of this system, and the difficulties of drawing such components in non-Cartesian (curvilinear) x'-space are pondered. Finally, the Jacobian Integration Rule for changing integration variables is derived. Appendix D discusses tensor densities and their rules of th e road. Special attention is given to the Levi-Civita ε tensor, including a derivation of all the εε contraction formulas and their covariant statements. It is noted that the curl of a vector is a vector density. Appendix E describes direct product and polyadic notations (including dyadics) and shows how to expand tensors (and tensor densities) of ar bitrary rank on an arbitrary basis. Appendix F deals with covariant derivatives and the affine connection Γ which tells how the tangent base vectors e n(x') change as x' changes. Everything is derived fro m scratch and the results provide the horsepower to make Chapter 15 go. The next four appendices provide demonstrations of most ideas presented in this paper. In each Appendix, a connection is made to continuum mechan ics, and the results are then derived by "brute force", by the covariant technique enabled by Appendix F, or by both methods. These Appendices were motivated by the continuum mechanics text of Lai, Rubin and Krempl (referred to as "Lai", see References). In each Appendix, it is shown how to expr ess the object of interest in arbitrary curvilinear coordinates. Maple code is provided for the general cal culations, and that code is then checked to make sure it accurately replicates the results quoted in Lai for spherical and cylindrical coordinates. Appendix G treats the dyadic object ( ∇v), where v is a vector. Appendix H does the same for the vector object divT where T is a rank-2 tensor. Overview 11 Appendix I deals with the vector Laplacian @B where B is a vector. Appendix J treats the object ( ∇T), where T is a rank-2 tensor. Appendix K (following Lai) discusses deformation tensors u sed in continuum mechanics. Those that are truly tensors may be used to construct cova riant constitutive equations describing continuous materials. The covariant time derivatives To n, TΔ n and T∇ n are derived ( n = 1 are objective Cauchy stress rates). Some examples of covariant constitutive equa tions involving these tensor objects are listed. Notations RHS, LHS refer to the right hand side and left hand side of an equation QED = which was to be demonstrated ("thus it has been proved") a ≡ stuff means a is defined by stuff. // indicates a comment on something shown to the left of // diag(a,b,c..) means a diagonal matrix with diagonal elements a,b,c.. det(A), A T = determinant of the matrix A, transpose of a matrix A Maple = a computer algebra system similar to Mathematica and MATLAB vectors (like v) are indicated in bold; all other tensors (like T) are unbolded. n^ = u n = unit vector pointing along the nth positive axis of some coordinate system ∂i = ∂/∂xi and ∂t = ∂/∂t are partial derivatives dt ≡ Dt ≡ D/Dt ≡ d/dt is a total time derivative, as in d tf(x,t) = ∂tf(x,t) + [∂f(x,t)/∂xi][∂xi/∂t] V,a means ∂aV which means ∂ V/∂xa, and V;a refers to the corresponding covariant derivative x • y is a covariant dot product = g abxayb, except where otherwise indicated when an equation is repeated after its initial appearance, its equation number is shown in italics Figures are treated as if they were equations in terms of equation numbering Components of a vector are written V i = (V)i (bold used only in the second form) Chapter N consists of Sections N.1, N.2 ..... Chapter 1: The Transformation F 12 1. The Transformation F: invertibility, coordinate lines, and level surfaces If x and x' are ele ments of the vector space RN (N-dimensional reals), one can specify a mapping x' = F(x) F: R N → RN ( 1 . 1 ) defined by a set of N continuous (C 2) real functions F i , each of N real variables, x' 1 = F1(x1, x2, x3... xN) x'2 = F2(x1, x2, x3... xN) ... x' N = FN(x1, x2, x3... xN) . ( 1 . 2 ) If all functions F i are linear in all of their arguments, then the mapping F: RN → RN is a linear mapping. Otherwise the mapping is non-linear. A mapping is often referred to as a transformation . We shall be interested only in transformations which are 1-to-1 and are therefore invertible. For such transformations, x' = F(x) x = F -1(x') or x' = x'(x) x = x(x' ) . ( 1 . 3 ) In the transformation x' = F(x), if x roams over the entire R N of x-space (the domain is RN), we may find that x' roams over only some subset of RN in x'-space. The 1-to-1 invertible mapping is then between the domain of mapping F which is all of RN, and the range of mapping F which is this subset. As just noted, it will be assumed that x' = F(x) is essentially invertible so x = F-1(x') exists for any x '. By essentially is meant there may be a few problem points in the transformation which can be "fixed up" in some reasonable manner so that x' = F(x) is invertible (see examples below). The functions F i must be C1 continuous to support the linearization derivatives appearing in Chapter 2, and they must be C2 continuous to support some of the differential operators expressed in curvilinear coordinates in Chapters 9-14. Chapter 1: The Transformation F 13 Example 1: Polar coordinates (N=2) (a) The transf ormation from Cartesian to polar coordinates is given by, x = (x1, x2 ) = (x,y) x' = (x1', x2') = (θ,r) // note that r = x 2' x = F -1(x') ↔ x = rcos( θ) x 1 = x2' cos(x1') y = r s i n ( θ) x 2 = x2' sin(x1' ) x' = F(x) ↔ r = x2+y2 x 2' = x12+x22 θ = tan-1(y/x) x 1' = tan-1(x2/x1) . (1.4) (b) The transformation is non-linear because at least one component function ( e.g., r = x2+y2 ) is not of the form r = Ax + By. In this transformation all functions are non-linear. (c) Here is a drawing showing the nature of this mapping: ( 1 . 5 ) The domain of x ' = F(x) (in x-space on the right ) is all of R 2, but the range in x'-space is the semi-infinite vertical strip shown in gray. Imitating the language of co mplex variables, we can regard this gray strip as depicting the principle branch of the multi-variable function x ' = F(x ). Other branches are obtained by shifting the gray rectangle left or right by multiples of 2 π. Still other branches are obtained by taking the other branch of the real function r = x2+y2 which produces down-facing strips. The principle branch plus all the other branches then fill up the E2 of x'-space, but we care only about the principle branch range shown in gray (this strip could be any strip of width 2 π, such as one from - π to π). (d) This mapping illustrates a "problem point" involving θ = tan -1(y/x). This occurs when both x and y are 0, indicated by the red dot on the right. The inve rse mapping takes the entire red line segment into this red origin point, so we have a lack of 1-to-1 goi ng on here, meaning that formally the function F is not invertible. This can be fixed up by eliminating the red line segment from the range of F, retaining only the point at its left end. Another problem is that both the left and right vertical edges of the gray strip map into the real axis in x-space, and that is fixed by removing the right edge. Thus, by doing a suitable Chapter 1: The Transformation F 14 trimming of the range, F can be made fudlly invertible. No one has ev er had major problems using polar coordinates due to these minor issues. Example 2: Spherical coordinates (N=3) (a) The transformation from Cartesian to spherical coordinates is given by, x = (x1,x2,x3 ) = (x,y,z) x' = (x1',x2',x3') = (r,θ,φ) x = F -1(x') ↔ x = r sin( θ) cos(φ) x 1 = x1'sin(x2')cos(x3') y = r sin( θ) sin(φ) x 2 = x1'sin(x2')sin(x3') z = r cos( θ) x 3 = x1'cos(x2') x' = F(x) ↔ r = x2+y2+z2 x 1' = x12+x22+x32 θ = cos-1(z/x2+y2+z2 ) x 2' = cos-1(x3/ x12+x22+x32 ) φ = tan-1(y/x) x 3' = tan-1(x2/x1) (1.6) (b) The transformation is non-linear because at least one component function( e.g., r = x2+y2+z2 ) is not of the form r = Ax + By + Cz. In this transformation, all three functions are non-linear. (c) Here is a drawing showing the nature of this mapping (1.7) The domain of x ' = F(x) (in x-space on the right) is all of R3, but the range in x'-space is the interior of an infinitely tall rectangular solid on the left we shall call an "office building". We could regard this office building as depicting the principle branch of the multi-variable function x' = F (x). Other branches are Chapter 1: The Transformation F 15 obtained by shifting the building left and right by multiples of 2 π, or fore and aft by multiples of π, or by flipping it vertically, taking the other branch of r = x2+y2+z2 . The principle branch plus all the other branch offices then fill up the R3 of x-space, but we care only about the principle bran ch office building whose walls are mostly shown in gray. (d) This mapping illustrates some "problem points". One is that entire green office building main floor (r=0) maps into the origin in x-space. This problem is fixed by trimming away the main floor keeping only the origin point of the bottom face of the office building. Another problem is that the entire red line segment ( θ = 0) maps into the red point shown in x-space. This is fixed by throwing out the back wall of the office building, retaining only a line going up the left edge of the back wall. A similar problem happens on the front wall (θ = π, blue) and we fix it the same way: throw out the wall but maintain a thin line which is the left edge of this front wall (this line is missing its bottom point). Thus, by doing a suitable trimming of the range, F is made fully invertible. Cartesian Space and Quasi-Cartesian Space (a) Carte sian Space. For the purposes of this document, a Cartesian Space in N dimensions is "the usual" Hilbert Space RN (or EN) in which the distance between two vectors is given by the formula d( x,y) = Σi=1N (xi-yi)2 ⇒ [d( x+dx,x)]2 = Σi=1N (dxi)2 metric tensor = diag(1,1,1....1) (1.8) as discussed in Chapter 4 below. The θ-r space in the above Example 1 would be a Cartesian space if it were declar ed that the distance between two points there were D' 2 = (θ-θ')2 + (r-r')2, but that is not the usual intent in using that space. As shown below, the metric tensor used there is g = diag (r2,1) and not diag(1,1). One might argue that our Cartesian Sp ace is in fact a Eu clidean space (hence EN) having Cartesian coordinates. A non-Cartesian space is sometimes referred to as a "curved space" (non-Euclidean) and the coordinates in such a space as "curvilinear coordinates". An example is the θ-r space above. With the Cartesian Space metric tensor as g C = 1 = diag(1,1....1), the above equations can be written d2(x,y) = gC ij(xi-yi)(xj-yj) and [d( x+dx,x)]2 = gC ij dxi dxj ≡ (ds)2 (1.9) metric tensor = gC = 1 = diag(1,1,1....1), gC ij = δi,j where repeated indices are implicitly summed, called the Einstein summation convention. (b) Quasi-Cartesian Space. We now define a Quasi-Cartesian Space (not an official term) as one which has a diagonal metric tensor G whose diagonal elements are independently +1 or -1 instead of all +1 as with gC. In a Quasi-Cartesian Space the two equations above become d2(x,y) = Gij(xi-yi)(xj-yj) and [d( x+dx,x)]2 = Gij dxi dxj ≡ (ds)2 (1.10) and of course this allows for th e possibility of a negative dist ance-squared (see Section 5.10). Notice that G-1 = G for any distribution of the ±1's in G. As shown later, this means that that covariant and contravariant versions of G are the same. Chapter 1: The Transformation F 16 The motivation for introducing this Quasi-Cartesian Space is to cover the case of special relativity which involves 4 dimensional linear transformations with G = diag(1,-1,-1,-1). Pictures A,B,C and D We shall usually work with one of four different "pictures" involving transformations. In each picture the spaces and transformations (and their associated objects) have certain names that prove useful in certain situations : ( 1 . 1 1 ) The matrices R and S are associated with transforma tion F as described in Chapter 2 below, while G and the g's are metric tensors. Systems not marked Cartesian could of course be Cartesian, but we think of them as general "curved" systems with strange metric tensors. And in genera l, all the full transformations might be non-linear. The polar coordinates example above was presented in the context of Picture B. Picture B is the right picture for studying curvilinear coordinates where for example x-space = Cartesian coordinates and x'-space = toroidal coordinates. Picture C is useful fo r making statements applying to objects in curved x- space where we don't want lots of primes floati ng around. Pictures A and D are appropriate for consideration of general transformations, as well as linear ones like rotations and Lorentz transformations. In Chapters 9-14 Picture M&S (Moon & Spencer) is introduced for the special purpose of displaying the differential operator expressions. This is Picture B with x' → u and g' →g on the left side. The entire rest of this Section uses the Picture B context. Coordinate Lines Suppose in x'-space one va ries a single coordinate, say x' i, keeping all the other coordinates fixed. In x'- space the locus of points thus created is just a straight line parallel to the x' i axis, or for a principle branch situation like that of the above examples, a straight line segment. When such a straight line or segment is Chapter 1: The Transformation F 17 mapped into x-space, the result is a curve known as a coordinate line . A coordinate line is associated with a specific x'-space coordinate x' i, so one might refer to the " x' i -coordinate line", x' i being a label. In N dimensions, a point x in x-space lies on a unique set of N coordinate lines with respect to a transformation F. Remember that each such line is associated with one of the x' i coordinates. In x'-space, a point x' lies on a unique intersection of straight lines or segments, and then this all gets mapped into x- space where point x = F-1(x') then lies on a unique intersection of coordinate lines. For example, in spherical coordinates we start with some (x,y,z) in x-space and compute the x' i = (r,θ,φ) in x'-space. Our point x in x-space then lies on the r-coordinate line whose label is r, it lies on the θ-coordinate line whose label is θ, and it lies on the φ -coordinate line whose label is φ (see below). In general a coordinate "line" is some non-planar curve in N-dimensional x-space, meaning that a coordinate line might not lie on an N-1 dimensional plane. In the 2D polar coordinates example below, the red coordinate line does not lie on a 1-dimensional plane (line). In the next example of 3D spherical coordinates, it happens that every coordinate line does lie on a 2-dimensional plane. But in ellipsoidal coordinates, another 3D orthogonal system, every coordinate line does not lie on a 2-dimensional plane. Some authors refer to coordinate lines as level curves , especially in two dimensions mapping the real and imaginary part of analytic functions w = f(z) (Ahlfors p 89). Example 1: Polar coordinates, coordinate lines Here are some coordinate lines for our prototype N=2 non-linear transformation, Cartesian to polar coordinates: ( 1 . 1 2 ) The red circle is a θ-coordinate line, and the blue ray is an r-coordinate line Chapter 1: The Transformation F 18 Example 2: Spherical coordinates, coordinate lines These coordinate lines are generated ex actly as descri bed above. In x'-space one holds two coordinates fixed while allowing one to vary. The locus in x'-sp ace is a line segment or a half line (in the case of varying r). In x-space, the corresponding coordinate lines are as shown. (1.13) The green coordinate line is a θ-coordinate line, since only θ is varying. The red coordinate line is an r-coordinate line, since only r is varying. The blue coordinate line is a φ -coordinate line, since only φ is varying. The point x indicated by a black dot in x-space lies on the unique set of coordinates lines shown. Appendix C gives an example of coordinate lin es for a non-orthogonal 2D coordinate system. Level Surfaces (a) Suppose i n x'-space one fixes one coordinate, say x' i, and varies all the other coordinates. In x'-space the locus of points thus created is just an (N -1 dimensional) plane perpendicular to the x i axis, or for a principle branch situation like that above, a rectangle or half strip in the case of r. Mapping this planar surface in x'-space into x-space produces a surface in x-space (of dimension N-1) called a level surface . The equations of the N different x i level surface types are a'i(n) = Fi(x1, x2.....xN) i = 1,2...N (1.14) Chapter 1: The Transformation F 19 where a' i(n) is some constant value selected for fixed coordinate x' i. By taking some set of closely spaced values for this constant, { a' i(1), a'i(2).....}, one obtains a family of level surfaces all of the same general shape which are closely spaced. For some different value of i, the shapes of such a family of level surfaces will in general be different. In general if f(x 1, x2.....xN) = k, the set of points x which make this equation true for some fixed k is called a level set , so a level set is a surface of dimension N-1. Thus, all our level curves are also level sets. In the polar coordinates example, since there are only 2 coordinates, there is no distinction between a level surface and a coordinate line. In the spherical coordinates example, there is a distinction. If one fixes r and varies θ and φ over their horizontal rectangle in side the office building, the level surface in x-space is a sphere. If one fixes θ and varies r and φ over a left-right vertical strip inside the office building, the level surface in x-space is a sphere is a polar cone If one fixes φ and varies r and θ over a fore-aft vertical strip inside the office building, the level surface in x-space is a half plane at azimuth φ. (b) In N dimensions there will be N level surfac es in x-space, each formed by holding some x' i fixed. The intersection of N-1 level surfaces (omitting say the x 3' level surface) will have all of the x' i fixed except x'3. But this describes the x' 3 coordinate line. Thus, each coordinate line can be considered as the intersection of the N-1 level surfaces associated with the other coordinates. One can see this happening on the spherical coordinates example: The green coordinate line is the intersection of two level surfaces: half-plane and sphere. The red coordinate line is the intersection of two level surfaces: half-plane and cone. The blue coordinate line is the intersecti on of two level surfaces: sphere and cone. Chapter 2: Linear Transformations R and S 20 2. Linear Local Transformations associated with F : scalars and two kinds of vectors 2.1 Linear Local Transformations We now shift to the Picture A context, where x-spac e is not necessarily Cartesian, (2.1.1) Consider again the possibly non-linear transformation x' = F (x) mapping F: RN→ RN. Imagine a very small neighborhood around the point x in x-space, a "ball" around x. Where the mapping is continuous in both directions, one expects a tiny x-space ball around x to map into a tiny x'-space ball around x' and vice versa. Here is a picture of this situation, (2.1.2) where everything in one picture is the mapping of the corresponding thing in the other picture. In particular, we show a small vector in x-space called d x which maps into a small vector in x'-space called d x'. Since F was assumed invertible, it must be inve rtible locally in these two balls. That is, given a dx above, one can determine d x', and vice versa. (Anticipating a fe w lines below, this means that the matrices S and R will be invertible so neither can have zero determinant.) How are these two differential vectors related? For a linear approximation, x' i + dx'i = Fi(x + dx ) ≈ Fi(x) + Σk( ∂Fi(x)/∂xk) dxk ⇒ dx'i = Σk( ∂Fi(x)/∂xk) dxk . ( 2 . 1 . 3 ) The last line shows an equals sign in the limit that dx k is a vanishing differential. Since F i(x) = x'i , dx' i = Σk(∂x'i/∂xk) dxk = Σk Rik dxk where R ik ≡ (∂x'i/∂xk) . (2.1.4) Doing the same operation in the other direction gives dxi = Σk( ∂xi/∂x'k) dx'k = Σk Sik dx'k where S ik ≡ (∂xi/∂x'k) . (2.1.5) Chapter 2: Linear Transformations R and S 21 One can regard R ik and Sik as elements of NxN matrices R and S. In vector notation then, d x' = R( x) dx R ik(x) ≡ (∂x'i/∂xk) R = S-1 // dx' i = Rij dxj d x = S( x') dx' Sik(x') ≡ (∂xi/∂x'k) S = R-1 // dx i = Sij dx'j . (2.1.6) It is obvious that matrices R and S are inverses of each other, just staring at the above two vector equations. One can verify this fact from the definitions of R and S using the chain rule (RS) ij = Σk RikSkj = Σk (∂x'i/∂xk) (∂xk/∂x'j) = Σk ∂x'i ∂xk ∂xk ∂x'j = ∂x'i ∂x'j = δi,j . (2.1.7) We could get rid of one of these matrices right now, perhaps keeping R and replacing S = R-1, but keeping both simplifies expressions encounter ed later, so for now both are kept. The letter R does not imply that matrix R is a rotation matrix, although it could be. According to the polar decomposition theorem (Lai p 110), any matrix R (detR ≠ 0) can be uniquely written in the form R = R U = V R where R is a rotation matrix (the same one in RU and V R) and U and V are symmetric positive definite matrices (called right and left stretch tensors) related by U = RTVR. Matrix S could of course be written in a similar manner. Matrices R( x) and S( x') are in general functions of a point in space x' = F(x). As one moves around in space, all the elements of matrices R and S are likel y to change. So R and S represent point-dependent linear transformations which are valid for the differentials shown. One might wonder at this point how the vector d x is related to its components dx i and the same question for d x'i and dx'i. As will be shown later in (6.6.9) and (6.6.15), d x = Σndxn un where the un are x-space axis-aligned basis vectors of the form u 1 = (1,0,0,..0) d x' = Σndx'n e'n where the e'n are x'-space axis-aligned basis vectors of the form e'n = (1,0,0,..0) ( 2 . 1 . 8 ) If x-space and x'-space were Cartesian, one could write u n = n^ and e'n = n^', but in general the un and e'n vectors do not have (covariant) unit length, as shown later in (6.5.3) and (6.4.1). The reader familiar with covariant "up and down" indices will notice that all indices are peacefully sitting "down" in the presentation so far (subscripts , no superscripts). As we carry out our various developmental tasks, that is where all indices sha ll remain until Chapter 7, whereupon they will start frantically bobbing up and down, seem ingly at will. [ Since rules are made to be violated, we have violated this one in some examples below where non- standard notation would be hard to swallow. ] Are there any "useful objects" that can be constructed from differentials dx and which might then transform according by R or S? The answer is yes, but first we discuss scalars. 2.2 Scalars A quantity is a scalar with respect to transformation F if it is the same in both spaces. Thus, any constant like π would be a scalar under any transformation. The mass m of a potato would be a constant under transformations that are rotations or translations. A function of space φ(x) is a "field" and it would be a "scalar field" if φ'(x') = φ(x). For example, temperature would be a scalar field under rotations. Notice Chapter 2: Linear Transformations R and S 22 that φ is evaluated at x, while φ' is evaluated at x' = F (x). As noted in Section 2.12, one could be more precise by referring to the objects described here as a "tensorial scalar" and a "tensorial scalar field". 2.3 Contravariant vectors If x and x' are spatial coordinates (time t is not one of the x i), then consider the familiar velocity vector, vi = dxi/dt ⇒ v = dx/dt . (2.3.1) Since dt transforms as a constant (scalar) under our selected transformation type, it seems pretty clear that velocity in x'-space can be related to velocity in x-space using the dx ' = R( x) dx rule above, so v' = R( x) v . ( 2 . 3 . 2 ) Even though the matrix R( x) changes as we move around, this linear transformation R is valid at any point x when applied to velocity. Momentum p = mv would work the same way, since mass m is a scalar (Newtonian mechanics). In contrast, unless R( x) is a constant in space (meaning from (2.8.7) that F(x) is linear), x' ≠ R(x) x, so in general x itself is not a contravariant vector although d x is. Any vector that transforms according to V' = R( x)V with respect to a transformation F (such as Newtonian velocity and momentum with respect to rotations) is called a contravariant vector . 2.4 Covariant vectors Much of physics i s described by differential equations involving the gradient operator ( the reason for the overbar is explained in the next Section) ∇¯i = ∂¯i = ∂/∂xi ( 2 . 4 . 1 ) which involves an "upside down" di fferential. Here is how this operator transforms going from x-space to x'-space, again according to the chain rule (implied sum on k, T means transpose) ∇¯ 'i = ∂¯ 'i = ∂ ∂x'i = ∂xk ∂x'i ∂ ∂xk = Ski∂¯k = ST ik ∂¯k = ST ik ∇¯k = (ST ∇¯)i ⇒ ∇¯' = ST ∇¯ . ( 2 . 4 . 2 ) One can think of ∇¯ as acting on a scalar field φ(x) = φ'(x'), and then the above becomes ∇¯ ' i φ'(x') = ∂ ∂x'i φ'(x') = ∂xk ∂x'i ∂ ∂xk φ(x) = ST ik ∇¯k φ(x) ⇒ ∇¯'φ'(x') = ST ∇¯φ(x) . ( 2 . 4 . 3 ) Chapter 2: Linear Transformations R and S 23 Since the differential is "upside down", one might expect ∇¯ to transform according to S = R-1 instead of R, but it is really ST that does the job. One could write ∇¯' = ∇¯ S in terms of row vectors. Vectors that transform according to V' = ST(x) V such as the gradient operator ∇¯ are called covariant vectors with respect to transformation F. An example of a covariant vector is the elect rostatic electric field obtained from the potential Φ E¯ = - ∇¯ Φ E¯i = - ∂¯iΦ = - ∂Φ/∂xi ( 2 . 4 . 4 ) 2.5 Bar notation In order to d istinguish a contravariant from a c ovariant vector, we shall (for a while) adopt this bar convention : contravariant vectors shall be written V with components V i and covariant vectors shall be written V¯ with components V ¯i. This is why overbars were placed on ∇ ¯ and ∂¯i and E¯ in the previous Section. We call this our "developmental notation", as distinct from the Standard Notation introduced in Chapter 7. The transformation rules for the two vector types can now be written this way: V' = R V contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 V¯' = ST V¯ covariant S ik(x') ≡ (∂xi/∂x'k) = ST ki(x') (2.5.1) One could imagine replacing S with some QT to make the second equation more like the first, but of course then RQT = 1 instead of RS = 1. In the Standard Notation, where there are four versions of the matrix R, we shall see in Section 7.5 that R → Ri j and S → Si j = Rji and S can be removed from the picture altogether. It is possible to do without either R or S and just write out all partial derivatives like ∂x'i ∂xk in place of Rik. Many authors do this, including Weinberg. We feel that the forms R ik and later Ri k are much more compact (3 symbols in place of 8) and reveal more clearly the matrix (but not tensor) nature of R and the fact that it is a linear transformati on that locally approximates the full x' = F (x). Later we shall have long strings of R objects which become extremely clut tered when derivatives are fully written out, see for example Section F.2 and following. Finally, R ik is a easy to type! 2.6 Origin of the names contravariant and covariant A justificatio n of the terms covariant and contravariant is presented in Section 7.14 , since the idea is more easily presented there than here. It seems that these terms were first used in 1851 (a half century before special relativity) in a paper (see Refs.) by J.J. Sylvester of Sylvester's Law of Inertia fame. Sylvester uses the words covariant and contravariant to describe the relations between a pair of "transformations". In much simpler notation than he uses, if those "transformations" (functions) are F( x) and G(x ) and if A is an 3x3 matrix, then Chapter 2: Linear Transformations R and S 24 the pair F(A x) and G(A x) are said to be covariant (or concurrent) the pair F(A x) and G(A-1x) are said to be contravariant (or reciprocal) The idea is that in comparing the way two things transf orm, if they both move the same way, then it is covariant, and if they move in opposite directions it is contrava riant. In Section 7.14 this idea is applied to the transformation of two objects, where one object is the component of a vector like V n and the other object is a basis vector onto which a vector is expanded. The connection is a bit distant, but the underlying concept carries through. Notations like y = F (Ax) would have mystified Sylvester in 185 1, although in this same paper he introduced two-dimensional arrays of letters and referr ed to them as "matrices". According to a web piece by John Aldrich of the University of Southampton, J.W. Gibbs in 1881 was the first person to use a single letter to represent a vector (he used Greek letters). It was not until 1901 when his student E.B. Wilson published Gibb's lectures in a Vector Analysis book that the idea was propagated to a wider circle. Wilson converted those Greek letters to bolded ones, (2.6.1) The Wilson/Gibbs book was reprinted seven times, the last being 1943. In 1960 it continued as a Dover book and is now available online as a public domain document (above from page 4). 2.7 Other vector types? Are there any other kinds of vectors with respect to a transformation F? There might be, but only the two types mentioned above are of interest to us in th is document. They are both called rank-1 tensors, and there are no other rank-1 tensor types in "tensor an alysis" (for rank-n tensors, see Section 7.10). Some authors refer to the rank of a tensor as the order of a tensor, and we shall sometimes use that term. In the Standard Notation introduced later, wher e contravariant vector components are written with indices up and covariant vectors with indices down, a nd where the notation is so slick and smooth and automatic, one sometimes imagines there are two kinds of vectors because there are two places to put indices, up and down. It is of course the other way around: the up/down notation was adopted because there are two rank-1 tensor types. Two particular (linear) transformation types of in terest are rotations and Lorentz transformations, each of which has a certain number of continuous parameters (3 and 6). As the parameters are allowed to vary over their ranges, the set of transformations can be viewed as elements of a continuous group ( SO(3) and SO(3,1) ). Each of these groups has exactly one "vector representation" ( "1" and "(1/2)⊕(1/2)" ). One should not imagine that somehow the "two-ness" of vector types under general transformations F is connected to there being two v ector representations of some particular group. It Chapter 2: Linear Transformations R and S 25 happens that the Lorentz group does have two "spinor representations" (1/2) ⊕0 and 0⊕(1/2), but this has nothing at all to do with our general notion of two ki nds of vectors. This subject is discussed in more detail in Section 5.14. 2.8 Linear transformations First, consider a scala r function f : Rn → R. The function f( x) is linear iff f(x +y) = f( x) + f( y) for all x and y in R n f(a x) = af (x) for all x in Rn and for all a in R . (2.8.1) These conditions imply that the function f( x) must have the form, f(x ) = Σi ai xi . where a i are constants independent of x . (2.8.2) One way to reach this conclusion is to assume a Taylor expansion for f(x ) about x = 0, f(x ) = f( 0) + Σi[∂if]x=0xi + Σij[∂ijf]x=0xixj + higher terms . (2.8.3) The (2.8.1) requirement that f(2 x) = 2f( x) eliminates the quadratic and higher terms and for x = 0 eliminates the first term f( 0) since f( 0) = 2f( 0), resulting in an expression of the form (2.8.2). When this same discussion is applied to the components F i(x) of the function F : Rn→Rn one concludes that the form of x' = F(x) for linear F is given by x'i = Fi(x) = ΣjFijxj where F ij are constants independent of x . (2.8.4) Applied to the differential vector d x this says dx' i = Σj Fij dxj . ( 2 . 8 . 5 ) Comparison with (2.1.4) shows that F ij = Rik so F linear ⇒ R = F and S = F-1 . (2.8.6) We then arrive at this miniature theorem and its converse: T h e o r e m : (a) If R(x ) = R does not vary with x, then d x' = Rd x ⇒ x' = Rx so x ' = F (x) is a linear transformation, namely F(x) = Rx . (b) If x' = F (x) is a linear transformation, then F(x) = R x where R is a matrix of constants which do not vary with x. ( 2 . 8 . 7 ) This is the situation for global rotations and Lorentz transformations. Chapter 2: Linear Transformations R and S 26 2.9 Vectors that are contravariant by definition A contravariant vector has been defined above as an y N-tuple which transforms the same way that d x transforms with respect to F, namely, d x' = R( x) dx. One might state this as { d x', dx } d x' = R( x) dx contravariant vector . (2.9.1) Suppose we start with an arbitrary N-tuple V and simply define V' ≡ RV. One would have to conclude that the pair { V', V } transforms as a contravariant vector. { V', V } V' ≡ R(x)V contravariant vector . (2.9.2) Conversely, one could start with some given V' and define V ≡ S(x) V' (recall S = R -1), and again one would conclude that { V', V } represents a vector that transforms as a contravariant vector. We refer to either process as producing a vector wh ich is "contravariant by definition". Creating a contravariant vector in this fashion is a fine thing to do, as long as the defined vector does not conflict with something that already exists. Example 1: We know that if F is non-linear, the vector x does not transform as a contravariant vector, because x' = R( x)x is not true, where x' = F (x). If we start with x and try to force { x', x} to be "contravariant by definition" by defining x ' ≡ R(x) x , this x' conflicts with the existing x' = F (x), so the method of contravariant by definition is unacceptable. (2.9.3) Example 2: As another example, consider an N-tuple in x'-space of three masses V' = (m1,m2,m3). The transformation is taken in this example to be regul ar rotations. Since masses are rotational scalars with respect to such rotations, we know that in an x-space rotated frame of reference we would find V = (m1,m2,m3). We could attempt to set up { V', V } as a vector that is "contravariant by definition" by defining V ≡ SV', but this conflicts with the existing fact that V = (m1,m2,m3), so the method of contravariant by definition is again unacceptable. (2.9.4) Example 3: This time F is a general transformation and we start with V' = e 'n which are a set of axis- aligned basis vectors in x'-space. We define vectors V = en according to e n ≡ Se'n. Then { e 'n, en } form a vector which is "contravariant by definition" and e'n = R en (R = S-1). Since the newly defined vector en does not conflict with some already-existing vector in x-space, the method of contravariant by definition in this example is acceptable. This is exactly what is done in the next Section with the tangent base vectors en. ( 2 . 9 . 5 ) 2.10 Vector Fields We considered above vectors like posit ion x (and dx ) and velocity v and the vector operator ∇¯, and we referred to a generic vector as V. Many vectors of interest (in fact, most) are functions of x, which is to say, they are vector fields. Examples are the electric and magnetic fields E(x) and B(x), or the average velocity of a small region of fluid V(x) or a current density J(x). Another example is the transformation F(x). Chapter 2: Linear Transformations R and S 27 We already mentioned scalar fields, such as temperature T( x) or electrostatic potential Φ(x). The way a scalar temperature field transforms going from x-space to x'-space is this T '(x') = T( x) where x' = F(x) . (2.10.1) If the transformation is a 3D rotation from frame S to frame S', then T ' is the temperature measured in frame S' at point x' and T is the temperature measured at the corresponding point x in frame S and of course there is only one temperature at that point so the numbers are equal. In x'-space one needs the prime on T ' because the functional form (how T ' depends on the x' i) is not the same as that of T (how T depends on the x i). For example, if transformation F is from 2D Cartesian to polar coordinates, then T '(r,θ) = T(x,y) = T(rcos θ,rsinθ) ≠ T(r,θ ) . ( 2 . 1 0 . 2 ) Contravariant and covariant vector fields transform as described above, but now one must show the argument for each field in its own space, and again x' = F(x) : V'(x') = R V(x) contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 V¯'(x') = ST V¯(x) covariant S ik(x') ≡ (∂xi/∂x'k) = ST ki(x') . (2.10.3) Similar transformation rules apply to tensors of any rank. For example (as we shall see in Chapter 5) the metric tensor g ab (developmental notation) is a rank-2 contrava riant tensor field and the transformation rule is this (implied summation on repeated indices), g'ab(x') = Raa'Rbb'ga'b'(x) or g' ab = Raa'Rbb'ga'b' (2.10.4) Often the coordinate dependence of g is suppressed, just as it is for R and S, as shown on the right above. Jumping momentarily into Standard Notation, in special relativity one has x' μ = Λμ νxν where F = R = Λ is a linear transformation, and one would then specify th e transformation of a contravariant vector field as V' μ(x'α) = Λμ ν Vν(xα) x 'μ = Λμ νxν (2.10.5) Comment . The word field in "vector field V(x)" is unrelated to the word field being an algebraic object with • and + properties, such as the field of the real numbers or the finite field {0,1}. However, the components of V and x in V(x) are normally elements of the real number field. 2.11 Names and symbols The m atrix Rik(x) = (∂x'i/∂xk) is called the Jacobian matrix for the transformation x' = F (x) , while the matrix S ik(x') = (∂xi/∂x'k) is then the Jacobian matrix of the inverse transformation x = F-1(x'). The determinant of the Jacobian matrix S will be shown in Section 8.6 to have a certain significance, and that determinant is called " the Jacobian " = det(S( x')) ≡ J(x'). Matrix R is sometimes called "the differential dF" of transformation x' = F (x) and is. Matrix S is then the differential of the inverse transformation x = F-1(x'). Chapter 2: Linear Transformations R and S 28 The author has anguished over what names to give the matrices R and S = R-1. One option was to use R = L, where L stands for the fact that this matrix is describing a Local coordinate system at point x, or a Linearized transformation. But L is commonly used for differential operators and angular momentum, so that got rejected. R is often called Λ in special relativity, but why go Greek so early? Another option is to use R = J for Jacobian matrix, but J looks too much like "an integer" or angular momentum or "the Jacobian". T for Transformation might have been conf used with the full transformation F, or Cauchy stress T. Our chosen notation R makes one think perhap s R is a Rotation, but that won't in general be the case. For the moment we will continue to use R and S, where recall RS = 1. We shall refer to R simply as "the R matrix for transformation F". The fact that vectors are processed by NxN matrices R and S puts that part of the subject into the field of linear algebra, and that may be the origin of the name tensor algebra as a generalization of this idea (tensors as objects of direct product algebras). Of course the differential calculus aspect of the subject is already highly visible, there are ∂ symbols everywhere (hence the name tensor calculus ). 2.12 Definition of the words "scalar", "vector" and "tensor" In Section 2.2 a "scalar" was defined as something that is invariant under some transformation F, and this was identified with a "rank-0 tensor". Similarly, a "vect or" is either a contravariant vector or a covariant vector and both of these are "rank-1 tensors". In Sec tion 5.6 certain "rank-2" te nsors will appear -- they are matrices that transform in a certain way under a tr ansformation F. In Section 7.10 tensors of rank-n will appear, and these are objects with n indices which transform in a certain manner under F. To be more precise and to provide protection agai nst the vagaries of "the literature", these objects probably should have been defined with the word "tensorial" in front of them. "tensorial scalar" ≡ rank-0 tensor with respect to some transformation F "tensorial vector" ≡ rank-1 tensor with respect to some transformation F "tensorial tensor" ≡ rank-n tensor with respect to some transformation F (2.12.1) As has been emphasized several times, a "tensori al tensor" is linked to a particular underlying transformation F, and one should really use the more precise term "tensorial tensor under F". In this paper, we generally omit the word "tenso rial" when discussing the above objects. This brings us into conflict with the following definitions whic h are often used: (Here, we use the term "expression" to indicate a number, a variable, or some combination of same.) • A "scalar" is a single expression, a 1-tuple. No invariance under any transformation is implied. • A "vector" is an N-tuple of expressions. No transformation rule is implied. • A "second order tensor" is a matrix of expressions. No transformation rule is implied. (2.12.2) • A "tensor" is an object with n indices, n = 2,3, 4... which includes the previous item. A tensor is therefore a collection of expressions which are labeled by n indices each of which goes 1 to N. No transformation rule is implied. To these definitions we can add another list: Chapter 2: Linear Transformations R and S 29 • A "scalar field" is a single function of x ( the x-space coordinates). No implication of invariance. • A "vector field" is an N-tuple of functions of x -- an N-tuple of scalar fields. No transform implied. • A "tensor field" of order n is a set of Nn scalar functions, for example, T abc... (x). Same. (2.12.3) In any discussion which includes relativity (special or general), the words scalar, vector and tensor would always imply the tensorial definitions of these word s. Continuum mechanics, however, seems to use the above alternate list of definitions, so that any ma trix is called a tensor. Usually such matrices are functions of space and should be called tensor fields, but everybody knows what is meant. In Section 7.15 we shall discuss the notion of an equation being covariant , which means it has the exact same form in different frames of reference whic h are related by a transformation. For example, one might have F = m a i n f r a m e S , a n d F ' = m' a' in frame S', where these frames are related by a static rotation. F and a are tensorial vectors with respect to this rotation, and m and m' are tensorial scalars, and m = m' for that reason. Both sides of F = m a transform as tensorial vectors. Since rotations are an invariance of Newtonian physics, any valid equation of motion must be "covariant", and this applies of course to particle, rigid body and continuum mechanics. In the latter field, continuum mechanics, one natu rally seeks out model equations which are covariant. In order to do this properly, one must know which te nsors are tensorial tensors, and which tensors are just tensors with either no transformation rule, or some transformation rule that does not match the tensor. Continuum mechanics has evolved special words to handle this situation. If a tensor is a tensorial tensor, it is said to be objective , or indifferent. In continuum mechanics an equatio n which is covariant is said to be frame-indifferent. In this document we shall follow the time-honored tradition of being inconsistent in our use of the words scalar, vector and tensor, but the reader is now at least warned. To further promote that tradition, we shall sometimes refer to tensorial tensors as "true" tensors. The notion of tensor densities described in Appendix D further co mplicates the nomenclature. One can have scalar densities and vector densities of various weights, for example. Appendix K explores a few commonly used tensors in continuum mechanics and determines which of these tensors actually transform as tensors (are object ive), and which tensors do not transform as tensors (are non-objective). Comment on "rank" . As noted in Section 2.7, some authors re fer to a rank-n tensor as an order-n tensor. The word rank has another common use which is unrelated to tensor rank. The rank of a square matrix is the number of linearly independent rows or columns. If an NxN matrix M has rank less than N, then det(M) = 0. Since our R and S matrices do not have ze ro determinant, they are both of full rank N. The same is true of the metric tensors of Chapter 5 below. Since g ≡ det(g¯ij) ≠ 0, g has full rank N. Chapter 3: Tangent Base Vectors 30 3. Tangent Base Vectors e n and Inverse Tangent Base Vectors u 'n 3.1 Differential Displacements This entire Section is in the context of Picture A , (3.1.1) In Fig (2.1.2) above showing dx and dx ', one has much freedom to "try out" different differential vectors. For any d x one picks at point x, one gets some d x' according to d x' = R( x) dx. Consider this slightly enhanced version of that figure (red curves added) ( 3 . 1 . 2 ) The point x in x-space (right side) can be regarded as lyi ng on some arbitrary 1-dimensional curve in RN shown on the right in red. Select d x to be the tangent to this curve at point x. That curve will then map into some (probably very different) curve in x'-space which passes through the point x '. The tangent to this curve at the point x' must be d x' = R( x) dx. A similar statement can be made starting instead with an arbitrary curve in x'-space. The tangent d x' there then maps into d x = S( x') dx' in x-space. The curves are in N-dimensional sp ace and are in general non-planar and the tangents are of course N dimensional tangents, so this 2D picture is mildly misleading. We now specialize such that the red curve on the left is a straight line parallel to an x'-space axis, which means the curve on the right is a coordinate line, Chapter 3: Tangent Base Vectors 31 ( 3 . 1 . 3 ) Admittedly the drawing does not strongly suggest that th e red line segment on the left is parallel to an axis in x'-space, but since those axes are not drawn, one cannot complain too strenuously. 3.2 Definition of the e n ; the en are the columns of S First, define a set of N basis vectors in x'-space which point along the positive axes of x'-space, e' n , n = 1,2...N ( e'n)i = δn,i e '1 = (1,0,0...) etc . (3.2.1) Assume that the d x' arrow above points in this e ' n direction so that d x' = e' n dx'n // no implied sum on n (3.2.2) where dx' n is a positive differential va riation of coordinate x' n along the e 'n axis in x'-space. The corresponding d x in x-space will be, d x = S d x' = S [ e' n dx'n] = [ S e'n] dx'n ≡ en dx'n (3.2.3) where this last equality serves as the definition of e n , e n ≡ Se'n . ( 3 . 2 . 4 ) Vector e n = en(x) points along d x in x-space and is tangent to the x' n- coordinate line there at point x. This vector en is generally not a unit vector, hence no hat ^ . Writing out the ith component of (3.2.4), ( e n)i = Σj Sij (e'n)j = Σj Sij δn,j = Sin ( 3 . 2 . 5 ) so that, with (2.1.5), ( e n)i = Sin = ∂xi/∂x'n or en = ∂x/∂x'n = ∂'nx . (3.2.6) This fact that ( e n)i = Sin says that the vectors en are the columns of the matrix S: S = [ e 1, e2, e3 .... eN ] matrix = N columns (3.2.7) Chapter 3: Tangent Base Vectors 32 We shall call these en vectors the tangent base vectors. The vectors exist in x-space and point along the various coordinate lines that pass through a point x . If the points on the x' n-coordinate line were labeled with the values of x' n from which they came, one would find that en points in the direction in which those labels increase. As one moves from x to some nearby point, the tangent base vectors all change slightly because in general S = S( x'(x)) and the e n = en(x) are the columns of S. Any set of basis vectors which depends on x in this way is called a local basis. In contrast, the corresponding x'-basis e'n shown above with ( e'n)i = δn,i is a global basis in x'-space since it is the same at any point x' in x'-space. Since det(S) ≠ 0 due to our assumption that F is invertible, the tangent base vectors are linearly independent and provide a basis for EN. One can of course normalize each of the en to be a unit vector e^n according to e^n = en/ |en|. Here is a traditional N=3 picture showing the tangent base vectors pointing along three generic coordinate lines in x-space all of which pass through the point x: (3.2.8) Comment on notation. Some authors refer to our en as gn or Rn or other. Later it will be shown that en•em = g¯'nm where g ¯'nm is the covariant metric tensor for x'-space, so admittedly this provides a reasonable argument for using gn so that g n•gm = g¯'nm. But then the primes don't match which is confusing: the gn are vectors in x-space, while g ¯' is a metric tensor in x'-space. We shall be using yet another g in the form g = det(g ¯nm) and a corresponding g'. Due to this proliferation of g objects, we stick with en, the notation used by Margenau and Murphy (p 193). A g-oriented reader can replace e → g as needed anywhere in this document. As for unit vector versions of the en, we use the notation e^n ≡ en/|en|. Morse and Feshbach use a n for this purpose (Vol I p 22). A g-person might use g^n . A related issue is what symbols to use for the "u sual" basis vectors in Cartesian x-space. As noted above in (2.1.8), we are using un with ( un)i = δn,i as "axis-aligned basis vectors" in x-space. If g ¯ = 1 for x-space, then these are the usual Cartesia n unit vectors (see (6.5.3) below that un • um = g¯nm). Many authors use the notation en for these vectors which then conflicts with our use of e n as the tangent base vectors. Morse and Feshbach use the symbols i, j, k for our Cartesian u1, u2, u3. Other authors use 1^, 2^, 3^ so then u n = n^ . Chapter 3: Tangent Base Vectors 33 Often the notation en is used by authors to represent some ge neric arbitrary set of basis vectors. For this purpose, we shall use the notation bn. 3.3 en as a contravariant vector The situation described above is this, d x' = e 'n dx'n x'-space (3.2.2) // no implied sum on n d x = en dx'n x-space (3.2.3) // no implied sum on n (3.3.1) and the full transformation F maps d x into d x'. Since d x is a contravariant vector, the linear transformation R also maps d x into dx '. Thus d x' = R( x) dx (2.1.6) so [ e' n dx'n] = R( x) [en dx'n] (3.3.1) so e' n = R( x) en . ( 3 . 3 . 2 ) We can regard the last line as a statement that the vector e n transforms as a contravariant vector under F. Written out in components one gets ( e'n)i = ΣjRij (en)j ⇒ δn,i= ΣjRijSjn (3.3.3) recovering the fact that RS = 1. This is an exampl e of a vector being "contravariant by definition", as discussed in (2.9.5). The two expansions (3.3.2) and (3.2.4) are easy to verify by showing that the components of both sides are the same: e' n ≡ Ren = Σi Rin ei since (e 'n)j = Σi Rin (ei)j = Σi Rin Sji = (SR)jn = δj,n = (e'n)j e n ≡ Se'n = Σi Sin e'i since (e n)j = Σi Sin (e'i)j = Σi Sin δi,j = Sjn = (en)j (3.3.4) 3.4 A semantic question: unit vectors Above it was noted t hat e'1 = (1,0,0....). Should this be called "a un it vector" ? It will be seen in (6.2.7) that in fact | e'1| = g¯'11 ≠ 1 where g ¯' is the covariant metric tensor in x'-space, and | e'1| is the covariant length of e'1. So e 'n is a unit vector in the sense that it has a si ngle 1 in its column vector definition, but it is not a unit vector in the sense that it does not (in ge neral) have unit magnitude ( it would if x'-space were Cartesian with g'=1).We take the magnitude = 1 requirement as the proper definition of a unit vector. For this reason, we refer to the e' n in x'-space as just "axis-aligned basis vectors" and they have no "hats". One wonders how such a vector should be depicted in a drawing, see Example 1 (b) below and also Section C.5 Chapter 3: Tangent Base Vectors 34 Example 1: Polar coordinates, tangent base vectors (a) The first s tep is to compute the matrix S ik(x') ≡ (∂xi/∂x'k) from the inverse equations: x = (x 1, x2 ) = (x,y) x' = (x1', x2') = (θ,r) // note that r chosen as the second variable x 2' x = F -1(x') ↔ x = rcos( θ) x 1 = x2' cos(x1') y = r s i n ( θ) x 2 = x2' sin(x1' ) (1.4) So S 11 = (∂ x/∂θ) = -rsinθ S12 = (∂ x/∂r) = cosθ S ik ≡ ( ∂xi/∂x'k) S21 = (∂ y/∂θ) = rcosθ S22 = (∂ y/∂r) = sinθ ( 3 . 4 . 1 ) S = ⎝⎛ ⎠⎞-rsinθ cosθ rcosθ sinθ // det(S) = -r R = S-1 = ⎝⎛ ⎠⎞-sinθ/r cosθ/r cos(θ) sinθ . [ Note: The above S and R are stated for the ordering 1,2 = θ,r . For the more usual ordering 1,2 = r, θ the colums of S should be swapped, and th e rows of R should be swapped. In the usual ordering, det(S) = +r.] The tangent base vectors e n can be read off as the columns of S according to (3.2.7), e1 = r(-sinθ,cosθ) = eθ = r e^θ // = r θ^ e2 = (cosθ,sinθ) = er = e^r . // = r^ (3.4.2) Notice that e θ in this case is not a unit vector. Below is a properly scaled drawing showing the location of the two x'-space basis vectors on the left, and the two tangent base vectors on the right. As just shown, the length of er is 1, while the length of eθ is 2. ( 3 . 4 . 3 ) The tangent base vectors are fairly familiar animals, since er = r^ and e θ = r θ^ in usual parlance. If one moves radially outward from point x, the er base vector stays the same, but eθ grows longer. If one moves Chapter 3: Tangent Base Vectors 35 azimuthally from x to some larger angle θ +Δθ, both vectors stay the same length but they rotate together staying perpendicular. (b) This is a good place to point out that vectors drawn in a non-Cartesian space can have magnitudes which do not equal the length of the drawn arrows. The "graphical arrow length" of a vector v is (v x2 + vy2)1/2, but that is not the right expression for | v| in a non-Cartesian space. For example, as will be shown below in (5.10.5), | eθ'| = |eθ| , so the magnitude of the vector e'θ shown on the left above is in fact | eθ'| = r = 2 and not 1, but the graphical length of the arrow is 1 since e'θ = (1,0). See Section C.5 for further discussion of this topic with a specific 2D non-orthogonal coordinate system. (c) In this example, two basis vectors e 'n in x'-space on the left map into the two e n vectors on the right according to en ≡ Se'n. If one were to apply the full mapping x = F-1(x') to each point along the arrows e'n, for some general non-linear F one would find that these arrows map into warped arrows on the right whose bases are tangent to those of the en. Those warped arrows lie on the coordinate lines. For this particular mapping, e'θ maps under F-1 into the warped gray arrow, while e'r maps into er. Example 2: Spherical Coordinates, tangent base vectors x = (x1, x2, x3 ) = (x,y,z) x' = (x1', x2',x3') = (r,θ,φ) x = F -1(x') ↔ x = rsin θcosφ y = r s i n θsinφ z = r c o s θ (1.6) S 11= (∂ x/∂r) = sinθcosφ S ik ≡ (∂xi/∂x'k) S12 = (∂ x/∂θ) = rcosθcosφ S13 = (∂ x/∂φ) = -rsinθsinφ S21= (∂ y/∂r) = sinθsinφ S22 = (∂ y/∂θ) = rcosθsinφ S23 = (∂ y/∂φ) = rsinθcosφ S31= (∂ z/∂r) = cosθ S32 = (∂ z/∂θ) = -rsinθ S33 = (∂ z/∂φ) = 0 ( 3 . 4 . 4 ) S = ⎝⎜⎛ ⎠⎟⎞ sinθ cosφ rcosθcosφ -rsinθsinφ sinθ sinφ rcosθsinφ rsinθ cosφ cosθ -rsinθ 0 R = ⎝⎜⎛ ⎠⎟⎞ sinθ cosφ sinθsinφ cosθ cosθcosφ/r cosθsinφ/r -sinθ/r -sinφ /(rsinθ) cosφ/(rsinθ) 0 where Maple computes R as S-1 and finds as well that: det(S) = r2 sinθ . Again, from (3.2.7) the tangent base v ectors are the columns of S, so Chapter 3: Tangent Base Vectors 36 er = (sinθ cosφ, sinθsinφ,cosθ) |e r| = 1 ≡ h'r eθ = r(cosθcosφ,cosθsinφ,-sinθ) |e θ| = r ≡ h'θ eφ = rsinθ(-sinφ,cosφ,0) |e φ| = rsinθ ≡ h'φ (3.4.5) and unit vector versions are then e^r = (sinθ cosφ, sinθsinφ,cosθ) = r^ er = r^ e^θ = (cosθcosφ,cosθsinφ,-sinθ) = θ^ e θ = r θ^ e^φ = (-sinφ,cosφ,0) = φ^ eφ = rsinθ φ^ (3.4.6) The unit vectors can be displayed in this standard picture, (3.4.7) Notice that ( r^, θ^, φ^) = (e^1, e^2, e^3) form a right-handed coordinate system at the point x = r. 3.5 The inverse tangent base vectors u 'n and inverse coordinate lines A complete swap x' ↔ x for a mapping x' = F (x) of course produces the "inverse mapping". This has the effect of causing R ↔ S in the above discussion. The tangent ba se vectors for the inverse mapping would then be the columns of matrix R instead of S. We shall denote these inverse tangent base vectors which exist in x'-space by the symbol u'n. Then: ( en)i = Sin = ∂xi/∂x'n // the tangent base vectors as above (3.2.6) S = [ e1, e2, e3 .... eN ] // are the columns of S (3.2.7) ( u' n)i = Rin = ∂x'i/∂xn // inverse tangent base vectors R = [ u'1, u'2, u'3 .... u'N ] // are the columns of R (3.5.1) By varying only x n in x-space holding all the other x i = constant, one generates the x n-coordinate lines in x'-space, just the reverse of the earlier discussion of this subject. Then inverse tangent base vectors u'n will then be tangent to these inverse coordinate lines. An example is given just below and another appears in Appendix C. Chapter 3: Tangent Base Vectors 37 The vector en was shown to transform as a contravariant vector into an axis-aligned basis vector e'n in x'- space (3.3.2) (3.3.3) (3.2.6) (3.2.1) e' n = R en ( e'n)i = ΣjRij (en)j (en)i = Sin (e'n)i = δn,i (3.5.2) The same thing happens here, only in reverse : u n = S u'n ( un)i = ΣjSij (u'n)j (u'n)i = Rin (un)i = δn,i (3.5.3) where now the u n are axis-aligned basis vectors in x-space. A prime on an object indicates which space it inhabits. The inverse tangent base vectors u'n are not the same as the reciprocal base vectors En introduced in Chapter 6 below. Example 1: Polar coordinates: inverse tangent base vectors a nd inverse coordinate lines It was shown earlier for polar coordinates that R = S -1 = ⎝⎛ ⎠⎞-sinθ/r cosθ/r cos(θ) sinθ (3.4.1) so the inverse tangent base vectors (expressed here as row vectors as usual to save space) are given by the columns of R as per (3.5.1), u' x = ( -sinθ/r,cosθ) // note near θ = 0 that u'x indicates a large negative slope u'y = (cosθ/r,sinθ) . // note near θ = 0 that u'y indicates a small positive slope (3.5.4) One expects u' x to be tangent to an inverse coordinate lin e in x'-space which maps to a line in x-space along which only x is varying, which is a horizontal line at fixed y (red below). Looking at the small θ region of the left graph in Fig (3.5.5) belo w, one sees slopes as just described above. For the polar coordinates mapping discussed near Fig (3.4.3), horizontal (red) and vertical (blue) lines in x'-space mapped into circles (red) and rays (blue) in x-space, and the tangent base vectors in x-space were tangent to the coordinate lines there. If one instead takes horizontal (red) and vertical (blue) lines in x- space and maps them back into coordi nate lines in x'-space, the picture is a bit more complicated. Since y = rsinθ, the plot of an x-coordinate line (x is varying, y fixed at y i) in x'-space has the form r = y i/sinθ, where y i denotes some selected y value (a red horizontal line), so plotting r = y i/sinθ in x'-space for various values of y i displays a set of inverse x-coordinate lines (red). Similarly r = x i/cosθ gives some y- coordinate lines (blue). Here is a Maple plot: Chapter 3: Tangent Base Vectors 38 x'-space ( θ,r) x-space (x,y) ( 3 . 5 . 5 ) Another example is given in Appendix C. Chapter 4: Cartesian Space 39 4. Notions of length, distance and scalar product in Cartesian Space This Section can be interpreted in either Picture B or Picture D where the x-space is Cartesian, G=1. Up to this point, we have dealt only with the vector space RN (a vector space is sometimes called a linear space), and have not "endowed" it with a norm, metric or a scalar product. Quantities like d xi above were just little vectors and x + dx was vector addition. Now, for the first time (officially), we discuss leng th and distance, such as they are in a Cartesian Space, as defined in Chapter 1. For RN one first defines a norm which determines the "length" of a vector, the first notion of distance in a limited sense. The "usual" norm is the L2 norm given by norm of x = || x || ≡ ( x 12 + x22 + .... + x N2 )1/2 ≡ | x | . (4.1) Now we have a normed linear space. One next defines the notion of the distance between two vectors. Although this can be done in many ways, just as there are many possible norms, for RN the "natural metric" is defined in terms of the above L2 norm, so that distance between x and y = metric = d( x,y) ≡ || x - y || = ( [x 1-y1]2 + [x2-y2]2 + .... + [x N-yN]2 )1/2 . (4.2) Now our space is both a normed linear space and a metric space, a combo known as a Banach Space. One finally adds the notion of a scalar product (inner product) in this way (x,y) ≡ Σ ixiyi ≡ x • y // = Σij δi,j xi yj (4.3) which of course implies this special case, (x,x) = x • x = Σ ixi2 = ||x||2 = | x |2 . ( 4 . 4 ) Our space has now ascended to the higher level of being a real Hilbert Space of N dimensions. All this structure is implied by the notation RN, our "Cartesian Space". The length of the vector dx in RN is given by length of dx = distance between vectors x+dx and x ≡ ds ≡ || dx || = Σi(dxi)2 . (4.5) To avoid dealing with the square root, one usually writes (ds)2 ≡ || dx ||2 = Σi(dxi)2 = (dx1)2 + (dx2)2 + ... + (dx N)2 = Σ i dxi dxi = Σij δi,j dxi dxj . ( 4 . 6 ) As shown in the next Section, one can interpret δ i,j as the metric tensor in Cartesian Space. Chapter 4: Cartesian Space 40 The cursory discussion of this Section is fleshed out in Chapter 2 of Stakgold where the concepts of linear spaces, norms, metrics and inner products are defined with precision. Stakgold compares our N dimensional Cartesian Hilbert Space to the N=∞ dimensional Hilbert Spaces used in functional analysis, where basis vectors might be Legendre polynomials P n(z) on (-1,1), n = 0,1,2... ∞. He has little to say, however, about curvilinear coordinate spaces in this particular book. Chapter 5: The Metric Tensor 41 5. The Metric Tensor 5.1 The Picture D Context The metric tensor is the heart of the machine of tensor analysis and we shall have a lot to say about it in this Chapter. Each Section is best presented in the context of one of our Pictures, and there will be some jumping around between pictures. We apologize for th is inconvenience and ask forbearance. Hopefully the Sections below will give the reader some expe rience with typical nitty-gritty manipulations. One advantage of the developmental notation over the standard notation is that matrix methods are easy to use, and they will be used below. Unless otherwise sp ecified, repeated indices are implicitly summed over (Einstein convention ). But sometimes we do show sum symbols Σ where extra clarity is needed. We now go to the Picture D context. Comparison with Picture B shows that primes must be placed on objects F, R and S related to the tran sformation from x-space to x'-space: ( 5 . 1 . 1 ) The various partial derivatives are de termined from their definitions, R' ik ≡ (∂x'i/∂xk) R" ik ≡ (∂x"i/∂xk) R ik ≡ (∂x"i/∂x'k) S' ik ≡ (∂xi/∂x'k) S" ik ≡ (∂xi/∂x"k) S ik ≡ (∂x'i/∂x"k) . (5.1.2) The unprimed S,R can be expressed in terms of th e primed objects this way (chain rule, implied Σ on a), Rik ≡ (∂x"i/∂x'k) = (∂x"i/∂xa) (∂xa/∂x'k) = R"ia S'ak ⇒ R = R" S' S ik ≡ (∂x'i/∂x"k) = (∂x'i/∂xa) (∂xa/∂x"k) = R'ia S"ak ⇒ S = R' S" . (5.1.3) 5.2 Definition of the metric tensor The metric or distance between vectors x and x+dx can be specified as done in Chapter 4 in terms of the norm of differential vector d x, metric(x +dx, x) = norm( [x +dx] - x) = norm(d x) ≡ | dx | ≡ ds , (5.2.1) Chapter 5: The Metric Tensor 42 with the caveat that for non-Cartesi an spaces this may not be an official norm, see Section 5.10 below. The squared distance (ds)2 must be a linear combination of products dx idxj just on dimensional grounds. The coefficients in this linear combination form a matrix called the metric tensor (later we show this matrix really is a tensor). Here for an N-dimensional space, (ds) 2 = Σi=1N Σj=1N [ metric tensor ] ij dxi dxj . (5.2.2) One "endows" a space with a certain metric tensor, and this in turn determines the distance ds = norm(d x) for any differential vector dx in that space. A metric tensor is specific to a space; it is a property of the space; it is part of the space's definition. Recall from the text near (1.10) that our special "quasi-Cartesian" space has a diagonal metric tensor G whose diagonal elements are independently either +1 or -1. Then distance ds is determined by (ds) 2 = ΣiΣj[Gij]dxidxj = ΣiΣj[Giiδi,j]dxidxj = Σi Gii dxidxi = Σi Gii (dxi)2. (5.2.3) How might one express this same ds in terms of the the coordinates x' i of x'-space in Picture D? We found in (2.1.6) that, for Picture A of Fig (2.1.1), d x = S(x ')dx' where S is a matrix which represents a general transformation x' = F (x) locally at a point. Translating this to the right side of Picture D, we have dx = S'( x') dx' since S is primed there (as is F and R). Therefore (ds) 2 = ΣiGiidxidxi = Σi Gii (ΣkS'ik dx'k) (ΣmS'im dx'm) = Σ kΣm { Σi Gii S'ikS'im } dx'k dx'm . (5.2.4) Now let's start all over again and define a metric tensor in x'-space just as we did in x-space, (ds')2 = ΣkΣm [ metric tensor ]' km dx'k dx'm . = Σ kΣm g¯'km dx'k dx'm . ( 5 . 2 . 5 ) We call the metric tensor in x'-spa ce g' because x' is primed, and we use an overbar because it will be shown below that g ¯'km are the components of a covariant rank-2 tensor g ¯' in x'-space. We now make an important hypothesis: the length of vector d x is the same as the length of the vector d x', which is to say, we assume that ds = ds'. We are a ssuming that ds is a scalar under transformation F', so (ds) 2 = (ds')2 . // hypothesis, (ds)2 is a scalar (5.2.6) Since equations (5.2.4) and (5.2.5) must be valid for any choice of d x', comparison of the two equations yields this relationship betw een the two metric tensors g ¯' and G : g¯'km ≡ { ΣiGiiS'ikS'im } . ( 5 . 2 . 7 ) Using G ij = Giiδi,j we obtain the matrix relati onship between metric tensors g ¯' and G : Chapter 5: The Metric Tensor 43 g¯'km ≡ { ΣiGiiS'ikS'im } = ΣiΣj S'T kiGijS'jm ⇒ g¯' = S'TG S ' . ( 5 . 2 . 8 ) We now repeat all the above steps for the transfor mation from x-space to x"-space on the left side of Picture D in Fig (5.1.1). We make the hypothesis that (ds)2 = (ds")2 and in doing so, we obtain a relationship between the metric tensor G for x-space and the metric tensor g ¯" for x"-space: g¯'' = S''TG S ' ' . ( 5 . 2 . 9 ) To summarize, there are three metric tensors for the three spaces in Picture D : g¯ = G g ¯' = S' T G S' g ¯" = S"T G S" . (5.2.10) where we have made the hypothesis that ds = ds' = ds". Concerning the invariance of (ds). In the above discussion, it was assu med in (5.2.6) that distance (ds)2 is the same in x'-space as it is in x-sp ace. As shown below in Section 5.10, this is part of a larger hypothesis that the covariant dot product of any two vectors gives the same number regardless of which space is used to compute the dot product: A • B = A ' • B'. This in turn implies that | A| = |A'| and in particular |dx | = |dx'| or ds = ds'. In our major application, where x-space is Cart esian and x'-space is that of some curvilinear coordinates, it is a requirement that | A | = | A ' |. The length of a vector in Cartesian physical space does not change simply because we choose to express that length in some curvilinear coordinates. Imagine that A is a velocity vector v. The speed | v| of an object is the same number whether one represents v in Cartesian or spherical coordinates. In special relativity it is a very well-verified hypothesis that dot products are scalars and that (ds)2 is a scalar under Lorentz transformations. In this context, ds is often written d τ (the so-called proper time). There are, however, applications of transformations where the scalarity of (ds)2 is not valid and in fact it is crucial that (ds)2 can change under a transformation. For example, in continuum mechanics one can think of the flow of a tiny bit of continuous matter as being modeled by a transformation x = F(x',t). In general F ( = F-1) is non-linear. At time t = 0 the geometry of this little blob is described by x'-space coordinates, and after a flow at time t = t it is de scribed by x-space coordinates. If once traces during this flow a little "dumbbell" vector between two very clos ely spaced particles in th e blob, one finds that d x' at time t = 0 becomes dx = Sd x' at time t = t where S is not simply a rotation. The whole point here is that during the flow, the distance vector between two close particles rotates and stretches in some manner, and in general (due to this stretch) |d x| ≠ |dx'|, so the geometric (ds)2 is definitely not invariant under the flow (ie, under the transformation F). On the other hand, as we shall see below, a certain curvilinear (ds)2 does remain invariant under the flow. The flow application is considered more in S ection 5.16 below, and still further in Appendix K. Chapter 5: The Metric Tensor 44 5.3 Inverse of the metric tensor The inverses of the three metric tensor s in (5.2.10) shall be indicates without an overbar, and we shall eventually show these matrices to be "contravariant" matrices and thus deserve no overbar. We thus now define three new g matrices as these inverses, and compute the inverses: g ≡ g¯-1 = G-1 = G // remember G just has +1 and -1 diagonal elements g' ≡ g¯' -1 = (S'T G S')-1 = S'-1 G (S'T)-1 = R' G R'T g" ≡ g¯" -1 = (S"T G S")-1 = S"-1 G (S"T)-1 = R" G R"T . (5.3.1) Here are the collected facts from above: g = G g' = R'G R' T g" = R" G R"T S = R' S" g¯ = G g ¯' = S'TG S' g ¯" = S"T G S" R = R" S' g¯g = 1 g ¯'g' = 1 g ¯"g = 1 (5.3.2) Comment: In the Picture C context but with a Quasi-Cartesian x(0)-space (g(0)=G), one could take the second column above and write it this way, g = RGRT g¯ = STGS g¯g = 1 (ds)2 = g¯km dxk dxm ( 5 . 3 . 3 ) where now the clutter of primes is gone. If x(0)-space is Cartesian so g(0) = G = 1, then g = RR T g¯ = STS. // if x(0)-space is Cartesian in Picture C (5.3.4) But we continue with Picture D shown in (5.1.1). 5.4 A metric tensor is symmetric Consider the following, w here A is any matrix and D is a diagonal matrix : N = ADAT ⇒ NT = (ADAT)T = ADTAT = ADAT = N ⇒ N symmetric (5.4.1) Replacing A with AT and N with M gives Chapter 5: The Metric Tensor 45 M = ATDA ⇒ MT = (ATDA)T = ATDT(AT)T = ATDA = M ⇒ M symmetric (5.4.2) Looking at (5.3.3), since g has the form of N and g ¯ the form of M with G diagonal, we conclude that both g and g ¯ are symmetric matrices, and the same is of course true fo r g' and g". Thus gab = gba g ¯ab = g¯ba g'ab = g'ba g ¯'ab = g¯'ba ( 5 . 4 . 3 ) 5.5 det(g) and g nn of a Cartesian-generated metric tensor are non-negative If we arrive at x'-space by a transformation F from a Cartesian x-space (as opposed to a Quasi-Cartesian one), we refer to the metric tensor g' in this x'-spa ce as being "Cartesian generated". In this case G = 1 and the metric tensors above are g = RRT and g ¯ = STS as in (5.3.4). Any matrix of either of these forms has positive diagonal elements a nd positive determinant: (A TA)aa = Σb (AT)abAba = Σb (A)baAba = Σb (Aba)2 ≥ 0 // diagonal elements ≥ 0 det(ATA) = det(AT) det(A) = det(A) det(A) = [ det(A) ]2 ≥ 0 // det ≥ 0 . (5.5.1) To show these results for the AA T form, just replace A →AT everywhere. Recall that transformation F maps RN → RN so the coefficients of the linearized matrices R and S are real, and elements of the metric tensor must therefore also be real. For a Quasi-Carte sian-generated metric tensor, these proofs are invalid since then g = RGRT and g¯ = STGS and G ≠1. 5.6 Definition of two kinds of rank-2 tensors We now switch to Picture A, (5.6.1) Recall the vector transformation rules from (2.5.1), V' = R V contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 V¯' = ST V¯ covariant S ik(x') ≡ (∂xi/∂x'k) = ST ki(x') (2.5.1) which can be written out in components (implied sum on a'), Chapter 5: The Metric Tensor 46 V'a = Raa' Va' contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 V¯'a = ST aa'V¯a' covariant S ik(x') ≡ (∂xi/∂x'k) = ST ki(x') . (5.6.2) A rank-1 tensor is defined to be a vector which transforms in one of the two ways shown above. Similarly, a (non-mixed) rank-2 tensor is defined as a matrix which transforms in one of these two ways: M'ab = Raa' Rbb' Ma'b' // contravariant rank-2 tensor M¯'ab = ST aa' ST bb' M¯a'b' // covariant rank-2 tensor (5.6.3) and again we put a bar over the covariant objects. Digression : Proof that (A-1)T = (AT)-1 for any invertible matrix A: ( 5 . 6 . 4 ) • det(A) = det(AT) • cof(AT) = [ cof(A)]T since [cof(AT)]ab = cof ( AT ab) = cof(A ba) = [cof(A)] ba = [cof(A)]T ab • (A-1)T = { [cof(A)]T / det(A) }T = [cof(AT)]T /det(AT) = (AT)-1 This fact is used many times in the manipulations below. 5.7 Proof that the metric tensor and its inverse are both rank-2 tensors The above r ank-2 tensor transformation rules (5.6.3) can be written in the following matrix form (something not possible with higher-rank tensors), M' = R M R T // contravariant rank-2 tensor M¯' = ST M¯ S // covariant rank-2 tensor (5.7.1) where recall . (5.7.2) But we now switch these rules to the Picture D context (upper arrow) where F maps x'-space to x"-space, Chapter 5: The Metric Tensor 47 (5.1.1) to obtain M" = R M' RT // contravariant rank-2 tensor (5.7.3) M¯" = ST M¯' S . // covariant rank-2 tensor Consider then this sequence of steps: 1 *G * 1 = 1 * G * 1 (S"R") G (S"R") T = (S'R') G (S'R')T // S"R" = 1 and S'R' = 1 S" (R"G R"T) S"T = S'(R'G R'T) S'T // regroup S" g" S"T = S' g' S'T // since g" = R"G R"T and g' = R'G R'T by (5.3.2) g" S"T = R" S' g' S'T // left multiply by S"-1 = R" g" = R" S' g' S'T R"T // right multiply by S"T,-1 = R"T (5.7.4) g" = (R" S') g' (S'T R"T) // regroup g" = (R" S') g' (R" S')T // (AB)T = BTAT g" = R g' RT // using long forgotten (5.1.3) With definition (5.7.3) this last result then shows that g' is a contravariant rank-2 tensor with respect to the transformation F taking x'-space to x"-space. Continuing on, g" = R g' R T g"-1 = (R g' RT)-1 g"-1 = ST g'-1 S // RT,-1= ST etc g¯" = ST g¯' S // g ¯' = g'-1 ( 5 . 7 . 5 ) and this last result shows that g ¯' is a covariant rank-2 tensor with respect to the transformation F taking x'-space to x"-space, again from (5.7.3). This is why we put a bar over this g from the start. These two metric tensor transformation statements can be converted to the Picture A context, Chapter 5: The Metric Tensor 48 g' = R g RT g' ab = Raa'Rbb'ga'b' // g is a contravariant rank-2 tensor g¯' = ST g¯ S g ¯'ab = ST aa'ST bb'g¯a'b' // g¯ is a covariant rank-2 tensor (5.7.6) Since RS = 1, the equations can be inverted to get g = S g' ST g ab = Saa'Sbb'g'a'b' g¯ = RT g¯' R g ¯ab = RT aa'RT bb'g¯'a'b' = Ra'aRb'bg¯'a'b' (5.7.7) Further variations of the above are obtained using RS = 1, gg ¯ = g'g¯' = 1 and g = g T (etc.) : Rg = g' S T g ¯' R g = ST g' ST g¯ = R g¯ S = RT g¯' g RT g¯' = S g ¯ S g' = RT . (5.7.8) Finally, if x-space is Cartesian so g = g ¯ = 1, one has, analogous to (5.3.4) for Picture C, g' = RRT // g = 1 g¯' = STS ( 5 . 7 . 9 ) Comment : If x-space and x'-space are both quasi-Cartesian, so g = g' = G, then (5.7.6) says G = RGRT. This does not imply that R must be independent of x. One could have G = R( x)GR( x)T. In particular, if both spaces are Cartesian so g = g' = 1, then one can have 1 = R( x)R(x)T if R(x ) is a real orthogonal matrix for every value of x, R-1(x) = RT(x). When R-1 = RT we refer to R as a "rotation", where we include possible parity transformations. Th en if g = 1 and g' = 1 one can have x' = F(x) = R( x) x where we refer to R(x ) as a local rotation . The rotation can be different at every point in space. An example would be R( x) = Rz(θ) = ⎝⎜⎛ ⎠⎟⎞ cosθ -sinθ 0 sinθ cosθ 0 0 0 1 where θ = θ(x) . (5.7.10) For N = 4 dimensions in special relativity where g = g' = G = diag(-1,1,1,1), one can have 4x4 boost and rotation matrices of the form shown below in (5.14.3) and (5.14.4) where the parameters b and r could be b(x) and r( x). Chapter 5: The Metric Tensor 49 5.8 Metric tensor converts vector types We continue in Picture A . Suppose V is a contravariant vector so V' = R V. Construct a new vector W with the following properties W = g ¯ V x-space W' = g ¯' V' x'-space . (5.8.1) Is vector W one of our two vector types, or is it neither? One must examine how it transforms under F: W' = g¯' V' = (S T g¯ S) (R V) = ST g¯ (SR)V = ST g¯ V = ST W . // using (5.7.6) (5.8.2) Therefore from (2.5.1) this new vector W is a covariant vector under F, so it should have an overbar, W¯ ≡ g¯ V and W¯' = ST W¯ . ( 5 . 8 . 3 ) This covariant vector W¯ can be regarded as the covariant partner of contravariant vector V. This shows the general idea that applying g ¯ to any contravariant vector produces a covariant vector! So this is one way to construct covariant vectors if we have a supply of contravariant ones. Conversely, starting with a known covariant vector W¯, one can construct a contravariant vector V ≡ g W¯ . Thus, every vector of either type can be thought of as having a partner vector of the other type. An obvious notation is to write W¯ as V¯ so no extra letter is needed. Then one has V¯ = g¯ V V = g V¯ // g ¯ = g-1 from (5.3.1) V¯i = g¯ijVj Vi= gijV¯j . (5.8.4) Then the vectors V and V¯ are the partner vectors, the first c ontravariant, the second covariant. 5.9 Vectors in Cartesian space Theore m: There is no distinction between a contravariant and a covariant vector in Cartesian space. ( 5 . 9 . 1 ) Proof : Pick a contravariant vector V. Since g ¯ = 1, V¯ ≡ g¯ V = V . But V ¯ is a covariant vector. Since V¯ = V, every contravariant vector is identical to its covariant partner in x-space . The vector components are numerically equal. The transformation rules (2.5.1) in this case are V' = R V V¯' = S T V¯ = ST V . ( 5 . 9 . 2 ) Since in general one does not have R = ST, one sees that in general V' ≠ V¯' , so the two partner vectors are in general not identical in x'-space even thoug h they are identical in Cartesian x-space. In the special case that R is a rotation, R = R-1,T (real orthogonal) and then R = R-1,T = ST so V ' = V¯' . Chapter 5: The Metric Tensor 50 5.10 The covariant dot product A • B and norm |A| For a C artesian space, Chapter 4 defined the norm as th e length of a vector, the metric as the distance between two vectors, and the scalar product (inner pr oduct) as the projection of one vector on another. The official definitions of norm, metric and scalar product require non-negativity: | x| ≥ 0, d( x,y) ≥ 0, and x • x ≥ 0. For non-Cartesian spaces, the logical extensions of these three concepts can result in all three quantities being negative. Neverthele ss, we shall use the term "covari ant scalar product" with notation A • B as defined below, as well as the notation | A|2 ≡ A • A where | A| will be called the length, magnitude or norm of A, even though these objects are not true scalar products or norms. In the curvilinear application of tensor analysis, where x-space is Cartesian, since the norm and s calar product are tensorial scalars, and since they are non-negative in Cartesian x-space, the problem of negative norms does not arise in either space. How do authors handle this problem? Some authors refer to A • A as "the norm" of A (e.g., Messiah last line of p 878 discussing special relativity) , which is our | A|2. For a general 4-vector A in special or general relativity, most authors just write A • A (AμAμ in standard notation), they note that the quantity is invariant under transformations, but don't give it a name. Whereas we use the bold • for this covariant dot product, most special relativity authors prefer to reserve this bold dot for a 3D spatial dot product, an d then the 4D dot product is written with some "less bold dot" such as A .B or A•B. Typical usage then in standard notation would be p•p = p μpμ = p02 - p•p (see for example Bjorken and Drell p 281). Without further ado, we define the "covariant scalar product" of two contravariant vectors (a new and different use of the word "covariant", but th e same as appears in Section 7.15) as A • B ≡ A aB¯a . // implied sum on a (5.10.1) The important fact about the dot product of two tensorial vectors is that it is a tensorial scalar, as we now show using (5.9.2) for the transformations of A and B and (2.1.6) that SR = 1 : A' • B' ≡ A' aB¯a' = (RA) a(STB¯)a = (RaiAi)(SjaB¯j) = (SjaRai)AiB¯j = (SR)jiAiB¯j = δ j,iAiB¯j = AiB¯i = A • B . ( 5 . 1 0 . 2 ) There are various equivalent ways to write th e dot product of (5.10.1) using (5.8.4) that V¯ = g¯ V and conversely that V = g V¯ , and also the fact (5.4.3) that g ab = gba : Chapter 5: The Metric Tensor 51 A • B ≡ AaB¯a = Aa(g¯abBb) = g¯abAaBb = (g¯baAa)Bb = A¯bBb = A¯aBa A • B ≡ AaB¯a = (gA¯)aB¯a = gab A¯b B¯a = gab A¯b B¯a . To summarize A • B = A aB¯a = A¯aBa = g¯abAaBb = gab A¯b B¯a . (5.10.3) In the special case that A = B, we use the shorthand norm notation (with caveat as noted above) and (5.10.2) to obtain, |A|2 ≡ A • A = A' • A' = |A'|2 . ( 5 . 1 0 . 4 ) Going back to Chapter 3 and the vectors e'n and en, a claim made at the start of Section 3.4 can now be verified: |e'n|2 = e'n • e'n = en • en = |en|2 ⇒ | e'n| = |en| . (5.10.5) Comment on Notation Consider again the definition (5.10.1) and the alternate form A ¯aBa shown in (5.10.3) A • B ≡ A aB¯a = A¯aBa . (5.10.3) The dot product involves the contravariant components of one vector and the covariant components of the other vector. In the dot product notation A • B , we have indicated each vector by its contravariant name just as a convention. We could just as well have indi cated one or both vectors by its covariant name, but the dot product indicated by whatever name would be the same: contravariant components of one vector and the covariant components of the other vector. Thus, A • B ≡ A iB¯i = A¯iBi = A¯ • B = A • B¯ = A¯ • B¯ . (5.10.6) We shall always use the first notation A • B since it is the simplest. Similarly, when B = A, |A|2 ≡ A • A = | A¯|2 . ( 5 . 1 0 . 7 ) We thus have the interesting fact that, if A and B are tensorial vectors for general x' = F(x), then A ≠ A¯ but A • B = A¯ • B for any B . ( 5 . 1 0 . 8 ) If B = A, this says A ≠ A¯ but A • A = A¯ • A = | A|2 = AiA¯i = gijAiAj . (5.10.9) Chapter 5: The Metric Tensor 52 We mention this notational issue to head off the following incorrect notion: A • B = g¯abAaBb ⇒ A¯ • B¯ = g¯abA¯aB¯b ≠ A • B . // wrong!!! A dot product application In applications in which (ds)2 is regarded as a scalar with respect to transformation F we have (ds')2 = d x' • dx' = (ds)2 = dx • dx (5.10.10) and ds = ds' is called "the invariant distance". Such applications include curvilinear coordinate transformations and relativity transformations. In special relativity, using the Bjorken and Drell notation noted above where g' μν = diag(1,-1,-,1,-1) and c = 1, one writes (Standard Notation), (dτ)2 = g'μν dx'μdx'ν = dx'μdx'μ = dx'• dx' = dx • dx = a Lorentz scalar = (dt)2 - dx • dx , xμ = (t, x) ( 5 . 1 0 . 1 1 ) and dτ is called "the proper time", a particular case of the invariant distance ds. Notice that (d τ)2 < 0 for a spacelike 4-vector dxμ, meaning one that lies outside the future and past lightcones (|d x| > |dt| ). [We now restore • to our covariant definition after temporarily usi ng it above for a 3-space Cartesian dot product. ] 5.11 Metric tensor and tangent base vectors: scale factors and orthogonal coordinates The context of Picture A continues, (5.11.1) Recall this fact from (3.2.7), S = [ e 1, e2, e3 .... eN ] (3.2.7) where the columns of S are the tangent base vectors. It follows from (5.7.6) that g¯' = ST g¯ S = [ e1, e2, e3 .... eN ]T g¯ [e1, e2, e3 .... eN ] (5.11.2) so Chapter 5: The Metric Tensor 53 e 1•e1 e1• e2 e1 • e3 ...... e 1• eN e 2•e1 e2• e2 e2 • e3 ...... e 2• eN g ¯' = e3•e1 e3• e2 e3 • e3 ...... e 3• eN (5.11.3) ........ e N•e1 eN• e2 eN • e3 ...... e N• eN since, e nT g¯ em = ( en)i g¯ij (em)j = g¯ij(en)i(em)j = en • em (5.11.4) using the covariant dot product shown in (5.10.3) . Taking the m,n component of (5.11.3) one gets g¯'mn = em • en or g ¯'mn = ∂'mx • ∂'nx // using (3.2.6) (5.11.5) which makes a direct connection between the covarian t metric tensor in x'-space and the tangent base vectors en in x-space. A less graphical derivation of this fact uses (5.7.6) and (3.2.5), g¯'nm = (STg¯S)nm = ST na g¯ab Sbm = g¯ab San Sbm = g¯ab (en)a(eb)n ≡ en • em . (5.11.6) Scale Factors and Orthgonal Coordinates We showed just above in (5.11.5) that em • en = g¯'mn. Regardless of whether or not the coordinates are orthogonal, one can define the scale factor h' n and unit vector e^n as follows, h' n2 ≡ |en|2 = en • en = g¯'nn ⇒ h'n = |en| = g¯'nn ⇒ e^n ≡ en/h'n = "unit vector", | e^n| = 1 . (5.11.7) If the tangent base vectors en are orthogonal , then clearly the metric tensor g ¯'mn = em • en must be diagonal. In fact, based on (5.11.7), th e covariant metric tensor must be g¯'mn = h'm2 δm,n . We claim that the contravariant metric tensor g' nm is then also diagonal with diagonal elements g' nn = h'n-2 . To verify this claim, we confirm that g ¯' g' = 1: Σm g¯'nm g'mk = Σm [δn,m h'm2][δm,kh'm-2] = Σm δn,mδm,k = δn,k = (1)nk Since the inverse matrix is unique as long as det(g') ≠0, g' mk = δm,kh'm-2 must be it! Chapter 5: The Metric Tensor 54 We have then established this simple fact: orthogonal coordinates ⇔ g¯'mn = h'm2 δm,n covariant g ' mn = h'm-2δm,n contravariant . (5.11.8) For an orthogonal coordinate system, at any point x in x-space, the tangents e n to the N coordinate lines passing through that point are orthogonal. Most examples below will involve such systems, with Appendix C providing a non-orthogonal example. In the Standard Notation of Chapter 7, the above association becomes orthogonal coordinates ⇔ g' mn = h'm2 δm,n covariant g 'mn = h'm-2δm,n contravariant . (5.11.9) The reason we put a prime on h' n is because it is associated with x'-s pace and its metric tensor g' in Picture A or B of Fig (1.11). The curvilinear coordinates are x'. When working with Picture C, however, we would call this scale factor h n because it is then associated with the metric tensor g and x-space. In Picture C, the curvilinear coordinates are x. In other Pictures, the curvilinear "space on the left" might be called ξ-space, and since this does not carry a prime, again one would write h n. Section 5.5 above showed that g ¯'nn ≥ 0 when x-space is Cartesian. This is the usual case for the curvilinear coordinates application, and so in this case the scale factors h' n are always real and positive. Note 1 . Different authors use different symbols for scal e factors. For example, Margenau and Murphy refer to them as the Q n. Morse and Feshbach and most modern works use h n. They never use a prime, because they always use Picture C or equivalent where x is the curvilinear coordinate. Note 2 : Some authors refer to the scale factors h' n as the Lamé coefficients , while other authors refer to Rij as the Lamé coefficients which they call h ji. (Lame, for PDF search with no accent mark) Chapter 5: The Metric Tensor 55 5.12 The Jacobian J The context of Picture A continues, (5.12.1) First of all, note that since RS = 1, det(S) = 1/det(R) . (5.12.2) Recall from (2.1.6) our definitions of R ik and Sik : R ik(x) = (∂x'i/∂xk) is called the Jacobian matrix for the transformation x' = F(x) Sik(x') = (∂xi/∂x'k) is then the Jacobian matrix of the inverse transformation x = F-1(x') . (5.12.3) There are two Jacobian matrices here, which are inverses of each other since RS = 1, and for each of these matrices we could define "a Jacobian" as the determinant of that matrix: J( F) = det(R) 1/ J( F) = 1/det(R) = det(S) = J( F -1) J( F-1) = d e t ( S ) . ( 5 . 1 2 . 4 ) Because our major interest is in curvilinear coordinates x', and because curvilinear coordinates are almost always defined by equations of the inverse form x = F -1(x') such as the following for polar coordinates, x = rcos θ x = F-1(x' ) ( 5 . 1 2 . 5 ) y = rsin θ, we shall officially define " the Jacobian" to be J = J(F -1). So, the Jacobian ≡ J(x') ≡ det(S( x')) = det( ∂x i/∂x'k) = 1/det(R( x(x')) = 1/ det(∂ x'i/∂xk) . (5.12.6) Note 1: Objects which relate to the transformation between x-space and x'-space cannot themselves be tensors because tensor objects must be asso ciated with a specific space, the way V(x) is a vector in x- space and V'(x') is a vector in x'-space. Thus S ij(x') = ∂ xi/∂x'k , although a matrix, is not a rank-2 tensor. Similarly, J(x '), while a "scalar" function, is not a rank-0 tensorial scalar. One does not ask how S and J themselves "transform" in going from x-space to x'-space. Note 2: An alternative notation used by some authors is this J(x,x') ≡ det(S( x,x')) = det( ∂xi/∂x'k) ( 5 . 1 2 . 7 ) Chapter 5: The Metric Tensor 56 as if x and x' were independent variab les. In our presentation, x is not an independent variable but is determined by x = F-1(x'). Just as one might write f '(x') = ∂f/∂x', we write J( x') = det(∂xi/∂x'k). The connection would be J( x') = J(x=F-1(x'),x' )) = J(x(x') ,x'). Note 3: Other sources often use the notation | M | to indicat e the determinant of a matrix. We shall use the notation det(M), and reserve | | to indicate the magnitude of some quantity, such as |J| below. The determinant of any NxN matrix S may be written ( εabc.. is the permutation tensor, see Section 7.7), det(S) = εabc...x Sa1 Sb2 ... SxN . // = ε abc...x S1a S2b ... SNx (5.12.8) For our particular S with S in = (en)i from (3.2.5) this becomes det(S) = ε abc...x (e1)a(e2)b....... ( eN)x ( 5 . 1 2 . 9 ) so J is related to the tangent base vectors by J = ε abc...x (e1)a(e2)b....... ( eN)x . (5.12.10) It was shown in (5.7.6) that g ¯' = S T g¯ S and g' = R g RT , these being the transformation rules for covariant and contravariant rank-2-tensors. Therefore det(g ¯') = det(S Tg¯S) = det(ST)det(g¯)det(S) = det(S)det(S)det(g ¯) = J2 det(g¯) det(g') = det(RgRT) = det(R)det(g)det(RT) = det(R)det(R)det(g) = J-2 det(g) or det(g ¯') = J2 det(g¯) ⇒ J2 = det(g ¯') / det(g ¯) = [det(S)]2 det(g') = J-2d e t ( g ) . ( 5 . 1 2 . 1 1 ) It is a tradition to define certain scalar (but not tensorial scalar) objects with the same name g and g', g( x) ≡ det(g¯(x)) = 1/det(g( x)) // in x-space g'( x') ≡ det(g¯'(x')) = 1/det(g'( x')) // in x'-space (5.12.12) so that J 2(x') = det(g ¯'(x')) / det(g ¯(x)) = g'( x') / g( x) . (5.12.13) Normally the argument dependence is suppressed and one then writes g ≡ det(g¯) = 1/det(g) g' ≡ det(g¯') = 1/det(g') J 2 = det(g ¯')/ det(g ¯) = g'/g ⇒ g' = J2 g . (5.12.14) As explained in Section D.1, the equation g' = J2 g says that g, instead of being a tensorial scalar, is a scalar density of weight -2. A tensorial scalar s has weight 0: s' = J0s = s. Chapter 5: The Metric Tensor 57 Warning : One must be careful to distinguish the scalars g and g' from the the matrices g and g'. It is usually clear from the context which meaning is implied. It is convenient to make the following definition, called the signature of the metric tensor, s = sign[det(g ¯)] = sign(g) // that is, s = either +1 or -1 . (5.12.15) Since g g ¯ = 1 by (5.3.2), one has det(g)det(g ¯) = 1 or so that sign[det(g ¯)] = sign[det(g)] . Since det(g ¯') / det(g ¯) = [det(S)] 2 by (5.12.11), one has sign[det(g ¯)] = sign[det(g ¯')]. Since g' g ¯' = 1 by (5.3.2), one has det(g')det(g ¯') = 1 or so that sign[det(g ¯')] = sign[det(g')] . Therefore: s = sign[det(g ¯)] = sign[det(g)] = sign[det(g ¯')] = sign[det(g')] = sign(g) = sign(g') . (5.12.16) Since transformation F is assumed invertible in its domain and range, one cannot have det(S)=0 anywhere except perhaps on a boundary. Since det(g ¯') = [det(S)] 2det(g¯) by (5.12.11), if we assume det(g ¯) is non- vanishing in the x-space domain of F, then det(g ¯') ≠ 0 everywhere in the range of F. The conclusion with this assumption is that the signature s is always well-defined. Obviously, the quantities sg and sg' are both positive, and since J 2 = g'/g one can write |J| = sg' / sg = | det(S) | = g'/g . (5.12.17) For the curvilinear coordinates app lication, x-space is Cartesian, det(g ¯) = g = 1, and thus s = 1 and then |J| = g' = | det(S) | . // curvilinear (5.12.18) For the relativity application, x-space is Minkowski space with det(g ¯) = g = -1 so s = -1 and |J| = -g' = | det(S) | . // relativity (5.12.19) Here then is a summary of the results of this Section: Chapter 5: The Metric Tensor 58 J( x') ≡ det(S( x')) = det( ∂xi/∂x'k) = 1/det(R( x(x')) = 1/ det( ∂x'i/∂xk) g ≡ det(g¯) g' ≡ det(g¯') g' = J2g ⇒ g is a scalar density of weight -2 (Section D.1) s ≡ sign[det(g ¯)] = sign[det(g)] = sign[det(g ¯')] = sign[det(g')] = sign(g) = sign(g') |J| = sg' / sg = | det(S) | = g'/g (5.12.20) Note : Weinberg p 98 (4.4.1) defines g = -det(g ij). This is the only one of Weinberg's conventions that we have not adopted, so in this paper it is always true that g ≡ + det(g ij) even though this is -1 in the application to special relativity. Carl Gustav Jacob Jacobi (1804 –1851) . German, Berlin PhD 1825 then went to Konigsberg, did much in a short life. Elucidated the whole world of elliptic integrals and functions, such as F(x,k) and sn(x;k), which occur even in simple problems like the 2D pendulum. Wiki claims he promoted Legendre's ∂ symbol for partial derivatives (used throughout this document) and made it a standard. Among many other contributions, he saw the significance of the obj ect J which now bears his name: "the Jacobian". The Jacobi Identity is another familiar item, a rule for non-commuting operators [ x,[y,z]] + [ z,[x,y]] + [ y,[z,x]] = 0 which finds use with quantum me chanical operators and matrices, and more generally with Lie group generators -- examples of which appear in Section 5.14 below. 5.13 Some relations between g, R and S in Pictures B and C (Cartesian x-space). In Picture B, which is Picture A with g = 1, (5.13.1) the statement of the rank-2 tensor transformation of g' and g ¯' becomes, as shown in (5.7.9), g' = RRT g¯' = STS ( 5 . 1 3 . 2 ) which can be written in a variety of ways, R T = (SR)RT = S(RRT) = S g' ⇒ R = g' ST ⇒ 1 = S g' ST ST = ST(RTST) = (STS)R = g ¯' R ⇒ S = RT g¯' ⇒ 1 = RT g¯' R . (5.13.3) In summary: Chapter 5: The Metric Tensor 59 g' = RRT RT = S g' R = g' ST 1 = S g' ST g¯' = STS ST = g¯' R S = RT g¯' 1 = RT g¯' R . (5.13.4) The diagonal elements of g ¯' and g' are given by g¯'nn = Σn ST niSin = Σn (Sin2) = Σi (∂xi/∂x'n)2 g'nn = Σn RniRT in = Σn (Rni2) = Σi (∂x'n/∂xi)2 . (5.13.5) If the x' i are orthogonal coordinates, then g ¯'nm = h'n2δn,m and g'nm = h'n-2δn,m as shown in (5.11.8) where the h' n are the scale factors defined in (5.11.7). These scale factors may then be expressed as, h' n2 = g¯'nn = Σi (∂xi/∂x'n)2 h ' n-2 = g'nn = Σi (∂x'n/∂xi)2 . (5.13.6) With notational changes x' n → ξn (curvilinear coordinates for M&F) and h' n → hn, these last two equations appear (for three dimensions) in Morse & Feshbach Vol I p 24 equation (1.3.4): (5.13.7) In Picture A (5.13.1) the curvilinear coordinates are called x' n and the Cartesian coordinates are x n. In Picture C, the curvilinear coordinates are called x n and the Cartesian coordinates are x n(0). To simplify notation, we use ξn in place of x n(0) for the Cartesian coordinates (these are totally different ξn from the ξn shown above in the M&F quote). ξn ≡ xn(0) (5.13.8) In this Picture C, equations (5.13.2,4,5,6) appear as follows (no primes on g's) : g = RR T g¯ = STS g = RR T RT = S g R = g ST 1 = S gST g¯ = STS ST = g¯ R S = RT g¯ 1 = RT g¯ R g¯ nn = Σn ST niSin = Σn (Sin2) = Σi (∂ξi/∂xn)2 gnn = Σn RniRT in = Σn (Rni2) = Σi (∂xn/∂ξi)2 h n2 = g¯nn = Σi (∂ξi/∂xn)2 h n-2 = gnn = Σi (∂xn/∂ξi)2 (5.13.9) Chapter 5: The Metric Tensor 60 In Picture B of (5.13.1), we have primes on things like g' and h n' because these primes match the names of the curvilinear coordinates which are x' n and which live in x'-space. In Picture C, since the curvilinear coordinates are now x n, objects like g and h n have no primes to match the fact that x n have no primes and live in x-space. And in Picture C, one has S in = ∂ξi/∂xn, where x n is the curvilinear coordinate and ξi the Cartesian coordinate. It is admittedly a bi t confusing, but it is just notation. Example 1: Polar coordinates: m etric tensor and Jacobian Picture C (5.13.8) continues (so now θ = x1 and r = x 2) and the metric tensor for polar coordinates will be computed in two ways. It was shown in (3.4.1) and (3.4.2) that S = ⎝⎛ ⎠⎞-rsinθ cosθ rcosθ sinθ = [ e1, e2 ] e1 = r(-sinθ, cosθ) e2 = (cosθ, sinθ) . (5.13.10) One way to compute g ¯ is this, using (5.13.9): ( 1= θ, 2=r ) g¯ = STS = ⎝⎛ ⎠⎞-rsinθ rcosθ cosθ sinθ ⎝⎛ ⎠⎞-rsinθ cosθ rcosθ sinθ = ⎝⎛ ⎠⎞ r2 0 0 1 ⇒ g ¯θθ = r2 g ¯rr = 1 h θ = r h r = 1 Another method uses (5.11.3) ( but in Picture C, so no prime on g ¯ ), ( 5 . 1 3 . 1 1 ) g¯ = ⎝⎜⎛ ⎠⎟⎞e1•e1 e1•e2 e2•e1 e2•e2 = ⎝⎛ ⎠⎞ r2 0 0 1 // det(g ¯) = r2 . Notice that this metric tensor is in fact symmetric , and that one of its elements is a function of the coordinates. The length2 of a small vector d x can be written using (5.2.5) (but in Picture C), (ds)2 = g¯km dxk dxm = g¯θθ dθ dθ + g¯rr dr dr = r2 (dθ)2 + (dr)2 . (5.13.12) The Jacobian is given by (5.12.6) ( but in Picture C so J( x') → J( x)), J(r,θ) = det(S(r, θ)) = det⎝⎛ ⎠⎞-rsinθ rcosθ cosθ sinθ = -r so |J| = r and g = J2 = r2, g = r . (5.13.13) Note that J2 = g'/g → g/1 = g converting (5.12.14) to Picture C. Chapter 5: The Metric Tensor 61 Example 2: Spherical coordinates: metric tensor and Jacobian As with Exam ple 1, Picture C is used, this time with (x 1, x2, x3) = (r,θ,φ) . In (3.4.4) it was found that S = ⎝⎜⎛ ⎠⎟⎞ sinθ cosφ rcosθcosφ -rsinθsinφ sinθ sinφ rcosθsinφ rsinθ cosφ cosθ -rsinθ 0 . (3.4.4) Comment : The matrix S looks the same in all Pictures. What differs in different Pictures are the names of the coordinates that go with the rows and columns of the matrix. Looking at Picture A of Fig (2.1.1) and equations (2.1.6), one sees that the matrix elements of S in Picture A are S ik = (∂xi/∂x'k) where x i are the Cartesian coordinates x,y,z and x' i are the curvilinear coordinates r, θ,φ. But in Picture C of Fig (5.13.8), one has instead S ik = ∂ξi/∂xk where ξ i are the Cartesian coordinates x,y,z and x i are the curvilinear coordinates r, θ,φ. The metric tensor from (5.13.9) is then given by a Maple matrix calculation as g¯ = S TS = ⎝⎜⎜⎛ ⎠⎟⎟⎞ 1 0 0 0 r2 0 0 0 r2sin2θ det(g ¯) = r4sin2θ (5.13.14) so that g¯ 11= g¯rr = 1 h 1 = hr = g¯rr = 1 g¯22= g¯θθ = r2 h 2 = hθ = g¯θθ = r g¯33= g¯φφ = r2sin2θ h 3 = hφ = g¯φφ = rsinθ . (5.13.15) The Jacobian using (5.12.6) is found by Maple to be, J(r,θ,φ) = det(S) = r 2sinθ . ( 5 . 1 3 . 1 6 ) An alternate calculation uses (5.12.14) converted to Picture C where J2 = g'/g → g/1 = g. Then, J = g = det(g¯) = r4sin2θ = r2sinθ. (5.13.17) Differential distance using (5.2.5) (but in Picture C) and (5.13.15) is: (ds)2 = g¯km dxk dxm = (dr)2 + r2(dθ)2 + r2sin2(dφ)2 (5.13.18) and if dφ = 0, this agrees with the pola r coordinates result (5.13.12). Chapter 5: The Metric Tensor 62 5.14 Special Relativity and its Metric Tensor: vectors and spinors In this Secti on the Standard Notation introduced below in Chapter 7 is used. In that notation R ij is written Ri j , contravariant vectors V i are written Vi, and covariant vectors V ¯j are written V j. It is a tradition in special and general relativity to use Gree k letters for 4-vector indices and Latin letters for spatial 3-vector indices. The (Quasi-Cartesian) metric tensor of special re lativity is frequently taken as G = diag(1,-1,-1,-1) and the ordering of 4-vectors as xμ = (t,x,y,z) where c=1 (speed of light) and μ= 0,1,2,3 ( Bjorken and Drell p 281). General relativity people often use G = diag(-1,1,1,1) ≡ η instead (Weinberg p 26). Still other authors use G = 1 and xμ = (it,x,y,z) where i is the imaginary i, but this approach does not easily fit into our tensor framework which is based on real numbers. A Lorentz transformation is a linear transformation Fμ ν (equation below has an implied sum on ν ) x'μ = Fμ ν xν = Rμ ν xν ⇒ xν is a contravariant vector // x' = F(x) (5.14.1) and a theory requirement is that invariant length be preserved, x' .x' = x .x = scalar. Special relativity also requires that the metric tensor G be the same in all frames, since no frame is special, so G' = G. But this says, in our old notation, that R G RT = G (which is (5.7.6) with g' = g = G). This condition restricts the (proper) Lorentz transformations to be rotations, boosts (velocity transformations), or any combination of the two. In particular, R G RT = G ⇒ det(R G RT) = det(G) ⇒ det(R) det(G) det(RT) = det(G) ⇒ [det(R)]2 (-1) = (-1) ⇒ det(R) = ±1 . (5.14.2) "Proper" Lorentz transformations have det(R) = det( F) = +1, and here are two examples. First, a boost transformation in the x direction, F μ v = ⎟⎟⎟⎟⎟ ⎠⎞ ⎜⎜⎜⎜⎜ ⎝⎛ 1 0 0 00 1 0 00 0 cosh(b) sinh(b)0 0 sinh(b) cosh(b) = exp(-ibK 1) where (K 1)μ ν = ⎟⎟⎟⎟⎟ ⎠⎞ ⎜⎜⎜⎜⎜ ⎝⎛ 0 0 0 00 0 0 00 0 00 0 0 ii (5.14.3) and second, a rotation transformation about the x axis, Fμ v = ⎟⎟⎟⎟⎟ ⎠⎞ ⎜⎜⎜⎜⎜ ⎝⎛ − ) cos( ) sin( 0 0) sin( ) cos( 0 00 0 1 00 0 0 1 r rr r = exp(-irJ 1) where (J 1)μ ν = ⎟⎟⎟⎟⎟ ⎠⎞ ⎜⎜⎜⎜⎜ ⎝⎛ − 0 0 00 0 00 0 0 00 0 0 0 ii . (5.14.4) The matrices K 1 and J1 are called generators and are a part of a set of six 4x4 matrices J i and Ki for i = 1,2,3. These 6 generator matrices satisfy a set of commutation relations known as a Lie Algebra, Chapter 5: The Metric Tensor 63 [ Ji, Jj] = +i εijkJk // [ A,B ] ≡ AB - BA [ Ji, Kj] = +i ε ijk Kk [ Ki, Kj] = -i εijkJk . ( 5 . 1 4 . 5 ) In these commutators, the generators J i and Ki can be regarded as abstract non-commuting operators, while the specific 4x4 matrices shown above for J 1 and K1 are just a "representation" of these abstract operators as 4x4 matrices. The six 4x4 generator matrices J i and Ki are (g = G = diag(1,-1,-1,-1) ) (J μν)α β = i ( gμαδν β – gναδμ β) (J1)α β ≡ (J23)α β = i ( g2αδ3 β – g3αδ2 β) and cyclic 123 (K 1)α β ≡ (J01)α β = i ( g0αδ1 β – g1αδ0 β) and cyclic 123 (5.14.6) where (-i)(J μν)αβ = ( gμαgνβ – gναgμβ) is a rank-4 tensor, antisymmetric under μ↔ν and α ↔ β. An arbitrary Lorentz transformation can be represented as Fμ v(r,b) = [ exp {– i ( r•J + b•K)} ]μ ν where the 6 numbers r and b are called parameters (rotation and boost) and this F is a combined boost/rotation transformation (note that eA+B ≠ eAeB for non-commuting matrices A,B). The product of two such Lorentz transformations is also a Lorentz transformation, and in fact the transformations form a continuous group known as the Lorentz Group, which then has 6 parameters. The first two commutators show n above (all J and all K) are each associated with a 3 parameter continuous group called the rotation group. The abstract generators of this group can be "represented" as matrices of any dimension, and are labeled by a numb er j such that 2j+1 is the matrix dimension. For example, the 2x2 matrix representation of the rotation group is labeled by j = 1/2, and is called the spinor representation and is associated in physics with the "intrinsic spin" of particles of spin 1/2 such as electrons. The vectors (spinors) in this case have tw o elements, and (1,0) and (0,1) are "up" and "down". Representations of the Lorentz group have labels {j 1, j2}, where j 1 is for the J-generated rotation subgroup, and j 2 for the K-generated rotation subgroup, and are usually denoted j 1⊗j2. Such a representation then has vectors containing (2j 1+1)(2j2+1) elements. In the case 1/2 ⊗1/2 there are 2*2=4 elements in a vector, and when these elements are linearly combined in a certain manner, they form the 4- vector object which one writes as Aμ such as xμ. This is the "vector representation" of the Lorentz group upon which is built the entire edifice of special relativity tensor algebra. Two other basic representations of the Lorentz group are these: 1/2 ⊗ 0 and 1/2 ⊗ 0. These are 2x2 matrix representations and they are different 2x2 representations. For each representation one can construct a whole tensor analysis world based on 2- vectors. Just as with the 4-vectors, one has contravariant and covariant 2-vectors. The two representations 1/2 ⊗ 0 and 1/2 ⊗ 0 are called spinor representations since they are each 2-dimensional. Since there are two distinct spinor representations, one needs some way of distinguishing them from each other. One representation might be called "undotted" and the other "dotted" and then there are four 2-vector types to worry about, which transform this way, V'a = Ra bVb V'a• = Ra• b•Vb• contravariant 2-vectors V'a = RabVb V' a• = Ra•b•Vb• covariant 2-vectors (5.14.7) where now dots on the indices indicate which Lorentz group representation that index belongs to. The 2x2 matrices Ra b and Ra• b• are not the same. A typical rank-2 tensor would transform this way, Chapter 5: The Metric Tensor 64 X' ab• = Ra a'Rb• b•' Xa'b•' . ( 5 . 1 4 . 8 ) This then is the subject of what is sometimes called Spinor Algebra as opposed to Tensor Algebra, but it is really just regular tensor algebra with respect to the two spinor representations of the Lorentz group. We have inserted this blatant digression just to show that the general subject of tensor analysis includes all this spinor stu ff under its general umbrella. In closing, Maple shows that the metric tensor G is indeed preserved unde r boosts and rotations. In Maple, evalm(Bx &* G &* transpose(Bx)) means B x G BxT and Maple is just verifying that B x G BxT = G and similarly R x G RxT = G : Chapter 5: The Metric Tensor 65 ( 5 . 1 4 . 9 ) 5.15 General Relativity and its Metric Tensor In general relativity a Picture of interest is Picture C x(0) → ξ, g(0) → η (5.15.1) but the x(0)-space is replaced by a Quasi-Cartesia n [see (1.10)] space with coordinates ξi with the metric tensor of special relativity which is now called η. This ξ-space represents a "freely-falling" coordinate system in which the laws of special relativity apply and the metric tensor is taken to be G = G ¯ = diag(-1,1,1,1) ≡ η. The xi are the coordinates of some other coordinate system. There is some transformation x = F(ξ) which defines the relationship between these two systems. The covariant metric tensor in x-space is written g ¯ = STGS = STηS. This is (5.7.6) for Picture C with g(0) = G = η, "quasi-Cartesian". Using the Standard Notation introduced in Chapte r 7 below, this is usually written as g ¯ = STηS // translation of (5.7.6) to our quasi-Cartesian Picture C g dn = ST ηdnS // developmental notation, g dn has "down" indices g μν = (ST)μ α ηαβ Sβ ν = Sα μ ηαβ Sβ ν // Standard Notation as in Chapter 7 Chapter 5: The Metric Tensor 66 gμν = (∂ξα/∂xμ) ηαβ (∂ξβ/∂xν) // e.g., Sα μ = (∂ξα/∂xμ) gμν(x) = (∂ξα/∂xμ) (∂ξβ/∂xν) ηαβ // Weinberg p 71 (3.2.7) . (5.15.2) The last line then defines the gravitational metric tensor in x-space based on the transformation ξ = F-1(x) = ξ(x). ( This and the following references are from the book of Weinberg, see References.) Newton's Second Law m a = f appears this way in general relativity, m (∂2xμ/∂τ2) = fμ - m Γμ νλ (∂xν/∂τ) (∂xλ/∂τ) // p 123 (5.1.11 following) (5.15.3) where f μ is an externally applied force, but there is th en an extra bilinear velocity-dependent term which represents an effective gravitational force (it acts on mass m) arising from spacetime itself. The object Γ μ νλ is called the affine connection and is rela ted to the metric tensor in this way. Γμ νλ = (1/2) gμσ( ∂νgλσ + ∂λgνσ – ∂σgνλ ) . // p 75 (3.3.7) (5.15.4) These very brief comments are only meant to convince th e reader that the equations of general relativity also have their place under the general umbrella of tensor analysis as discussed in this document. The fact that Γ μ νλ is not a mixed rank-3 tensor is demonstrated for example in (F.6.3). 5.16 Continuum Mechanics and its Metric Tensors One can describe (Lai) the forward "flow" of a continuous blob of matter by x = x(X,t) where X = x(X,t0). A "particle" of matter (imagine a tiny cube) that starts at location X at time t 0 ends up at x at time t. Two points in the flow separated by d X at t0 end up separated by some d x at t. The relation between these separation vectors is given by d x = F d X where F is called the deformation gradient . F describes how a particle starting say with a cubic shape at t 0 gets deformed into some parallelepiped (3-piped) shape at t as shown in Fig (5.16.7) below. If we examine d x = F d X we find that | d x | ≠ | dX | since the vector d X typically gets rotated and stretched as d X → dx during the flow. [ This flow is further described in Appendix K. ] The finite-duration flow x = x(X,t) from time t 0 to time t can be thought of as a (generally non-linear) transformation of the form x = F(X,t), where we can regard t as an a dded parameter. Recall from Chapter 2 that a general transformation was annotated as x' = F(x) or x = F-1(x') and the linearized transformation was d x' = R d x or dx = S d x', as in (2.1.6). To be compatible w ith Lai notation which uses symbol F for the deformation gradient, and to make things clearer , we shall replace our Chapter 2 transformation name F by the name F-1, and we shall identify x'-sp ace with X-space. We then have this modified version of Picture A, Chapter 5: The Metric Tensor 67 (5.16.1) In this flow we assume Cartesian coordinates in both x-space and X-space, so the metric tensors which determine physical distance in these spaces are both 1. Before continuing, it is helpful to have an example of a fluid flow situation. Example: A 2D Fluid Flow based on the Polar Coordinates Transformation We first quote Fig (3.4.3) with a few enhancements ( F ≡ F-1) : (3.4.3) (5.16.2) The tiny white square on the left maps into a tiny white 2-piped shown on the right under this transformation. The edges of this 2-piped in x-space (on the right) are dr and rd θ, there is no confusion about that, and the area is dA = rdrd θ. The question at hand is how one should interpret the edges of the tiny white square on the left, and what is the x'-space metric tensor g ¯' on the left? It is shown in Section C.5 that there are two distinct interpretations and each has its own use. We call the two interpretations the Cartesian View and the Curvilinear View. The view we are familiar with is the Curvilinear View where g ¯' = ⎝⎛ ⎠⎞ r2 0 0 1 . In this view, the top edge of the white square on the left has length ds' = rd θ and this matches the corresponding edge of the 2- piped on the right, which is ds = rdθ . This is the view in which hypothe sis (5.2.6) is valid (that ds' = ds), so all our tensor machinery is applicab le, such as the f act (5.7.9) that g ¯' = STS. One might call this the covariant view, since distance ds is a scalar under the transformation. Here is a bit more detail. For the upper edge of the white square on the left, we can set d x' = dθ e'θ= (dθ r) e^'θ. This maps into the northeast edge of the 2-piped on the right which is then d x = (rdθ) e^θ. One then verifies that Chapter 5: The Metric Tensor 68 (ds')2 = dx' • dx' = g'θθ dθ dθ = r2 dθ dθ = (rdθ)2 // Curvilinear View (ds)2 = dx • dx = (rdθ ) e^θ • (rdθ) e^θ = (rdθ)2 ( 5 . 1 6 . 3 ) In the Cartesian View , we instead use g ¯' = ⎝⎛ ⎠⎞ 1 0 0 1 on the left. In this case, the edges of the white square are dr and dθ , just as if these were regular Cartesian coordinates. As explained in Appendix C, one advantage of the Cartesian View is that, for a general coordinate system, it is the only view that can be drawn on a piece of paper (2D) allowing the reader to have some comprehension of what is going on (and this is even more true in 3D). A disadva ntage of the Cartesian View is that ds ≠ ds' so it is "non- covariant". In the Cartesian View we find, for our same top edge vector, (ds')2 = dx' • dx' = 1 dθ dθ = (dθ)2 // Cartesian View (ds)2 = dx • dx = (rdθ ) e^θ • (rdθ) e^θ = (rdθ)2 ( 5 . 1 6 . 4 ) and so ds ≠ ds' which says | d x | ≠ |dx'|. To arrive at our Example model for a flui d flow, we redraw Fig (5.16.2) renaming x' = (θ,r) to be X = (X,Y), and make corresponding changes elsewhere in the figure: ( 5 . 1 6 . 5 ) The tiny white square on the left is a blob of (2D) fl uid at time t = 0 and after flowing a while it ends up in the location of the white 2-piped on the right at time t = 1. This is just a simple 2D flow example making use of a transformation we are already familiar with (see Comment below). For a general 3D forward flow transformation x = F(X,t) one finds that the differential white cubic blob at t = 0 ends up in a 3-piped at t = t which is both rota ted and stretched relative to the star ting cube. Due to this stretch, one finds in general that |d x| ≠ |dX| where these are vector lengths relative to the Cartesian View metric tensor, which is in fact the metric tensor that determines physical distance in both X-space and x-space. Thus, the g ¯' = ⎝⎛ ⎠⎞ 1 0 0 1 metric tensor for X space has this we ll-defined meaning -- it relates to actual physical distance in X-space. However, it is the curvilinear view metric tensor g ¯' = ⎝⎛ ⎠⎞ r2 0 0 1 = STS whose Chapter 5: The Metric Tensor 69 matrix elements determine such th ings as how much edges stretch in the flow and how much areas and volumes change during the flow, which we shall examine below. Comment: The Example presented above really makes no fl ow sense since there is no time parameter in the flow transformation! The purpose of the Example is merely to illustrate the idea of two metric tensors in X-space, and to do so with a transformation that is familiar from our "curvilinear coordinates" study, and to tie in the notions of the two Views fro m Appendix C. The Example could be made more reasonable using a contrived transformation like this, general t t = 0 t = 1 x = F(X,t): x = (1-t)X + t Y cos(X) x = X x = Ycos(X) y = (1-t)Y + t Y sin(X) y = Y y = Ysin(X) . (5.16.6) In this case, at t = 0 we have the correct x = F(X,t=0) = X and at the end of the flow (t = 1) we have the transformation equations shown in Fig (5.16.5). Having worked through this contrived example, we can now draw a figure showing the general mapping of a differential blob during a flow from time t = 0 to time t = t. This figure is taken from Chapter 8 where the X-space left side is called x'-space as in (15.6.1) and the initial "blob" is shown drawn in the Cartesian View: (this picture happens to use the Chapter 7 Stan dard Notation for coordinates, contravariant = upper index) Flow X-space (= x'-space) t = 0 Flow x-space t = t (8.2.2) (5.16.7) We might be interested in knowing how much the dx' 3 edge is stretched going fr om time t=0 to time t=t, and by how much the volume changes, and so on. A ll these geometric questions are posed and answered in Chapter 8, and we shall use some of those results below. Chapter 5: The Metric Tensor 70 Here then is a translation table comparing the contin uum mechanics notation of Lai to the tensor notation used earlier in our document (and in the above Example). continuum mechanics our document ( Forward Flow X → x ) X, x ↔ x', x x = x(X,t) = F(X,t) ↔ x = F(x',t) = F-1(x',t) // Lai p 70 (3.1.4) d x = F d X ↔ d x = S d x' // as in (2.1.6) // Lai p 86 (3.7.6) or p 105 (3.18.3) F ↔ S F-1 ↔ R // R = S-1 x = Cartesian ↔ g ¯ = 1 X = Cartesian ↔ g¯' = 1 // Cartesian View C = FTF ↔ g ¯' = STS // Curvilinear View, as in (5.7.9) // Lai p 114 (3.23.2) C-1 = F-1(F-1)T ↔ g' = RRT // since g' = g ¯'-1 and S-1 = R , as in (5.7.9) ( 5 . 1 6 . 8 ) Thus, the deformation gradient F is just th e S matrix associated with transformation F = F-1. The Curvilinear View metric tensor g ¯' = STS appears in Lai as C = FTF which is known as the right Cauchy- Green deformation tensor (manifestly symmetr ic, so a viable metric tensor). [The left Cauchy-Green deformation tensor is B = FFT ]. Given the above flow situation, it is then possible to add two more transformations F1 and F2 which take X-space and x-space to independent sets of curvilinear coordinates X' and x': ( a new x' here) (5.16.9) and we then have an interesting trip le application of the notions of Ch apter 2 to a real-world situation. This drawing is the implicit subject of Section 3.29 (p131-138) of Lai. The flow transformation of interest here is x' = F2(x) = F2(F(X)) = F2(F(F1-1(X'))) ≡ G(X') . (5.16.10) In (reverse) dyadic notation the de formation gradient is written F = ( ∇x) where ∇ means ∇(X)so that d x = F d X = (∇x) dX F ij = (∇x)ij = ∂j(X)xi = ∂xi/∂Xj . (5.16.11) Chapter 5: The Metric Tensor 71 The (∇x) notation is explained in (E.4.4), and in Appendix G the object ( ∇v) for an arbitrary vector field v(x) is expressed in general curvilin ear coordinates. Consider again Fig (5.16.7) displayed above: Flow X-space (= x'-space) t = 0 Flow x-space t = t (8.2.2) (5.16.7) We can identify the mapping shown in this picture with our Flow situation of table (5.16.8). Chapter 8 discusses in much detail how length, volume and area transform under a general transformation. The length, area and volume magnitudes on the left are called d L'n = dx'(n), dA'n and d V', while the corresponding quantities on the right are called d x(n), dA¯(n) and dV, where the first two items are vectors. Cribbing the results of (8.4.g.2) and conver ting them from standard notation to developmental notation, we have | d x(n)|/ dL'n = h'n = [g¯'nn]1/2 = the scale factor for edge d x(n) | d A¯(n)|/ dA'n = (1/h'n) |J| = (1/h' n) g'1/2 = [g'nn g']1/2 = [cof(g ¯'nn)]1/2 |dV| / d V' = |J| = g'1/2 // where g' ≡ det(g¯'ij) = J2 , g¯' = STS (8.4.g.2) (5.16.12) We can then translate these thr ee lines into our Flow context: | dx(n)| / | d X(n)| = h'n = [g¯'nn]1/2 = [(FTF)nn]1/2 = [Cnn]1/2 // Lai p 114 (3.23.6-8) dAn| / |dA0n| = [g'nng']1/2 = [cof(g ¯'nn)]1/2 = [cof(FTF)nn)]1/2 = [cof C nn)]1/2 // Lai p 129 (3.27.11)* |dV| / |dV 0| = |J| = g'1/2= [det(g ¯'ij)]1/2 = [det(FTF)]1/2 = |det(F)| // Lai p 130 (3.28.3) (5.16.13) w h e r e Chapter 5: The Metric Tensor 72 edge area volume X-space : d X(n) dA0n dV0 time t 0 x-space : d x(n) dAn dV time t (5.16.14) Thus, for example, the volume change of a "flowing" particle of continuous matter is given by the Jacobian |J| = |detF| associated with the deformation gradient tensor F. We put quotes on "flowing" only because this might be a particle of solid steel th at is momentarily moving and deforming a very small amount during an oscillation or in response to an applied stress. In the middle line of (5.16.13) we state that | d An| / | d A0n| = [cof(FTF)nn)]1/2 and quote Lai p 129 (3.27.11) for verification. However, what Lai (3.27.11) actually says (slightly translated to our notation) is this: dA(n)/dA(n) 0 = det(F) | (F-1)T un | un = unit base vector, ( un)i = δn,i (5.16.15) which seems a far cry from our result [cof(FTF)nn)]1/2. But consider, using F-1 = R from table (5.16.8), | (F -1)T un |2 = | RT un |2 = [RTun]i[RTun]i = Rni Rni = (RRT)nn = g'nn (5.16.16) so det(F) | (F -1)T un | = g'1/2 g'nn1/2 = [cof(g ¯'nn)]1/2 = [cof(FTF)nn)]1/2 . (5.16.17) Here det(F) = det(S) = g' 1/2 from (5.12.19) and we have used Theorem 1 (8.4.f.1) converted to developmental notation, g' g' nn = [cof(g ¯'nn)] . // more generally, (detA) A-1 = cof(A) if A = AT (5.16.18) It might be noted that the Lai book does in fact use our "developmental notation" in that all indices are written "down" (when indices are shown), but no ove rbars mark covariant objects. Here are a few examples: Lai notation Developmental notation Standard Notation d A0 = dX(1)x dX(2) (3.27.1) d A¯0 = dX(1)x dX(2) (dA0)i= εijk [dX(1)]j [dX(1)]k [divT] i = ∂jTij (4.7.3) [divT] i = ∂¯jTij [divT]i = ∂jTij (5.16.19) Of course when Cartesian coordinates are assume d, the up and down position makes no difference. Lai writes tensors in bold face such as F for the deformation gradient noted above, or T for the stress tensor. Perhaps this is done to emphasize the notion of a tensor as an operator, as in our Section E.7. Lai writes a specific matrix as [ T], but a matrix element is T ij. Notation is an ongoing burden and each area of physics seems to have its own accepted conventions. Chapter 6: Reciprocal Base Vectors 73 6. Reciprocal Base Vectors E n and Inverse Reciprocal Base Vectors U' n 6.1 Definition of the E n This entire Chapter uses the Picture A context, (6.1.1) Although various definitions are possible, we shall define the reciprocal tangent vectors En in the following manner [ implied sum on i, and en = ∂'nx from (3.2.6)] En ≡ g'ni ei = g'ni ∂'ix ⇒ en = g¯'niEi . // since g ¯' = g'-1 (6.1.2) Comment : Notice how this differs in structure from th e rule for forming a covariant vector from a contravariant one, V¯n = g¯'niVi . ( 6 . 1 . 3 ) In the last equation, the right side is a linear combination of vector components Vi, while in the previous equation (6.1.2) the right side is a linear combination of vectors ei. In this case, i is a label on ei , whereas in the other case i is an index on V i. Labels and indices are different. Since the tangent base vectors ei are contravariant vectors in x-space by (3.3.2), and since En is a linear combination of the ei, the En are also contravariant vectors in x- space. Notice that in the definition En ≡ g'ni ei, these two x-space vectors are related by the metric tensor of the other space, x'-space. One can express the components of En in two ways, ( En)k ≡ g'ni (ei)k = Skig'ni // ( ei)k ≡ Ski by (3.2.5) = g' niSki = Rnigik // R nigik = g'ni(ST)ik = g'niSki by (5.7.8) Rg = g' ST (6.1.4) Taking i→c and then k →i (and g is symmetric), ( En)i = g'ncSic = gicRnc . // sum on second indices (6.1.5) Applying R to both sides of En ≡ g'ni ei in (6.1.2) gives En transformed into x'-space, E'n = g'nk e'k so ( E'n)i = g'nk (e'k)i = g'nkδk,i = g'ni . (6.1.6) Chapter 6: Reciprocal Base Vectors 74 Using the axis-aligned basis vectors un in x-space, (u n)m = δn,m, we also have this dot product, En • um = Rnm . (6.1.7) Proof : En • um = g¯ab(En)a(um)b = g¯ab (gacRnc)(δm,b) = (g¯magac) Rnc = δm,c Rnc = Rnm . 6.2 The e n and En Dot Products and Reciprocity (Duality) Three covariant dot products are of great interest . The first is this (see (5.7.6) for last step) en • em = g¯ij (en)i (em)j = g¯ij Sin Sjn = ST ni g¯ij Sjm = ( ST g¯ S)nm = g¯'nm . (6.2.1) The second is En • em = g¯ij (En)i (em)j = g¯ij gic Rnc Sjm = δj,c Rnc Sjm = Rnj Sjm = (RS)nm = δn,m (6.2.2) and the third is En • Em = g¯ij (En)i (Em)j = g¯ij gic Rnc gjb Rmb = δj,c Rnc gjb Rmb = Rnj gjb Rmb = R nj gjb RT bm = (R g RT)nm = g'nm . // last step is (5.7.6) (6.2.3) To summarize the last three results and (6.1.2), en • em = g¯'nm ⇒ | en| = g¯'nn = h'n (scale factor) En = g'ni ei En • em = δn,m en = g¯'ni Ei En • Em = g'nm ⇒ | En| = g'nn . (6.2.4) When the relation between two sets of vectors en and Em is as shown in (6.2.4), the vectors are said to be reciprocal or dual to each other. Since the en already have the name "tangent base vectors", we refer to the En as the "reciprocal base vectors". The notion of reciprocal or dual vector sets is studied in more detail in the notes just below. Notice that the two relations on the right in (6.2.4) are implied by the three dot product relations on the left. For example, since the en are a complete set, we can expand En with temporarily unknown coefficients α nk to get En = αnkek. Dotting both sides with Em then reveals the coefficients: g'nm = En • Em = (αnkek) • Em = αnk (ek • Em) = αnk δk,m = αnm and therefore we find En = αnkek = g'nkek = g'niei as shown top right in (6.2.4). Chapter 6: Reciprocal Base Vectors 75 Whereas en and Em exist in x-space, e'n = R en and E'm = R Em exist in x'-space -- these are just the transformed contravariant vectors using the rule (2.3.2), v' = R( x)v. Recall from (3.2.1) that the e'n are axis-aligned unit vectors in x'-space, ( e'n)i = δn,i. Using the fact (6.1.6) that ( E'n)i = g'nk (e'k)i we find that, E'n • e'm = g¯'ij (E'n)i(e'm)j = g¯'ij g'ni δm,j = g¯'im g'ni = δn,m . (6.2.5) This is consistent with (6.2.4) and (5.10.2) that A' • B' = A • B, E'n • e'm = En • em = δn,m . (6.2.6) The other two dot products above work this same way. We can then rewrite (6.2.4) for the primed vectors, e'n • e'm = g¯'nm ⇒ | e'n| = g¯'nn = h'n (scale factor) E'n = g'ni e'i E'n • e'm = δn,m e'n = g¯'ni E'i E'n • E'm = g'nm ⇒ | E'n| = g'nn . (6.2.7) As an exercise, we can ve rify the top left equation in this manner, knowing ( e'n)i = δn,i, e'n • e'm = g¯'ij (e'n)i (e'm)j = g¯'ij δn,i δm,j = g¯'nm The vectors en and En are examples of the general notion of dua l vector sets which we now describe. Notes on Reciprocity (Duality) 1. Suppose some set of vectors bn forms a complete basis for x-space. Can one find a set of vectors Bn that have the property Bm • bn = δm,n ? // duality relation; Ba and ba are reciprocal (6.2.8) As shown below, the answer is normally "yes", and the vectors Bn are uniquely determined by the bn. One says that the set { Bn} is "dual to" the set { bn} and vice versa. If we regard bn as a basis, then Bn is the "dual basis", and Bm • bn = δm,n is the "duality relation". Another terminology is that the vectors Bn are "reciprocal to" the vectors bn and vice versa. 2. It was shown in (6.2.2) that En • em = δn,m so the En and the en vectors are dual to each other. The en are the tangent base vectors, and the En are "reciprocal to" the en which is why we call them the reciprocal base vectors. 3. One can solve for the Bn in terms of the bn. Each Bn has N components, so there are N2 unknowns. The duality relation Bm • bn = δm,n is a set of N2 equations. This is basically a Cramer's Rule problem in Chapter 6: Reciprocal Base Vectors 76 N2 variables. Since the bn form a complete basis, one can expand Bm on the bn with some coefficients we will call w' mn ( at this point w' mn is unknown), Bm = w'mnbn . ( 6 . 2 . 9 ) Then from (6.2.8), δm,k = Bm • bk = w'mn bn • bk . // bn • bk = g¯ij(bn)i(bk)j . (6.2.10) Define matrix W' by, W' nk ≡ bn • bk ( 6 . 2 . 1 1 ) and note that W' nk is symmetric. Then (6.2.10) says δm,k = w'mnW'nk or w'W' = 1 or w' = W'-1 . (6.2.12) Assuming for the moment that detW' ≠ 0, the solution is given by w' = W'-1 . The (5.6.4) "Digression" showed that (A-1)T = (AT)-1 for invertible A, so w'T = (W'-1)T = (W'T)-1 = W'-1 = w' and therefore w' is symmetric as well. Since W' is known from (6.2.11), w' = W'-1 and the Bm = w'mnbn of (6.2.9) have been found. Finally, Bm • Bn = w'mibi • w'njbj = w'miw'nj bi • bj = w'miw'njW'ji = w'miδni = w'nm . (6.2.13) 4. In the case that bn = en and Bn = En, one finds that W' nk = en • ek = g¯'nk , the covariant metric tensor. Then w' = W'-1 must be the contravariant metric tensor w' mn = g'mn = Em • En. Then Bm = w'mnbn says En = g'mn en which agrees with our original definition (6.1.2) of the En. Moreover, det(W') = det( g ¯'ab) = g' of (5.12.14), so as long as g' ≠0, one has detW' ≠ 0. 5. For the general bn case, one can imagine that the bn are the tangent base vectors for some transformation x" = Fb(x) (with some linearized R b ≠ R where R goes with F), then the issue of detW' ≠ 0 boils down to g' ≠ 0 for that transformation. In the case that x-space is Cartesian, we know that J2 = det(g¯')/ det(g ¯) = g'/g = g'. If this transformation Fb is invertible, then J ≠ 0 everywhere, so g' ≠ 0 and then detW ≠ 0. 6. If the bm are true tensorial vectors, then the Bm = w'mnbn will be as well (linear combination) and then Bn • bm is a tensorial scalar. Therefore if Bn • bm = δn,m in x-space, then so also B'n • b'm = δn,m in x'- space, where bm' = R bm and B'n = RBn. For example, E'n • e'm = δn,m in x'-space where em' = R em and E'n = REn . 7. In Section 6.5 we shall encounter another dual pair Un • um = U'n • u'm = δn,m which is associated with the inverse transformation x = F-1(x'). Chapter 6: Reciprocal Base Vectors 77 8. One major significance of the equation Bn • bm = δn,m is that it allows the following expansions: V = Σn kn Bn where k m = V • bm V = Σn cn bn where c m = V • Bm ( 6 . 2 . 1 4 ) so that for example bm • V = bm • [Σn kn Bn] = Σn kn bm • Bn = Σn kn δm,n = km. These expansions are explored in Section 6.6 below for the two dual sets En, en and Un, un. 9. Saying that the basis bn is complete implies that any vector V can be expanded in this basis. Consider the second expansion shown in (6.2.14). Installing c n = V • Bn into the expansion one finds, V = Σn (V • Bn) bn so Vb = Σn [ΣaVa(B¯n)a] (bn)b // from (5.10.3) = Σ aVa { Σn( B¯n)a (bn)b } . The last equation can only be valid if Σn( B¯n)a (bn)b = δa,b // completeness relation for the dual set { bn, Bn } (6.2.15) and this is the formal statement that the basis bn is complete. We can verify this completeness claim for { en, En} as follows Σn (E¯n)a(en)b = Σn (g¯ En)a(en)b = Σnc g¯ac(En)c(en)b // from (5.8.1) = Σnc g¯ac [ Σk gckRnk] [Sbn] // from (6.1.5) and (3.2.6) = Σ nck SbnRnk gkc g¯ca = [SRgg ¯]ba = [ 1 1 ] ba = 1ba = δb,a . (6.2.16) 10. As a final step, we want to write the completeness relation (6.2.15) in matrix form. To do this, we first take a short digression. If b is a column vector, and if bT is the corresponding row vector version of b, then (bT)i = bi. If a and b are vectors of dimension N, and if it happens that ( a)i(b)j = cij , ( 6 . 2 . 1 7 ) then one can write ( a)i(bT)j = ci j . (6.2.18) Chapter 6: Reciprocal Base Vectors 78 This equation can be written in matrix form as, abT = c . ( 6 . 2 . 1 9 ) Graphically, here is an example for dimension N=2 which shows how abT= c agrees with ( a)i(b)j = cij : abT = ⎝⎛ ⎠⎞ a1 a2 (b1 b2) = ⎝⎛ ⎠⎞ a1b1 a1b2 a2b1 a2b2 = a matrix = ⎝⎛ ⎠⎞ c11 c12 c21 c22 . (6.2.20) This notion can be trivially generalized to apply to a sum of vectors. Let a[n} and b[n] each be a set of N vectors labeled by n. Then the following component equation Σn (a[n])i(b[n])j = cij ( 6 . 2 . 2 1 ) can be written in matrix form as Σ n a[n] ( b[n])T = c . (6.2.22) Now, since the completeness relation (6 .2.15) has the form of (6.2.21) Σn( B¯n)a (bn)b = δa,b // completeness relation for the dual set { bn, Bn } (6.2.15) it can be expressed in the following compact matrix form from (6.2.22), where c = 1 is the identity matrix, Σn B¯n bnT = 1 // completeness (matrix form) . (6.2.23) In the special case that bn = en this reads Σn E¯n enT = 1 // completeness (matrix form) . (6.2.24) 6.3 Covariant partners for e n and En A covariant partner vector is formed according to (5.8.4) which says V¯ = g¯ V or V¯i = g¯ijVj, so e¯n = g¯ en ⇒ ( e¯n)i = g¯ij(en)j = g¯ijSjn // (3.2.6) = [ g ¯S ]in = [RT g¯']in = Rjig¯'jn // (5.2.8) (6.3.1) E¯n = g¯ En ⇒ ( E¯n)i = g¯ij(En)j = g¯ijgjcRnc = δi,cRnc = Rni // (6.1.5) (6.3.2) The components of the four vectors are then ( en)i = Sin (3.2.6) ( En)i = gicRnc = g'ncSic (6.1.5) (e¯n)i = g¯ijSjn = Rjig¯'jn (6.3.1) ( E¯n)i = Rni (6.3.2) (6.3.3) Chapter 6: Reciprocal Base Vectors 79 In (3.2.7) we wrote S = [ e1, e2, e3 .... eN ] as a representation of the ( en)j = Sjn , showing that the tangent base vectors are the columns of matrix S. In the equation ( E¯n)i = Rni the index order is not reversed as it is in ( en)j = Sjn, so one concludes that the E¯n as are the rows of matrix R R = ⎣⎢⎡ ⎦⎥⎤E¯1 E¯2 E¯3 ... E¯N = [ E¯1, E¯2, E¯3 .... E¯N ]T ( 6 . 3 . 4 ) Now recall how vectors transform, V' = R V contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 V¯' = ST V¯ covariant S ik(x') ≡ (∂xi/∂x'k) = ST ki(x') (2.5.1) Here we regard both V and V¯ as existing in x-space, while V' and V¯' are the corresponding vectors in x'- space. As shown in the example (3.4.3), tangent base vectors en are vectors that one draws in x-space. They exist in x-space. The relation En ≡ g'ni ei. of (6.1.2) says the En vectors are just linear combinations of the en vectors, so the En vectors also exist in x-space and that is where one would draw them. Based on this fact and the previous paragraph, we conclude the perhaps obvious fact that the vectors en, e¯n, En,E¯n all exist in x-space. What are the corresponding vectors in x'-space that one gets from transformation x' = F(x) ? We just apply the transformation rules quoted above to find out: en' = R en ⇒ ( en')i = Rij (en)j = RijSjn = [RS]in = [1]in = δi,n (6.3.5) in agreement with (3.2.1). Next, En' = R En ⇒ ( En')i = Rij (En)j = RijgjcRnc = [RgRT]in = g'in // (5.7.6) (6.3.6) For the covariant vectors we find e¯n' = STe¯n ⇒ ( e¯n')i = Sji (e¯n)j = Sji g¯jkSkn = [STg¯S]in = g¯'in // (5.7.6) (6.3.7) E¯n' = STE¯n ⇒ ( E¯n')i = Sji (E¯n)j = Sji Rnj = [RS]ni = δn,i (6.3.8) The conclusions are ( en')i = δi,n ( En')i = g'in ( e¯n')i = g¯'in ( E¯n')i = δn,i ( 6 . 3 . 9 ) Chapter 6: Reciprocal Base Vectors 80 6.4 Summary of the basic facts about e n and En For use below, we package various results above into a single block. Here we use i,j for indices, and n.m for basis vector labels, ( e¯n)i = g¯ij (en)j (E¯n)i = g¯ij (En)j [ e¯n = g¯ en E¯n = g¯ En] (5.8.4) (e'n)i = Rij(en)j (E'n)i = Rij(En)j [ e'n = R en E'n = R En ] (e¯'n)i = Sji(e¯n)j (E¯'n)i = Sji(E¯n)j [ e¯'n = ST e¯n E¯'n = ST E¯n ] (2.5.1) ( en)i = Sin ( En)i = gijRnj = g'njSij (en')i = δi,n ( En')i = g'ni (e¯n)i = g¯ijSjn = Rjig¯'jn (E¯n)i = Rni ( e¯n')i = g¯'ni ( E¯n')i = δn,i (6.3.3) (6.3.9) en • em = g¯'nm ⇒ | en| = g¯'nn = h'n (scale factor) En = g'ni ei En • em = δn,m en = g¯'ni Ei En • Em = g'nm ⇒ | En| = g'nn . (6.2.4) e'n • e'm = g¯'nm ⇒ | e'n| = g¯'nn = h'n (scale factor) E'n = g'ni e'i E'n • e'm = δn,m e'n = g¯'ni E'i E'n • E'm = g'nm ⇒ | E'n| = g'nn . (6.2.7) ( E¯n)i(en)j = δi,j [ Σ n E¯n enT = 1 ] (6.2.16) and (6.2.24) (6.4.1) 6.5 Repeat the above for the inverse transformation: definition of the U 'n Section 3.5 introduced the inverse tangent base vectors called u'n . That is to say, the u'n are the tangent base vectors of the inverse transformation x = F-1(x'). Whereas the forward tangent base vectors en exist in x-space, the inverse tangent base vectors u'n exist in x'-space, so that is why we have put a prime on the u'n . Recall from (6.1.2) that En ≡ g'ni ei where En were the reciprocals of en . For the inverse transformation we define in exact analogy (g ↔ g') some vectors U'n reciprocal to the u'n , U'n ≡ gni u'i ⇒ u'n = g¯ni U'i . // since g ¯ = g-1 (6.5.1) This is just another application of the generic dualit y discussion of Section 6.2 with generic basis vectors called bn and Bn. Everything goes along as in the previous S ections, but with the following changes to account for the fact that we are now doing the inverse transformation x = F-1(x') : g'↔ g R ↔ S e n → u'n e'n → un En → U'n E'n → Un (6.5.2) Chapter 6: Reciprocal Base Vectors 81 and similarly for barred vectors. We can then manually translate the block of data presented in (6.4.1) with the above rules to get: ( u¯'n)i = g¯'ij (u'n)j (U¯'n)i = g¯'ij (U'n)j [ u¯'n = g¯' u'n U¯'n = g¯' U'n] (un)i = Sij(u'n)j (Un)i = Sij(Un')j [ un = S u'n Un = S Un' ] (u¯n)i = Rji(u¯'n)j (U¯n)i = Rji(U¯'n)j [ u¯n = RT u¯'n U¯n = RT U¯'n ] ( u'n)j = Rjn ( U'n)i = g'ijSnj = gnjRij (un)i = δi,n ( Un)i = gni (u¯'n)i = g¯'ijRjn = Sjig¯jn (U¯'n)i = Sni ( u¯n)i = gni ( U¯n)i = δn,i u'n • u'm = g¯nm ⇒ | u'n| = g¯nn = hn (scale factor) U'n = gni u'i U'n • u'm = δn,m u'n = g¯ni U'i U'n • U'm = gnm ⇒ | U'n| = gnn un • um = g¯nm ⇒ | un| = g¯nn = hn (scale factor) Un = gni ui Un • un = δn,m un = g¯ni Ui Un • Um = gnm ⇒ | Un| = gnn ( U¯'n)a(u'n)b = δa,b or Σn U¯'n u'nT = 1 (6.5.3) It is helpful to keep all these eight vector sym bol names in mind (and each has a covariant partner) x'-space x-space ( 6 . 5 . 4 ) axis-aligned basis vectors e'n un ( e'n)i = δn,i ( un)i = δn,i dual partners to the above E'n Un (E'n)i = g'ni ( Un)i = gni tangent base vectors u'n en ( u'n)i = Rin ( en)i = Sin reciprocal base vectors U'n En ( U'n)i = g'iaSna ( En)i = giaRna = g naRia = g' naSia and recall that Xn • xm = δn,m for each of the four dual pairs (two primed, two unprimed). 6.6 Expanding vectors on different sets of basis vectors x-space expansions on un and Un Assume that V is some generic N-tuple V = (V1,V2....VN). There are various ways to expand V onto basis vectors. One way is to expand on the axis-aligned basis vectors un, which recall live in x-space, V = V1 u1 + V2 u2 +... = ΣnVnun where Un • V = Vn . (6.6.1) The components V n are Un • V because Un • um = δn,m. From (5.8.4) one can write V n = gnmV¯m Chapter 6: Reciprocal Base Vectors 82 ( regarded here as a definition of the V ¯m) so one finds that V = ΣnVnun = Σn gnmV¯m un = ΣnV¯m gmn un = ΣnV¯m Um (6.6.2) and thus another expansion for V is this V = V¯1 U1 + V¯2 U2 +.. = ΣnV¯nUn where un • V = V¯n . (6.6.3) Comments: 1. If V is not a contravariant vector, one can still define V ¯n = g¯nmVm, but V¯n won't be a covariant vector. A familiar example is that x n is never a contravariant vector if F is non-linear, but we can still talk about the components x ¯n ≡ g¯nmxm . In Standard Notation, x n → xn and x¯n → xn and we do not hesitate to use these two objects even though they are not tensorial vectors. 2. If V is a contravariant vector, the expansion above V = ΣnVnun displays the contravariant components of V. The second expansion V = ΣnV¯m Um is still an expansion for contravariant vector V, but it displays the components of the covariant vector V¯ which is the "partner" to V by V¯n = g¯nmVm. It would be incorrect to write this second expansion as V¯ = ΣnV¯m Um since that would say g ¯V = ΣnV¯m Um which is just not true. We comment later on how this situ ation changes a bit in the Standard Notation. x-space expansions on en and En Another possibility is to expand V on the tangent basis vectors en, and we denote the components just momentarily as αn, V = α1 e1 + α2 e2 +... = Σn αn en . (6.6.4) Using en • Em = δn,m one finds that α n = En • V = ( En)k Vk = Rnk Vk = V'n // g ¯ = 1 so A•B = g¯abAaBb = AkBk (6.6.5) Therefore, the expansion is V = V'1e1 + V'2e2 +... = Σn V'n en where En • V = V'n . (6.6.6) If it happens that the N-tuple V = (V1,V2....VN) transforms as a contravariant vector, then V n are the contravariant components of that vector, and V' n are the contravariant components of V' in x'-space. On the other hand, if V is not a tensorial vector, so V n are not components of a contravariant vector, we can still define V' n ≡ Rnk Vk, but then the V' n are not the contravariant components of V'. Writing V' n = g'nmV¯'m the above expansion can be expressed as Chapter 6: Reciprocal Base Vectors 83 V = Σn V'n en = Σn,m g'nmV¯'m en = Σm V¯'m Σn g'mn en = Σm V¯'m Em (6.6.7) so V = V¯'1E1 + V¯'2E2 +... = Σ n V¯'n En where en • V = V¯'n (6.6.8) Summary of x-space expansions: V = V1 u1 + V2 u2 +... = ΣnVn un where Un • V = Vn Un = gni ui V = V¯1 U1 + V¯2 U2 +... = ΣnV¯n Un where un • V = V¯n V = V'1 e1 + V'2 e2 +... = Σn V'n en where En • V = V'n En = g'ni ei V = V¯'1 E1 + V¯'2 E2 +... = Σn V¯'n En where en • V = V¯'n (6.6.9) Expanding on unit vectors. The covariant lengths of the different basis vectors are given by |en| = |e'n| = g'¯nn | un| = |u'n| = g¯nn |En| = |E'n| = g'nn | Un| = |U'n| = gnn . (6.6.10) Using these lengths, one can define uni t vector versions of all the basis vectors and then rewrite the above expansions as expansions on the unit vectors w ith lower case coefficients. For example, using e^n ≡ en/g'¯nn ( 6 . 6 . 1 1 ) the third expansion above becomes (script font for unit-vector components) V = V'1e^1 + V'2e^2 +... = Σn V'n e^n where g'¯nn En • V = V'n = g '¯nn V'n . (6.6.12) An example of a case where this last expansion would be useful is in the use of spherical curvilinear coordinates, where for example e^1 = r^. The N-tuple ( V'1, V'2 ... V'N), although related to contravariant vector V (V'n = Rnk Vk), is not itself a contravariant vector since it does not obey the rule V'n = Rnk Vk . In fact V'n = Rnk Vk ⇒ (1/ g'¯nn ) V'n = Rnk (1/ g¯kk ) Vk ⇒ V'n = Rnk (g'¯nn /g¯kk )Vk ( 6 . 6 . 1 3 ) x'-space expansions Having done x-space expansions, we turn now to x'-s pace expansions. These can be obtained from the x- space expansions by this set of rules, g'↔ g R ↔ S u'n → en un → e'n U'n → En Un → E 'n V'n ↔ Vn V¯'n ↔ V¯n ( 6 . 6 . 1 4 ) and here then are the x'-space expansions: Chapter 6: Reciprocal Base Vectors 84 V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where E'n • V' = V'n E'n = g'ni e'i V ' = V¯'1 E'1 + V¯'2 E'2 +... = ΣnV¯'n E'n where e'n • V' = V¯'n V' = V1 u'1 + V2 u'2 +... = Σn Vn u'n where U'n • V' = Vn U'n = gni u'i V' = V¯1 U'1 + V¯2 U'2 +... = Σn V¯n U'n where u'n • V' = V¯n (6.6.15) 6.7 Another way to write the E n The reciprocal base vectors were defined above as linear combinations of the tangent base vectors, all in the general Picture A context, Ek ≡ g'ki ei . ( 6 . 7 . 1 ) It is rather remarkable that there is another way to write Ek in terms of the ei that looks completely different. In this other way, it turns out that Ek is expressed in terms of all the ei except ek and is given by (only valid in Picture B where g=1) Ek = det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing k = 1,2,3...N (6.7.2) This is a generalized cross product (see Section A.4) of N-1 vectors, since ek is missing, so there are N-2 "crosses". The above multi-cross-pr oduct equation is a shorthand for ( Ek)α ≡ det(R) (-1)k-1εαabc...x (e1)a(e2)b ...... ( eN)x // κ and ( eκ)K are missing (6.7.3) Here ε is the totally antisymmetric tensor with N indices. If κ is the kth letter of the alphabet (k = 2 ⇒ κ = b ), then κ is missing from the list of summation indices of ε, and the factor ( ek)κ is missing from the product of factors, so there are then N-1 factors. This cross product expression for En is derived in Appendix A . This is all fairly obscure sounding, but can be br ought down to earth by writing things out for N = 3, where the formula reduces to this cyclic set of equations, E1 = det(R) e2 x e3 E2 = det(R) e3 x e1 E3 = det(R) e1 x e2 . ( 6 . 7 . 4 ) These equations can be verified in a simple manner. To show an equation is true, if suffices to show that the projections of both sides on the three en are the same, since the en form a complete basis as noted earlier. For the first equation one needs then to show that E1 • en = det(R) e2 x e3 • en for n = 1,2,3 . (6.7.5) If n=2 or n=3, both sides vanish, according to en • Em = δn,m on the left, and according to geometry on the right, which leaves just the case n = 1. In this case the LHS = 1, so one has to show that e2 x e3 • e1 = 1/det(R) = det(S). But Chapter 6: Reciprocal Base Vectors 85 e1 • e2 x e3 = (e1)i (e2 x e3)i = (e1)i εijk (e2)j(e3)k = εijk (e1)i(e2)j(e3)k = εijk Si1Sj2Sk3 = det(S) QED // (3.2.5) says ( en)i = Sin (6.7.6) The other two equations of the set can be verified in the same manner. Here is a picture (N=3) drawn in x-space for a no n-orthogonal coordinate sy stem. The vectors shown here form a distorted right handed coordi nate system which has det(R) > 0. (6.7.7) The reader is invited to exercise his or her right hand to confirm the directions of the arrows, E1 = det(R) e2 x e3 E2 = det(R) e3 x e1 E3 = det(R) e1 x e2 . (6.7.8) 6.8 Comparison of e ¯n and En Recall from (6.3.1) that e¯n is the covariant partner of en where e¯n = g¯ en . Recall from (6.1.2) that En is the reciprocal of en where En ≡ g'ni ei . The comparison of these two defini ng equations is interesting, ( e¯n)i ≡ g¯ik(en)k // matrix acts on vector index (6.8.1) ( En)i ≡ g'nk (ek)i . // matrix acts on ek label (6.8.2) If x-space is Cartesian, then e¯n = en as usual, but of course En ≠ en in this case since normally g' ≠ 1. We mention this to head off a possible confusion when the Standard Notation is introduced in the next Chapter and the above two equations become ( see (7.4.1) and (7.13.1) ), ( en)i ≡ gik(en)k // Standard Notation, g lowers an index (6.8.3) ( en)i ≡ g'nk (ek)i . // Standard Notation, k is a label on ek, not an index (6.8.4) Chapter 6: Reciprocal Base Vectors 86 The mapping to standard notation does not include e¯n → en, for example. One fact about the standard notation is that, unlike the developmental notation, one cannot look at a vector in bold like en and determine whether it is contravariant or covariant. On ly when the index is disp layed can one tell. One can think of en as representing both its contravarian t self and its covariant partner ( en is a tensorial vector). 6.9 Handedness of coordinate systems: the e n , the sign of det(S), and Parity 1. Handedness of a Coordinate System. Let bn be a complete set of basis vectors in an N dimensional vector space, where the bn are not necessarily of unit length, and are not necessarily orthogonal. Consider this quantity B ≡ det ( b1, b2.....bn ) = εabc..x (b1)a (b2)b.... (bN)x = b1• [b2 x b3......x bN] (6.9.1) where the generalized cross product is discussed in Section A.4. This basis bn defines a "coordinate system" in that we can expand a position vector as follows x = Σn x(b) n bn ( 6 . 9 . 2 ) where the x (b) n are the "coordinates" of point x in this coordinate system. We make the following definition: system bn is a "right handed coordinate system" iff B > 0 system bn is a "left handed coordinate system" iff B < 0 . (6.9.3) One of course wants to show that for N = 3 this definition corresponds to one's intuition about right and left handed systems. For N= 3 , B ≡ det ( b1, b2, b3 ) = εabc (b1)a (b2)b(b3)c = b1• [b2 x b3] (6.9.4) Suppose the bn are arranged as shown in this picture, where the visual intention is that the corner nearest the label b1 is closest to the viewer. (6.9.5) With one's high-school-trained right hand, one can see that b2 x b3 points in the general direction of b1 (certainly b2 x b3 lies somewhere in the half space of the b2, b3 face plane which contains b1), and so the Chapter 6: Reciprocal Base Vectors 87 quantity B = b1• [b2 x b3] > 0. So this is an example of a right-handed coordinate system. The figure shown is a skewed 3-piped which can be regarded as a distortion of an orthogonal 3-piped for which the bn would span an orthogonal coordinate system in which one would have b^1 = b^2 x b^3. 2. The x'-space e'n coordinate system is always right handed. In our standard picture of x-space and x'- space, the coordinate system in x'-space is spanned by a set of basis vectors e'n where ( e'n)i = δn,i , as discussed in Chapter 3 (a). This system is "right handed" because B = εabc..x (e'1)a (e'2)b.... (e'N) = εabc..x δ1aδ2b....δxN = ε123...N = +1 > 0 (6.9.6) where we use the standard normalization of the ε tensor as shown. Notice that this conclusion is independent of the metric tensor g' in x-space. 3. The x-space un coordinate system is always right handed. The basis un where ( un)i = δn,i in x-space is right handed for the same reason as shown in the abov e paragraph, and for any g. When g=1 in x-space, the un form the usual Cartesian right-handed orthonormal basis in x-space. See Section 3.4 concerning the meaning of "unit vector". 4. The x-space en coordinate system handedness is determined by the sign of det(S). Our x-space coordinate system of great interest is that spanned by the en basis vectors, where ( en)i = Sin as in (3.2.5). One has B = εabc..x (e1)a (e2)b.... (eN) = εabc..x Sa1 Sb2.... SxN = det(S) . (6.9.7) Therefore, using σ ≡ sign(detS) and the Jacobian J = det(S) of (5.12.6), one has system en is a "right handed coordinate system" iff det(S) = J > 0 or σ = +1 system en is a "left handed coordinate sy stem" iff det(S) = J < 0 or σ = –1 . (6.9.8) 5. The Parity Transformation. The identity transformation F = 1 results in matrix S I = I with detS I = +1. In this case the en form a right-handed coordi nate system, and in fact en = un. The parity transformation F = -1, on the other hand, results in S P = -I with det(S P) = (-1)N and en = -un. When N is odd, the parity transformation converts the right-handed un system to a left-handed en system. If S is some matrix which does not change handedness, meaning detS > 0, then S' = SS P does change handedness for odd N, since in this case detS' = detS det S P = (-1)N detS. So given some S' that changes handedness for N=odd, one can regard it as "containing the parity transformation" which, if removed, would result in no handedness change. When N is even, handedness stays the same under F = -1. 6. N=3 Parity Inversion Example. Since x' = -x under the parity transform F = -1, parity is a reflection of all position vectors through the origin. Objects sitting in x-space, such as N-pipeds, whether or not the origin lies inside the object, are "turned inside out" by the parity transformation, but the inside of the object still maps to the inside of the parity-t ransformed object under this transformation. Chapter 6: Reciprocal Base Vectors 88 Consider this crude picture which shows on th e left a right-handed 3-piped in x-space where e1 and e2 happen to be perpendicular, and the ba ck part of the 3-piped is not drawn. This 3-piped is associated with some transformation S [ ( en)i = Sin ] with detS > 0. (6.9.9) Now consider S' = SS P = SPS = -S. For this S', the 3-piped appear s as shown on the right. The two pipeds here are related by a parity transformation, all po ints inverting through the origin. On the right, the volume of the 3-piped lies toward the viewer from the plane shown. The circled dot on the left represents the out-facing normal vector of the 3-piped face area wh ich is facing the viewer, and this normal is in the direction – e1xe2. This is called a "near face" in Appe ndix B since it touches the tails of the en. After the parity transformation, this same face has become the back face on the inverted 3-piped shown on the right, with out-facing normal indicated by the X. The direction of this normal is + e1xe2 . In general, an out-facing "near face" area points in the - En direction, and Appendix A shows that E3 = e1 x e2 / det(S). On the left we have E3 = e1 x e2 / |det(S)| so the face there just mentioned points in the - E3 = – e1xe2 direction. On the right we have E3 = e1 x e2 / det(S') = - e1 x e2 /|det(S)|, so the face there points in the -E3 = +e1xe2 direction. 7. The sign of det(S) in the curvilinear coordinates application. For a given ordering of the x' coordinates, det(S) will have a certain sign. By changing the x' i ordering to any odd permutation of the original ordering (for example, swap two coordinates), det(S) will negate because two columns of S ik(x') ≡ (∂xi/∂x'k) will be swapped. In the curvilinear coordinates application it is therefore always possible to select the ordering of the x' i coordinates to cause det(S) to be positive. One always starts with a right- handed Cartesian system n^ for x-space, and det(S)>0 then guarantees that the en will form a right-handed system there as well. Since the underlying transfor mation F is assumed invertible, one cannot have det(S)=0 anywhere in the domain x (or range x') of x' = F(x), and therefore det(S) cannot change sign anywhere in the space of interest. For graphical reasons, we have selected coor dinates in the "wrong order" in both the polar coordinates examples (called Example 1) and in the e lliptic polar system studied in Appendix C, which is why detS < 0 for both these systems. Chapter 7: Standard Notation 89 7. Translation to the Standard Notation In this Chapter we discuss the "translation" from our developmental notation (all lower indices; overbars for covariant objects) to the Standard Notation used in tensor analysis. The developmental notation has served well in the di scussion of scalars and vectors, tensors of rank-0 and rank-1. For pure (unmixed) tensors of rank-2 it does especially well, allowing the use of matrix algebra to leverage the use of familiar matrix theo rems such as det(ABC) = det(A)det(B)det(C) and A -1 = cof(AT)/det(A). The transformation of the contravariant me tric tensor is cleanly expressed as g' = R g RT, and so on. The notation in fact works fine for unmixe d tensors of any rank, but runs into big trouble with "mixed" tensors as shown in the next Sections. We continue to use Picture A: 7.1 Outer Products It is possible to form larger tensors from smaller ones using the "outer product" method. For example, consider, Tab ≡ UaVb ( 7 . 1 . 1 ) where U and V are assumed to be contravariant vectors. One then has T 'ab = U'aV'b = (Raa'Ua') (Rbb'Vb') = Raa' Rbb' Ua'Vb' = Raa' Rbb' Ta'b' (7.1.2) so in this way a contravariant rank-2 tensor like M in (5.6.3) has been successf ully constructed from two contravariant vectors. Similarly, T¯ab ≡ U¯aV¯b ⇒ T ¯'ab = ST aa' ST bb' T¯a'b' (7.1.3) so the outer product of two covariant vectors transforms as a covariant rank-2 tensor, again as in (5.6.3). More generally, one can combine any number of cont ravariant tensors in an outer product fashion to obtain a new tensor. Here are a few examples: T abc = UaAab T abcd ≡ AabBcd Tabcde ≡ AabBcdUe ( 7 . 1 . 4 ) Chapter 7: Standard Notation 90 7.2 Mixed Tensors and Notation Issues Suppose we take the "outer product" of a contravariant vector with a covariant vector, [ ... ] ab ≡ UaV¯b ( 7 . 2 . 1 ) where we are not sure what to call this thing, so we ju st call it [...]. Here is how this new object transforms (always: with respect to the underlying transformation x' = F(x) ) [ ... ]' ab = U'aV¯'b = (Raa'Ua') (ST bb'V¯b') = Raa' ST bb' Ua'V¯b' = Raa' ST bb' [...]ab . (7.2.2) This object transforms as a contravariant vector on the first index (ignoring the second), and as a covariant vector on the second index (ignoring the first). This is an example of a "mixed" rank-2 tensor. Extending this outer product idea, one can make ela borate tensor objects with an arbitrary mixture of "contravariant indices" and "covariant indices". For example [.....] abcd = UaV¯b XcY¯d . ( 7 . 2 . 3 ) To write down the transformation rule for such an object, one must know which indices are contravariant and which are covariant. It is totally clear how the obj ect transforms, looking at the right side of (7.2.3), but somehow this information has to be embedded in the notation [.....] abcd because once this object is defined, the right hand side might not be immedi ately available for inspection. Worse, there may be no right hand side for a mixed tensor, because not all mi xed tensors are outer products of vectors (they just transform as if they were). Just as we can use the idea V¯ ≡ g¯ V to convert a contravariant vector to its covariant partner, we can similarly use g ¯ to convert the 1st or 3rd index on [.....] abcd from contravariant to covariant. We could apply two g¯'s with the proper linkage of indices to convert them both at once. So given the ability of g ¯ to change any index one way, and g to change it the other way, one can think of the 4-index object [.....] abcd as a family of 16 different 4-index objects, each corresponding to a certain choice for the indices being one type or the other. We know how to interconvert between these 16 objects just applying g or g ¯ factors. So how does one annotate which of the 16 objects [.....] one is staring at for some choice of index types? Here is a somewhat faceti ous possibility, the Morse Code method W – ab ≡ UaV¯b W - - abcd = UaV¯b XcY¯d . (7.2.4) Instead of having a bar over the entire object, in the first case the bar it is placed just over the right side of the W to indicate that b is a covariant index, while no bar means the first index is contravariant. The second example shows how horrible such a notation would be. We really want to put some kind of notation on the individual indices , not on the object! Here is a notation that is slightly better than the Morse code option, though similar to it, W ab-cd- = UaVb- XcYd- ( 7 . 2 . 5 ) Chapter 7: Standard Notation 91 Here overbars on covariant indices distinguish them . Now one can dispense with the overbars on covariant vectors as well, putting the overbar on the index, for example V ¯a = g¯abVb → Va- = g a-b-Vb . There are several problems with this scheme. One is that in the spinor application of tensor analysis used in special relativity, dots are placed on certain i ndices, such as in (5.14.8) , and these would conflict with the proposed overbars. A more subs tantial reason is that this last not ation is hard to type (or typeset, as one used to say), it looks cluttered with all the overbars, and the subscripts are already hard to read without extra decorations since they are in a smaller font than the main text. 7.3 The up/down bell goes off This is where a bell went off somewhere, perhaps in the mind of Gregorio Ricci in the 1880-1900 time frame (1900 snippet quoted in (7.10.8) below). Someone might have said: suppose, instead of using overbars on indices or some other decoration, we distinguish covariant indices by making them be superscripts instead of subscripts. Superscripts are as easy to type as subscripts, and the result is fairly easy to read and totally unambiguous. We would then have for our ongoing example of (7.2.4 and 5), Wab cd = UaVbXcYd // a path not taken This is almost what happened, but the up/down decision went the other way and we now have: superscripts = contravariant = up subscript = covariant = down (7.3.1) and then we get this translation for (7.2.4), W ab-cd- = UaVb- XcYd- → Wa bc d = UaVbXcYd // the path taken (7.3.2) and this has become The Standard Notation . Perhaps the reason for this choice was that the covariant gradient ∂n object appeared more commonly in equations than idealized objects such as d x, and ∂n already used a lower index. A downside of this particular up/down decision is th at every student has be be confused by the fact that his or her familiar position, velocity and momentum vectors that always had subscripts suddenly have superscripts in the Standard Notation. The silver lini ng is that this shocking change alerts the student to the fact that whatever subject is being studied is going to have two kinds of vectors. Despite appearances, it is not completely obvious how one should translate the whole world as presented in the previous six Chapters into this new notation. There are quite a few subtle details that will be discussed in the following Sections. Chapter 7: Standard Notation 92 7.4 Some Preliminary Translations; rais ing and lowering tensor indices with g In the rest of this entire Chapte r, anything to the left of a → arrow is in "developmental notation", while anything to the right of → is in "Standard Notation". 1. Basic translations. So we start translating some of the results above: s → s // a scalar V a → Va // a contravariant rank-1 tensor (vector) V¯a → Va // a covariant rank-1 tensor (vector) M ab → Mab // a contravariant rank-2 tensor M¯ab → Mab // a covariant rank-2 tensor gab → gab // the contravariant rank-2 metr ic tensor g (and similarly for g') g¯ab → gab // the covariant rank-2 metric tensor g ¯ g is inverse of g ¯ → gab is inverse of g ab . (7.4.1) As noted earlier, one "feature" of the Standard Notation is that it is no longer sufficient to specify an object by a single letter. One has to somehow indicat e the index nature by showing index positions. Thus, "g" stands for all four metric tensors g ab , gab, ga b and gab. The pure covariant metric tensor is g ab or perhaps g ** . At first this seems a disadvantage of the nota tion, but one then realizes that the true object really is "g", and it has four different "representations" and the notation makes this very clear. In this same spirit, we have the following translation for a bolded vector: V → V V¯ → V . ( 7 . 4 . 2 ) The reason is that the overbar is no longer used to denote covariancy. The above lines show a subtle change in the interpretation of the bolded symbol V in the standard notation: the single symbol V stands for both the developmental vector V and for its developmental covariant partner vector V¯. The new symbol V is both contravariant with components Vn and it is covariant with components V n. ( 7 . 4 . 3 ) 2. Raising and Lowering Tensor Indices with g As for converting a vector from one type to the other, the relations (5.8.4) become, V ¯a = g¯abVb → Va = gabVb // g ab "lowers" a contravariant index Va = gab V¯b → Va = gab Vb // gab "raises" a covariant index (7.4.4) In the Standard Notation, the nature of a vector (c ontravariant or covariant) is determined by the up or down position of the index. Looking at g abVb = Va, we see that " g ab lowers the index on Vb to give V a " Chapter 7: Standard Notation 93 which of course just means that g ¯abVb = V¯a in the developmental notation. It means nothing more, and nothing less. Similarly, gab Vb = Va says that " gab raises the index on V b to give Va " and this is just the Standard Notation way of saying g ab V¯b = Va. The key idea here is that g ab lowers a tensor index of a tensor, and gab raises a tensor index (both with an implied sum as above). We know from the outer product idea shown in (7.1.1) that we can construct a covariant rank-2 te nsor from two covariant rank- 1 tensors in this way. Tab ≡ UaVb (7.1.1) This is in Standard Notation now, so unlike (7.1.1), all three objects are covariant tensors. We can then use gca to raise the a index on both sides of this equation, gcaTab = Tc b // implied sum on a -- always implied sums on repeated indices! gca(UaVb) = ( gcaUa)Vb = UcVb ⇒ Tc b = UcVb . ( 7 . 4 . 5 ) Here, T c a is a "mixed" rank-2 tensor, the first index is c ontravariant, the second index is covariant. This fact matches the nature of th e right side of the equation UcVb. We could have but did not deal with mixed tensors in the developmental notation because the no tation could not really handle it well, as shown in (7.2.4). So in developmental notation, we dealt only with "pure" rank-2 tensors like M ab and M¯ab. As a next step, we can apply g db to both sides of (7.4.5), g db Tc b = Tcd gdb (UcVb) = Uc(gdbVb) = UcVd ⇒ Tcd = UcVd ( 7 . 4 . 6 ) We end up with a contravariant rank-2 tensor that is the outer product of two contravariant rank-1 tensors. This is in fact the equation (7.1.1 ) expressed in Standard Notation. Now let's do both index raising operations at the same time: T ab ≡ UaVb g ca gdb Tab = Tcd gca gdb UaVb = ( gcaUa)(gdbVb) = UcVd ⇒ Tcd = UcVd ( 7 . 4 . 7 ) Chapter 7: Standard Notation 94 This shows that we are free to ra ise or lower corresponding indices on bot h sides of a tensor equation at will, and we don't have to write out th e details. Thus if the first of these is true, then the other three forms are also valid: Tab ≡ UaVb ⇒ Ta b = UaVb Tab = UaVb Tab = UaVb (7.4.8) This idea works with any rank-2 tensor because any rank-2 tensor transforms under x' = F(x) in the same way that the outer product of two vectors transforms . If M is some arbitrary contravariant rank-2 tensor, then gcaMab = Mc b and so on, repeating all of the above. The idea applies as well to rank-3 tensors and rank-n tensors as we shall describe below in Section 7.10. For example, if T abc = UaVbWc, or if M is some arbitrary rank-3 tensor M abc, then T abc ≡ UaVbWc M abc g daTabc = Td bc gdaMabc = Md bc (gdaUa)VbWc = UdVbWc ⇒ Td bc = UdVbWc . ( 7 . 4 . 9 ) In this case, the validity of the first equa tion implies the validity of the 7 others: Tabc ≡ UaVbWc ⇒ Ta bc ≡ UaVbWc T ab c ≡ UaVbWc T abc ≡ UaVbWc Tab c ≡ UaVbWc Tabc ≡ UaVbWc Ta bc ≡ UaVbWc and finally Tabc ≡ UaVbWc (7.4.10) To summarize, in this new Standard No tation, the covariant metric tensor g ab becomes an "index lowering operator" and the cont ravariant metric tensor gab becomes an "index raising operator". This is a huge advantage of the Standard Notation. It pretty much eliminates the need to think, something universally appreciated. In a certain obscure sense, it is like double entry accounti ng (credits and debits), where the notation itself serves as a check on the accuracy of bookkeeping entries. The g raising and lower rules may be repr esented generically in this manner, gaa' [----a'---] = [---- a---] gaa' [----a'---] = [----a- - - ] ( 7 . 4 . 1 1 ) where [----a'---] represents an arbitrary tensor expression (perhaps just one tensor) with lots of tensor indices indicated by dashes. Each of these dash in dices could be up or down, it does not matter. An example of the second line is gaa'Ma'bc = Ma bc . Chapter 7: Standard Notation 95 3. The invariant distance and covariant dot product : Based on the discussion so far, we can write the followi ng translations, starting from (5.2.5) and (5.10.3), dxi → dxi (ds)2 = g¯ab dxa dxb → gabdxadxb A•B = g¯ab Aa Bb → A•B = gab Aa Bb . (7.4.12) Using the raising and lowering idea above, we can write the dot product in other ways: A•B = gab Aa Bb = Aa( gabBb) = AaBa A•B = gab Aa Bb = gba Aa Bb = (gba Aa)Bb = AbBb // g is symmetric A•B = AbBb = Ab (gbcBc) = gbcAbBc . (7.4.13) To summarize : A•B = gab Aa Bb = AaBa = AaBa = gabAaBb = B •A . (7.4.14) The general idea is this: any tensor index on any tensor object can be raised by gab and can be lowered by gab. Remember that a tensor object lives in some space like x-space, so we shall have to ponder what to do for our matrices S ab and Rab which live half in x-space and half in x'-space, a subject we defer to Section 7.5. 4. Contraction Rules When an index is summed with one index down and the other up, one says that the two indices are contracted . For example, in the expressions AaBa or AaBa, index a is contracted. In g abAaBb both indices a and b are contracted. Tilt Reversal Theorem. The tilt of any pair of contracted indi ces can be reversed without affecting a n y t h i n g . ( 7 . 4 . 1 5 ) A proof is given below in Section 7.11. An example is A aBa =AaBa as shown above in (7.4.14). Contraction Theorem . In analyzing the transformation nature of a tensor expression, one can simply ignore all contracted indices and look only at remain ing indices to determine the tensor nature of the o b j e c t . ( 7 . 4 . 1 6 ) A proof is given below in Section 7.12. As examples, in the cases AaBa o r gabAaBb, if we ignore contracted indices, there are no indices left, so these objects must transform as a tensor with no indices, Chapter 7: Standard Notation 96 which is a tensorial scalar. Thus A•B above is a scalar, something we already knew from (5.10.2). The object AaMab transforms as a covariant vector, and the object Ma a transforms as a scalar. 5. What about having g raise and lower indices on itself ? After all, g is a valid tensor. We find g abgbc = gac // lower the first index of rank-2 tensor gbc g acgbc = gb a // lower the second index g cbgb a = gca . // lower both indices (lower second, then lower first) (7.4.17) We know from (7.4.1) above that g ab and gab are inverses. Thus, it must be true that gabgbc = (1)ac = δac = δa,c gbcgac = (1)b a = δb a = δb,a . (7.4.18) Here δac and δb a are just cosmetically nice ways to wr ite the Kronecker deltas shown. Comparing (7.4.17) and (7.4.18) we learn that gac(x) = δac = δa,c g b a(x) = δb a = δb,a . ( 7 . 4 . 1 9 ) Thus, the mixed forms of the metric tensor are trivia l and are not functions of the nature of x-space. 7.5 Dealing with the matrices R and S ; various Rules and Theorems 1. Translations of R and S Consider the translation of this partial derivative in to the new up/down notation. Since the differential d x element is contravariant and is now written dxi : (∂x' i/∂xk) → (∂x'i/∂xk) . (7.5.1) In terms of "existence", this object has one leg in each space of Picture A of (1.11). The gradient operator ∂/∂x k is an x-space object, while x'i is an x'-space object. Since ( ∂x'i/∂xk) does not live in x-space or in x'-space exclusively, but straddles the two spaces, it cannot possibly be a tensor of any kind. Recall that a tensor object must be entirely within a space, it cannot have body parts hanging out into other spaces. Nevertheless , it seems clear that each of the two indices has a well-defined nature. We showed in (2.4.2) that the gradient transforms as a covariant vector , so we regard k as a covariant index and write ∂/∂x k = Chapter 7: Standard Notation 97 ∂k. And of course dx'i is a contravariant vector, so i is a cont ravariant index. Here then is the proper translation starting with (2.5.1), Rik ≡ (∂x'i/∂xk) → Ri k ≡ (∂x'i/∂xk) = ∂kx'i . (7.5.2) To summarize, Ri k is not a mixed rank-2 tensor, though it looks just like one. Therefore, Ri k can never appear in a tensor equation -- it just appears in the equations that show how tensors transform. However, each of the two indices of R has a well-defined tr ansformational nature, and we place them up and down in the proper manner. It is not atypical for an object to have up and dow n indices but the object is not a tensor. As was noted below (2.3.2), the canonical example is that for a non-linear transformation x' = F(x), xi has a contravariant index but is not (does not transform under F as) a contravariant vector. Consider now the translation of the transformati on rule for a contravariant vector from (2.5.1), V' a = RabVb → V'a = Ra bVb . // contravariant (7.5.3) Even though R is not a tensor, we see that index b is contracted and is thus neutralized from the evaluation of the tensor nature of the RHS. This leaves upper index a as the only free index, indicating that the RHS is a contravariant vector, and this of course then matches the LHS. However, it is a contravariant vector in x'-space, which will be furthe r discussed in (7.5.9) below. Main point: we can deal with the indices on R just as we deal with indices on true tensors. Notice that, even though both sides of V' a = Ra bVb have the same "tensor nature" (both sides are a contravariant vector component in x'-s pace) one cannot ask how the equation V'a = Ra bVb "transforms" under a transformation. That question can only be as ked about equations constr ucted of objects all of which are tensors in the same space. Here V and half of R are in x-space, and V' and the other half of R are in a x'-space. There is no object called R', as if R were in x-space and R' were in x'-space. We can repeat the above discussion for S instead of R. We omit the words and just show the translations, quoting expressions from (2.1.6) and (2.5.1) : (∂x i/∂x'k) → ( ∂xi/∂x'k) S ik ≡ (∂xi/∂x'k) → Si k ≡ (∂xi/∂x'k) = ∂'kxi V¯'a = ST abV¯b = Sba V¯b → V' a = Sb aVb // covariant (7.5.4) 2, Transformation rule for rank-2 tensors For a contravariant rank-2 tensor, the translati on of the transformation rules (5.6.3) are, M' ab = Raa'Rbb'Ma'b' → M'ab = Ra a'Rb b'Ma'b' contravariant rank-2 tensor ( 7 . 5 . 5 ) M¯'ab = Sa'aSb'bM¯a'b' → M' ab = Sa' aSb' bMa'b' . covariant rank-2 tensor Applying this to the metric tensor M = g we find these translations of (5.7.6), Chapter 7: Standard Notation 98 g' = R g RT ⇒ g'ab= Raa'Rbb'ga'b' → g'ab = Ra a'Rb b'ga'b' ( 7 . 5 . 6 ) g ¯' = ST g¯ S ⇒ g¯'ab = Sa'aSb'b g¯a'b' → g'ab = Sa' aSb' b ga'b' . The inverses of the above equations appear in (5.7.7) and translate this way , g = S g' ST ⇒ g ab = Saa'Sbb'g'a'b → gab = Sa a'Sb b'g'a'b ( 7 . 5 . 7 ) g¯ = RT g¯' R ⇒ g¯ab = Ra'aRb'bg¯'a'b' → gab = Ra' aRb' b g'a'b' . By raising and lowering indices on the first result in (7 .5.5), and using the tilt reversal rule (7.4.15), we obtain these four transformation rules : M' ab = Ra a' Rb b' Ma'b' M'a b = Ra a' Rbb' Ma' b' M'ab = Raa' Rb b' Ma'b' M'ab = Raa' Rbb' Ma'b' . (7.5.8) One sees then a family of four tensors associated with M. The first is contravariant, the last is covariant, and the other two are mixed. Comparison of the last line of (7.5.8) to the last line of (7.5.5) suggests that Raa' = Sa' a , and we will formally prove th is fact in (7.5.13) below. 3. Raising and lowering indices on R and S Although R and S are not tensors, one can still raise and lower their two i ndices using metric tensors, but things are a little different from the tensor situation, the reason being that R and S each have one foot in x- space and the other foot in x'-space. Object R a b = (∂x'a/∂xb) was considered above. One could lower the a index using g' ** since x'a is in x'- space and is an up index. The index in ∂/∂xb = ∂b is really a lower index (gradient), so one could in effect raise it using g** (no prime) because ∂/∂xb is in x-space. So when raising and lowering indices on Ra b one has the unusual situation that one must use g' when acting on the first index, and g when acting on the second. With this in mind, we can now wr ite three other index configurations of Ra b : R a b = ( ∂x'a/∂xb) // original object (formerly R ab) Rab = Ra b' gb'b = ( ∂x'a/∂xb) // g pulls up the second index of Ra b Rab = g'aa'Ra' b = ( ∂x'a/∂xb) // g' pulls down the first index of Ra b Rab = g'aa'Ra' b' gb'b = (∂x'a/∂xb) . // both actions at once (7.5.9) Although the g and g' factors can be placed anywhere, we have put g' factors on the left of R, and g factors on the right, each next to its appropriate leg of R. Chapter 7: Standard Notation 99 In each case, examination of the corresponding partial derivative shows that that the index sense matches on both sides. For example, in Rab = (∂x'a/∂xb) = ∂bx'a, both indices are contravariant on both sides. Remember that Rab is not a contravariant rank-2 tens or due to its dual-space nature. In the same manner, we arrive at these index configurations for Sa b : S a b(x) = ( ∂xa/∂x'b) // original object (formerly S ab) Sab ≡ Sa b' g'b'b = ( ∂xa/∂x'b) // g' pulls the second index up Sab ≡ gaa'Sa' b = ( ∂xa/∂x'b) // g pulls the first index down Sab ≡ gaa'Sa' b' g' b'b = (∂xa/∂x'b) // both actions at once (7.5.10) For object S, the metric tensor g raises or lowers the first index of S, while g' raises or lowers the second index of S. This is just the reverse of what happens for object R as reported above. This is not surprising since one gets S ↔R when one swaps x-space ↔ x'-space. 4. Inverse of R and S (and Comment on Transpose) We have showed the translation R ab → Ra b . In the standard notation, imagine that there is some inverse R-1 defined by (R-1)c aRa b = δc b. The chain rule says that (∂x c/∂x'a) (∂x'a/∂xb) = δc b or Sc a Ra b = δc b (∂x' c/∂xa) (∂xa/∂x'b) = δc b or Rc a Sa b = δc b . (7.5.11) Therefore from the first line it must be that (R -1)c a = Sc a. The second line shows that (S-1)c a = Rc a. Using the above rules for raising and lowering indices on both sides of an equation, we obtain these four versions of the developmental notation fact that R-1 = S and S-1 = R : (R -1)ik = Sik (S-1)ik = Rik (R-1)i k = Si k (S-1)i k = Ri k (R-1)ik = Sik (S-1)ik = Rik (R-1)ik = Sik (S-1)ik = Rik . (7.5.12) In (7.5.11) we see a certain "up tilt" matrix multiplica tion which will be explained below in Section 7.8. Chapter 7: Standard Notation 100 5. The R-S Tilt Theorem: Sa b = Rba which is the same as: ( ∂xa/∂x'b) = (∂ x'b/∂xa) (7.5.13) Notice in this rule that the indices are reflected in a vertical line running between the indices: Sa|b = Rb|a. Proof : This proof is a bit long-winded, but brings in many earlier results and is a good exercise: δba" Sa a" = Sa b // introduce a δ . Remember all g's are symmetric. (g' bb' g' a"b') Sa a" = Sa b // since g' ab and g'ab are inverses, see (7.4.18) g' bb' δb' b" Sa a" g' a"b"= Sa b // reorder and use g' a"b' = δb' b"g' a"b" g' bb' (Rb' a' Sa' b") Sa a" g'a"b"= Sa b // δb' b" = (Rb' a' Sa' b") from (7.5.11) g' bb' Rb' a' (Sa a" Sa' b" g'a"b") = Sa b // regroup (7.5.14) g' bb' Rb' a' (gaa') = Sa b // gaa' = Sa a" Sa' b" g'a"b" from (7.5.7) (g' bb' Rb' a' gaa') = Sa b // regroup R ba = Sa b . // use last line of (7.5.9) Notice that the above theorem says S a b = (∂ xa/∂x'b) = (∂x'b/∂xa) = Rba . (7.5.15) Similar results can be derived for other index positions (or we can just raise and lower indices!) to get Sa b = Rba = ( ∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rba = ( ∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rba = ( ∂xa/∂x'b) = (∂x'b/∂xa) Sab = Rb a = ( ∂xa/∂x'b) = (∂x'b/∂xa) . (7.5.16) Here index a is always in x-space, while index b is in x'-space. 6. Determinants of R and S. We showed earlier that R ij→ Ri j and Sij→ Si j. The determinants det(R) and det(S) translate as follows: det(R) = εabc... R1aR2b....RNx → det(R* *) = εabc... R1 aR2 b....RN x det(S) = εabc... S1aS2b.....SNx → det(S* *) = εabc... S1 aS2 b....SN x or Chapter 7: Standard Notation 101 det(R) = εabc... Ra1Rb2....RxN → det(R* *) = εabc... Ra 1Rb 2....Rx N det(S) = εabc... Sa1Sb2.....SxN → det(S* *) = εabc... Sa 1Sb 2....Sx N (7.5.17) where ε is the bookkeeping permutation tensor discussed in Section 7.7 below. Reader Exercise: Write det(R* *) = (εabcR1 aR2 bR3 c) and det(S* *) = (εa'b'c' Sa' 1Sb' 2Sb' 3 ). We know that in either developmental or sta ndard notation det(R)det(S) = det(RS) = det(1) = 1. The exercise is to verify this directly using the N=3 first expression in (D.10.37) for εabcεa'b'c' along with (7.5.11). 7. Jacobian and related Here are translations of a few equations from Section 5.12: (5.12.2) det(S) = 1/det(R) → det(S i j) = 1/ det(Ri j) (7.5.18) (5.12.6) J = det(S) → J = det(S i j) (7.5.19) (5.12.12) g = det(g ¯) → g = det(g ij) g' = det(g ¯') → g' = det(g' ij) (7.5.20) (5.12.14) 1/g = det(g) → 1/g = det(g ij) 1/g' = det(g') → 1/g' = det(g'ij) (7.5.21) (5.12.14) g' = J2 g → g' = J2 g |J| = g'/g → |J| = g'/g (7.5.22) 7.6 Orthogonality Rules, Inversion Rules, Cancellation Rules 1. Orthogonality Rules: Theorem (7.5.13) that S a b = Rba can be used to eliminate S in various forms of SR = 1: SR = 1 → Sa b Rb c = δa c ⇒ R ba Rb c = δa c ⇒ R ba Rb c = ga c from (7.4.19) . (7.6.1) Although R is not a tensor, the object R ba Rb c is a tensor of the mixed type Ma c. Thus, both sides of this last equation transform as this type of mixed rank-2 tensor. In this last equation, we can lower index a on both sides, raise index c on both sides, and reverse the tilt of the b contraction to get this result, again using (7.4.19), R b a Rbc = gac = δac . ( 7 . 6 . 2 ) We now repeat this process starting instead with RS = 1: Chapter 7: Standard Notation 102 RS = 1 → Ra b Sb c = δa c ⇒ Ra b Rcb = δa c ⇒ Ra b Rcb = ga c Rab Rc b = gac = δac . ( 7 . 6 . 3 ) Collecting the four forms written with δ we obtain the four orthogonality rules for R: 1: R ba Rb c = δa c 2: Rb a Rbc = δac // sum is on first index 3: R a b Rcb = δa c 4: Rab Rc b = δac // sum is on second index (7.6.4) For any general transformation x' = F(x), these rules are valid. Going the reverse direction, if we replace all Rxy with Sy x, the four orthogonality rules of (7.6.4 ) can be rederived in this manner, 1: R ba Rb c = Rba Scb = ScbRba = (SR)ca = (1)ca = δca = δa c // down-tilt 2: Rb a Rbc = Rb aSc b = Sc bRb a= (SR)c a = (1)c a = δc a = δac // up-tilt 3: Ra b Rcb = Ra b Sb c = (RS)a c = (1)a c = δa c // up-tilt 4: R ab Rc b = RabSbc = (RS)ac = (a)ac = δac // down-tilt All these rules just say RS = SR = 1. Notice in the la st four lines the notion of up-tilt and down-tilt matrix multiplication. This subject is di scussed in Section 7.8 below. Comment : The orthogonality rules are easy to memorize using these few facts: • The sum is either on both first indices, or on both second indices • the tilts of the two R's are opposite each other. 2. Inversion Rules. The first rule makes this claim: [---- a-----] = Ra b [--------b------] ⇔ Rab [----a-----] = [--------b------] (7.6.5) sum on 2nd index b sum on 1st index a so if one wants to invert the left equation for [--------b------], the result is as shown on the right. Notice that the R indices are reflected through a horizontal line segment between the indices. This is different from the fact (7.5.13) that Sa b = Rba where indices are reflected through a vertical line segment between the indices. Proof : Start with [----a-----] = Ra b' [--------b'------]. Apply R ab to both sides, then use (7.6.4) : R ab[----a-----] = R ab Ra b' [--------b'------] = δb b' [--------b'------] = [--------b------] QED Chapter 7: Standard Notation 103 The rule is also valid if all tilts are reversed. Thus, [---- a-----] = R ab [--------b------] ⇔ Ra b [----a-----] = [-------- b------] (7.6.6) sum on 2nd index b sum on 1st index a Proof : Start with [---- a-----] = R ab' [--------b'------]. Apply Ra b to both sides, then use (7.6.4) : R a b[----a-----] = Ra b Rab' [--------b'------] = δbb] [--------b'------] = [-------- b------] QED Example of (7.6.5) : V'a = Ra bVb ⇔ RabV'a = Vb or Vb = RabV'a (7.6.7) Example of (7.6.6) : V'a = RabVb ⇔ Ra bV'a = Vb or Vb = Ra bV'a (7.6.8) Example: Rank-3 tensor transformation inversion : M'abc = Ra iRb jRc kMijk // rank-3 transformation rule (contravariant) Mabc = RiaRjbRkcM'ijk // its inverse (7.6.9) Notice that, in the inverse transformation, the R indices are up-tilted and the summation indices are strangely on the left side. Using the raising and lowering rules of (7.5.9), one sees that the corresponding covariant forms are obtained just by moving indices vertically on both sides. Thus. M' abc = RaiRbjRckMijk // rank-3 transformation rule (covariant) Mabc = Ri aRj bRk cM'ijk // its inverse (7.6.10) 3. Cancellation Rules The first rule makes this claim: R a b [----b-----] = Ra b [--------b------] ⇔ [----b-----] = [--------b------] (7.6.11) where the two R factors can be cancelled out. Proof : Start with Ra b' [----b'-----] = Ra b' [--------b'------]. Apply R ab to both sides, then use (7.6.4) : RabRa b' [----b'-----] = R abRa b' [--------b'------] ⇒ δ b b' [----b'-----] = δb b' [--------b'------] ⇒ [----b-----] = [--------b------] QED The rule is also valid if all tilts are reversed. Thus, Rab [----b-----] = R ab [--------b------] ⇔ [---- b-----] = [-------- b------] (7.6.12) The proof is similar to that shown above. Chapter 7: Standard Notation 104 7.7 About δ and ε The Kronecker δ is sometimes written in different wa ys to make things "look nice", δab = δa b = δba = δb a = δa,b . (7.7.1) Eq (7.4.19) showed that that one can rega rd the above sequence of equalities as saying gab = ga b = gba = gb a = δa,b ( 7 . 7 . 2 ) where these g objects are mixed versions of the symmetric rank-2 metric tensor g ab. There is no "δ tensor", it is the g metric tensor, but tradi tion is to write the diagonal objects using the δ symbol. The object ε abc... is a bit more complicated. It can at first be regarded as a mere bookkeeping device, in which context it is usually called "t he permutation tensor". It appears for example in the expansion of a determinant in N dimensions, det(M) = εabc...x M1aM2b.....MNx = εabc...x Ma1Mb2.....MxN (7.7.3) or in an ordinary cross product in Cartesian space, A a = εabcBbCc . ( 7 . 7 . 4 ) This permutation tensor ha s the usual properties that ε 123...N = +1, that ε changes sign when any two indices are swapped, and that ε vanishes if two or more indices are the same. This permutation "tensor" is not really a tensor since one would regard it as being the same in x-space or x'-space. Whether indices are written up or down on this ε is immaterial. At another level, however, εabc...x with N indices (the same ε symbol is used) is a covariant rank-N tensor density of weight -1 known as the Levi-Civita tensor. This subj ect is addressed in Appendix D in much detail. In what we call the Weinberg convention, individual indices of ε can be raised and lowered by g as discussed in Section 7.4 subsection 2, just as with any tensor. Therefore, in Cartesian space with g = 1, indices on ε are raised and lowered with no consequence [ see (5.9.1) or (7.4.4) ], and then one can identify any form of ε as being the permutation tensor. For example, εabc = εabc = εab c and so on. In a non-Cartesian x-space, however, one would say that εab c = gbb'εab'c ≠ εabc. In the Weinberg convention, one sets ε123..N = ε'123..N = 1 and εabc..x = ε'abc..x has the properties of the permutation tensor described above and these pr operties are the same in x-space as in x'-space. Then for general g≠ 1, εabc..x (lower indices) is NOT the permutation tensor. Th e bottom line is that one must be aware of the space in which one is working (the Picture). The ε appearing above in the determinant expansion (7.7.3) is always just the permutation tensor, but in the cross product that is not the case, and one would properly write A a = εabcBbCc ( 7 . 7 . 5 ) Chapter 7: Standard Notation 105 and conclude that the cross product of two ordinary contravariant vectors is a covariant vector density (Section D.8). Again, in Cartesian space where one often works, this would be the same as A a = εabcBbCc = εab cBbCc , but the "properly tilted form" A a = εabcBbCc reveals the tensor nature of the object A a. As mentioned below in Section 7.15, this "covariant" equation would appear as A' a = ε'abcB'bC'c in x'-space, but since A' a is a covariant vector density, A' a ≠ RabAb, and in fact A' a = J RabAb. The permutation tensor εabc...x and the contravariant Levi-Civita tensor εabc...x are both "totally antisymmetric" which just means ε changes sign if any pair of indices is swapped. In fact, as discussed in Section D.3, there IS only one antisymmetric tensor of rank N apart from a multiplicative scalar factor, and εabc...x is it. This fact simplifies various calculations. Technically, εabc...x is a totally antisymmetric tensor density , but normally it is just called "the to tally antisymmetric tensor". As implied by (D.5.1), the covariant Levi-Civita tensor εabc...x is also totally antisymmetric and is therefore a multiple of εabc...x. The reader is invited to peruse Appendix D at so me appropriate time for mo re about tensor densities and the ε tensor. 7.8 Covariance and Matrix Multiplication Before continuing the process of translation from developmental to standard notation, we digress momentarily to consider the notion of covariance in developmental notation. As we shall discuss in more detail below in Section 7.15, an equation is said to be covariant under the transformation x' = F(x) if it has "the same form" in both x-space and x'-space. The "same form" means that the equation looks the same but everything is primed in x'-space. Example 1: Newton's Law F = m a is covariant under rotations ( x' = F(x) = R x), and in x'-space this law takes the form F' = m' a' which has the same form as the equation in x-space F = m a. Once we know that F and a are contravariant vectors and m is a scalar, this conclusion is automatic from (2.3.2), F = ma ⇒ F' = m' a' proof: F' = R F = R(m a) = m R a = m a' = m' a' (7.8.1) where m = m' follows since mass is a scalar under ro tation. Thus, Newton's Law has the same form when it is examined in two frames of re ference related by a rotation. It is covariant under rotations. Example 2: Consider the equation A•B = π where A and B are contravariant vectors and • is the covariant dot product shown in (5.10.3), A•B ≡ g¯abAaBb. It was shown in (5.10.2) that the quantity A•B transforms as a scalar under general transformation x' = F(x) so that A'•B' = A•B. Since the number π is also a scalar under any transformation (it is a constant), one could say that π' = π (it is the same number 3.14159 in x'-space and x-space), so A•B = π ⇒ A'•B' = π' , equation is covariant. (7.8.2) What we see here is that an equation is covariant IFF both sides of the equation transform as the same tensorial tensor type under the transformati on of interest. In Example 1, both sides of F = ma transform as Chapter 7: Standard Notation 106 contravariant vectors under rotations , and in Example 2 both sides of A•B = π transform as scalars under a general transformation. Example 3: Consider the outer product equation (7.1.1) T ab = UaVb where U and V are contravariant vectors. We show in (7.1.2) that T ab transforms as a contravariant rank-2 tensor. Both sides of this equation transform in this way, so in x'-space the equation becomes T' ab = U'aV'b. The equation is therefore covariant under the transformation x' = F(x) with d x' = Rd x. Approaching this example in a s lightly different manner, suppose we define Tab ≡ UaVb where U and V are contravariant vectors. We then ask: Is T ab a contravariant rank-2 tensor? Line (7.1.2) shows that the answer is yes, T 'ab = U'aV'b = (Raa'Ua') (Rbb'Vb') = Raa' Rbb' Ua'Vb' = Raa' Rbb' Ta'b' (7.8.3) which matches the transformation rule as stated in (5.6.3). Example 4: Suppose A and B are tensorial contravarian t rank-2 tensors. Is the equation AB = C covariant? If it were, we would have to show that in x' -space we have A'B' = C' where C is a contravariant rank-2 tensor. To investigate, we use the rule (5.7.1) which states how a contravariant rank-2 tensor transforms in terms of Picture A shown in (5.7.2) : A'B' = (RAR T)(RBRT) = RA(RTR)BRT // since A and B are contra rank-2 tensors (7.8.4) C' = RCR T = RABRT // assuming C is also a contra rank-2 tensor and AB = C If it were true that RTR = 1, one would find from the first line above that A'B' = RABRT = RCRT = C' and the answer would be yes, the equation AB = C is covariant. However, for a general Picture A transformation with metric tensor g in x-space and g' in x'-space, what we know about R comes from (5.7.6) : g' = R g RT . Even if g = 1 so x-space is Cartesian, this says g' = RRT, but this tells us nothing about RTR. So for a general transformation, we have RTR ≠ 1 and so the equation AB = C is NOT covariant. [In the special case that R is a rotation, so RT = R-1 (real orthogonal), then RTR= R-1R = 1.] As with Example 3, we can reformulate the current example in a different manner. Suppose we define C ≡ AB and specify that both A and B are contravariant rank-2 tensors. In this case, is C a contravariant rank-2 tensor? If it were, we would have to have (A'B') = R(AB)RT from (5.7.1). But we showed above that, since RTR ≠ 1, we end up with (A'B') ≠ R(AB)RT . Therefore, C ≡ AB is not a contravariant rank-2 tensor. Could C be a covariant rank-2 tensor? If it were, we would need to have (A'B') = ST(AB)S from (5.7.1). But above we show that A'B' = RA(RTR)BRT and this is completely different from (A'B') = ST(AB)S. Thus, C is not a covariant rank-2 tensor. Since C has two indices, the only way it could be a tens orial tensor is if it is either a contravariant or a covariant rank-2 tensor, but we have ju st ruled out both these possibilities. Therefore C ≡ AB is not a tensorial tensor of any kind whatsoever, even though A and B are tensorial tensors. Chapter 7: Standard Notation 107 Matrix Rule #1. In developmental notation, if A and B ar e contravariant rank-2 tensors, the matrix product AB is (in general) not a rank- 2 tensor and is in fact not any kind of tensor. The equation C = AB is not covariant. Mimicking the above discussion, the reader can show that the same conclusion applies to C = A¯B, C = AB ¯ and C = A ¯B¯ : in none of these cases is C a tensor of any kind, and all these equations are non-covariant. Similarly, the Rule applies to X = ABC or X = ABCD and so on. (7.8.5) For this reason, we shall never ask how to transform an equation like X = ABC... from developmental to standard notation. Equations which are non-covariant ar e simply of no interest, and can never describe a physical relationship, as explained below in Section 7.15. The attentive reader might ask: What about the equation g' = R g R T which has the form X = ABC. And if g = 1, what about g' = RRT whose form is X = AB? In both these cases, the left hand side is a contravariant rank-2 tensor. These e quations do not violate the Matrix Ru le #1 above because the matrices R and RT are not tensors of any kind, as noted below (7.5.2). Furthermore, one does not ask whether g' = R g RT is covariant or not because it is an equation re lating objects in different spaces and not all objects in the equation are tensors. We now consider the notion of matrix multiplica tion using mixed rank-2 tensors. Since we never introduced such mixed tensors in our developmental notation, we have this discussion entirely in the Standard Notation. Consider C i j = Ai kBk j . // implied sum on k (7.8.6) The indices k have the right adjacen cy so one could think of this as being a matrix equation C = AB where all three objects are "down-tilt rank-2 mixed te nsors". Down-tilt just means the two indices are tilting down like i j. We can ask again our questions of Exampl e 4. If A and B are rank-2 tensors, is C = AB covariant? And if we define C ≡ AB, is C a rank-2 tensor? The answer to both questions is yes. To show that AB = C is c ovariant, we start with the second line of (7.5.8) with R bb' = Sb' b from (7.5.13) : M' a b = Ra a' Sb' b Ma' b' (7.5.8) and we apply this transformation rule to both A and B to get A' i kB'k j = (Ri a' Sb' k Aa' b') (Rk a" Sb" j Ba" b") = Ri a' Sb' k Rk a" Sb" j Aa' b'Ba" b" = Ri a' (Sb' k Rk a") Sb" j Aa' b'Ba" b" = Ri a' (SR)b' a" Sb" j Aa' b'Ba" b" = Ri a' δ b' a" Sb" j Aa' b'Ba" b" = R i a' Sb" j Aa' b'Bb' b" = Ri a' Sb" j (AB)a' b" = Ri a' Sb" j Ca' b" = C ' i j . // using (7.5.8) a third time with M = C (7.8.7) Chapter 7: Standard Notation 108 Thus we have shown that AB = C ⇒ A'B' = C' so our down-tilt ma trix equation is covariant. If we define C ≡ AB where A and B are down-tilt mixed rank-2 tensors, then C will be a rank-2 down-tilt tensor providing we can show that (A'B')'a b = Ra a' Sb' b (AB)a' b = Ra a' Sb' b Ca' b . But this is just what was shown above (albeit with different indices), so yes, C is also a down-tilt mixed rank-2 tensor. One way to clarify the intention of C = AB is to write the matrix equation as C dt = AdtBdt where the notation A dt means the down-tilt mixed rank- 2 tensor having components Ai j. In other words, Ci j = Ai kBk j ⇔ (C dt)ij = (Adt)ik(Bdt)kj (7.8.8) where the matrices C dt, Adt and Bdt can be regarded as ordinary linear algebra matrices. For example, the identity det(XY) = det(X)det(Y) then says det(C dt) = det(A dt)det(Bdt) or det(C* *) = det(A* *)det(B* *) (7.8.9) where we use wildcards to indicate the position of the i ndices of the elements which appear in the matrix whose determinant we are computing. It is easy to show that the conclusions reached above apply similarly to an all up-tilt matrix equation C ij = Aik Bkj . // implied sum on k (7.8.10) We thus arrive at: Matrix Rule #2. In Standard Notation, it is reasonable to use matrix notation in the following two situations involving mixed rank-2 tensors: C i j = Ai kBk j C dt = AdtBdt dt = down-tilt Cij = AikBkj C ut = AutBut ut = up-tilt (7.8.11) In special relativity the down-tilt matrix form is most often used, and one just writes C = AB without bothering with the dt clarifying subscripts. Th is is consistent with the usual statement x'μ = Λμ νxν to describe a Lorentz transformation acting on the contravariant vector xν (where Λ = our R ). Note : In the matrix equation Ci j = Ai kBk j containing "down-tilt" mixed ra nk-2 tensors, the k index sum "tilts up". 7.9 Matrix Inverse, Transpose and Determinant Matrix Inverses Consider the standard notation matrix equation AB=1 where (assuming det(A) ≠ 1) we can write B = A-1. As demonstrated above, AB = 1 can only be a covari ant equation if A and B are both down-tilt or both up-tilt mixed rank-2 tensors. Then we are talking about either A dtBdt = 1dt or AutBut = 1ut, and the corresponding B dt = (A-1)dt and But = (A-1)ut, all these being matrix equations . Thus Chapter 7: Standard Notation 109 AdtBdt = 1dt B dt = (A-1)dt Ai kBk j = δi j Bi j = (A-1)i j AutBut = 1ut B ut = (A-1)ut A ikBkj = δij B ij = (A-1)ij . (7.9.1) In this context, we have shown that if A is a mixed rank-2 tensor, then (A -1) is a mixed rank-2 tensor as well, assuming it exists. In (7.5.11) we considered S = R -1 and R = S-1 and reached the conclusions shown in (7.9.1) for the cases A = S and R = B. It happens that in this special case, S and R are not tensors, but the results are still valid. It was shown in (7.4.19) that 1 dt is really g dt, the down-tilt form of the metric tensor g, and similarly for 1ut : 1 dt = gdt (1)i j = gi j = δi j = δi,j 1ut = gut (1) ij = gij = δij = δi,j . (7.9.2) Thus first equation in (7.9.1) can be written in covariant form A dtBdt = gdt where all three matrices are down-tilt mixed rank-2 tensors. A nd this is also true for A utBut = gut . Transpose Matrices It is possible to define the "covariant transpose" MT (italic T) of a matrix M in Standard Notation, and things work out as follows, where M is any rank-2 tensor, (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)a b = Mba (RT)a b = Rba (ST)a b = Sba (MT)ab = Mb a (RT)ab = Rb a (ST)ab = Sb a (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba . // covariant transpose (7.9.3) The general rule is that, in each equation, the indices are reflected in a vertical line separating them. The notation is "covariant" in that one can raise a nd lower indices at will in each equation and thereby create a new valid equation. The drawback of this notation is that, when the indices are tilted, M T in general differs from what we might call the "matrix transpose" MT of a matrix M. When one transposes a matrix, one normally means to swap the rows and co lumns, and that means to swap the indices. One would then write (M T)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)a b = Mb a (RT)a b = Rb a (ST)a b = Sb a (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba (MT)ab = Mba (RT)ab = Rba (ST)ab = Sba . // matrix transpose (7.9.4) Chapter 7: Standard Notation 110 The middle two lines (tilted indices) are very non-covariant in form since for example (MT)a b = Mb a has the up and down sense of index a and b not even matching. The use of the MT matrix transpose is in theorems like det(M) = det(MT) where rows and columns are swapped without changing the determinant. Comments: 1. The issue here is that one might have (M T)a b = Mba ≠ (MT)a b = Mb a . If M is an x-space tensor and if the x-space metric tens or is g = 1, then up and down index positions don't matter and our problem goes away, (M T)ab = Mba = (MT)ab = Mba so then M T = MT. ( 7 . 9 . 5 ) 2. For R and S: As shown in (7.5.9), g' acts on the first index of R while g acts on the second index. Conversely as shown in (7.5.10), g acts on the first i ndex of S while g' acts on the second index. So up and down indices are the same for R and S only if both g = 1 and g' = 1. Only in this special case, which implies a (local) rotation for x' = F(x), does one have RT = RT and ST = ST . This situation arises in Appendix E where we have x" = FM(x) and R and S (there called M a nd N) are rotations. (7.9.6) 3. As it turns out, regardless of g and g', one has det(M) = det(MT) = det(MT) ( 7 . 9 . 7 ) for any index positions on the matrices. Th e traditional fact that det(M) = det(M T) is true for any kind of square matrix, while the fact that det(M) = det(MT) is proven in (D.12.20). Since T and T are the same when indices are both up or both down, the interesting cas e is when indices are tilted. Here is an outline, det(M T) = det([MT]i j) = det(M ji) = det(g jj'Mj' i'gi'i) = det(g dnMdtgup) = d e t ( g dn)det(Mdt)det(gup) = g det(M dt) (1/g) = det(M dt) = det(Mi j) = det(M) . 4. As shown in the next few paragraphs, in either up-tilt or down-tilt notations one can write, RR T = RTR = 1 SST = STS = 1 RS = SR = 1 R T = R-1 = S ST = S-1 = R . (7.9.8) Thus, both the R matrix and S matrix are "covariant real-orthogonal" for any underlying transformations x' = F(x), even when R and S are not rotation matrices. Since the matrix elements are real, R and S are also "covariant unitary". Chapter 7: Standard Notation 111 Restatement of Orthgonality Conditions We shall generally avoid using the covariant trans pose notation since it leads to results which can be confusing although correct, but there will be times when we use it. To restate the potential confusion, in developmental notation the statement RRT = 1 (or RT = R-1, meaning "real orthogonal") implies that R is a "rotation", where we include in this term the possi bility of axis reflections. But in Standard Notation, RRT = 1 is valid for any matrix R associated with a general transformation x' = F(x), rotations and non- rotations alike. To see why this is so, ju st write out orthogonality rules from (7.6.4) : #4 Rab Rc b = δac ⇒ Rab (RT)bc = δac ⇒ RRT = 1 (up-tilt) . #3 R a b Rcb = δa c ⇒ Ra b (RT)b c = δa c ⇒ RRT = 1 (down-tilt) . #2 R b a Rbc = δac ⇒ (RT)ab Rbc = δac ⇒ RTR = 1 (up-tilt) . #1 R ba Rb c = δa c ⇒ (RT)a bRb c = δa c ⇒ RTR = 1 (down-tilt) . (7.9.9) The equations on the right are just restatements of the orthogonality rules. The same equations are valid with R→ S, for example RRT = 1 ⇒ (RRT)-1 = 1 ⇒ (RT)-1R-1 = 1 ⇒ (R-1)TR-1 = 1 ⇒ STS = 1 . To totally distinguish the above situations, we could use this rather unpleasant notation, [RRT = 1]DN ⇔ R is a "rotation" DN = Developmental Notation [RRT = 1]SN,ut for any R associated with F SN,ut = Standard Notation, up-tilt [RR T = 1]SN,dt for any R associated with F SN,dt = Standard Notation, down-tilt (7.9.10) But such notations are not necessary if one understands that an equation like RRT = 1 implies all up-tilt or all down-tilt matrices. How to recognize a rotation How then does one recognize a rotation matrix directly in Standard Notation? We translate using the rule Rik→ Ri k shown in (7.5.2), [RRT= 1]DN ⇒ RikRjk = δi,j → Ri kRj k = δi,j . A simpler alternate form for recognizing a rotation can be obtained as follows, Chapter 7: Standard Notation 112 Ri kRj k= δi,j // now apply Sa i to both sides Sa iRi kRj k= Sa iδi,j (SR) a kRj k = Sa j // but SR = 1 δ a kRj k = Sa j = Rja // right side from (7.5.13) R j a = Rja // if R is a rotation Therefore, in Standard Notation a "rotation" can be identified in either of these two ways, R i kRj k = δi,j or Rj a = Rja // "rotation" . (7.9.11) Matrix form of the transformation rule for a rank-2 tensor Instead of trying to use this covariant transpose T superscript in the standard notation, we can get rid of T in the developmental notation and then do our translation to standard notation. For example we start in developmental notation, M' = R M R T ⇒ M' ad = RabMbc(RT)cd = RabMbcRdc = RabRdcMbc (7.9.12) Then we make the conversion using (7.5.2) R ik → Ri k and (7.4.1) M ab → Mab : M'ad = RabRdcMbc → (M')ad = Ra bRd cMbc (7.9.13) and this then is the Standard Notation rule for the way a contravariant rank-2 te nsor transforms, as was shown in the first line of (7.5.8). However, notice that, where we make use of the "tilt reversal theorem" (7.4.15), (M') ad = Ra bRd cMbc // covariant, so can lower index d on both sides so (M') a d = Ra bRdcMbc = Ra bRdcMb c = Ra bMb cRdc = Ra bMb c(RT)c d = (RMRT)a d . (7.9.14) Thus we can write M' = RMR T in standard notation provided we assume that we are using all down-tilt matrices. The same equation resu lts if all matrices are up-tilt: (M') ad = Ra bRd cMbc so (M') ad = RabRd cMbc = RabMbcRd c = RabMbc(RT)cd = (RMRT)ab (7.9.15) We then arrive at this conclusion: Chapter 7: Standard Notation 113 Fact: The transformation rule for a rank-2 tensor M can be written in matrix notation in the Standard Notation as M' = RMRT all up-tilt or all down-tilt (7.9.16) provided it is understood that all matr ices are up-tilt or all are down-tilt. The result is very similar to what it was in Developmental Notation: M' = RMRT. The use of the covariant transpose thus causes this equation to maintain its form when we go to Standard Notation. Transpose in Dirac Notation In the Dirac notation dot products are represented as a • b = <a | b> = b • a = <b | a>. In general Dirac theory (as used in quantum mechanics) one has < a|b> = < b|a>*, but all our scalar products are real so we can ignore complex conjugation. Consider then, < a | M | b> ≡ <a | M b> = a • (Mb) = ai(Mb)i = ai[ Mijbj] = ai Mij bj = b j Mij ai = bj (MT)j i ai = bj [MTa]j = b • (MTa) = < b| MTa> = < b| MT |a> . (7.9.17) This proves: Fact: In Dirac notation < a | M | b> = < b | MT | a> for any rank-2 tensor M, where T is the covariant t r a n s p o s e . ( 7 . 9 . 1 8 ) If a and b are vectors in Cartesian space with g = 1, then < a | M | b> = < b| MT |a> since then MT = MT. A fuller description of Dirac notation is presented in (E.7.4). Summary of Covariant Transpose Equations We summarize the above facts which relate to th e covariant transpose in Standard Notation, det(M) = det(M T) determinant matrix transpose theorem (7.9.7) RS = SR = 1 so R-1 = S and S-1 = R relation between S and R from their definition (7.5.11) RRT = RTR = 1 ⇒ RT = R-1 = S orthogonality rules (R is covariant real orthog.) (7.9.8) SST = STS = 11 ⇒ ST = S-1 = R orthogonality rules (S is covariant real orthog.) (7.9.8) M' = RMRT how a (tilted) rank-2 tensor transforms (7.9.16) <a | M | b> = < b | MT | a> Dirac matrix elements of operator M (7.9.17) (7.9.19) In these equations M T is the covariant transpose shown in (7.9.3). It is assumed that all matrix products involve matrices which are either all down-tilt or all up-tilt. The various orthogonality rules like RRT = 1 are valid for any R matrix, not just for a rotation. Chapter 7: Standard Notation 114 Determinant of a Matrix In developmental notation one writes det(A) = ε abc... A1aA2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.20) where the A ij are components of the contravariant rank-2 tensor A and where ε is the permutation tensor discussed above in Section 7.7. We have argued above that the notion of a rank-2 te nsor being a matrix in Standard Notation is only viable for mixed rank-2 tensor of either the down-tilt or up-tilt variety. Thus, the matrix determinants of interest in Standard No tation would be these: det(A dt) = det(A* *) = εabc... A1 a A2 bA3 c..... = εabc... Aa 1 Ab 2Ac 3..... (7.9.21) det(A ut) = det(A **) = εabc... A1a A2bA3c..... = εabc... Aa1 Ab2Ac3..... (7.9.22) Determinants of a rank-2 tensor A rarely appear in this document, but they do appear for the non-tensor objects R and S as shown in (7.5.17), as needed for the Jacobian in (5.12.6). For more information on determinants of rank-2 tensors see Section D.12. 7.10 Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita The most general tensor of rank n (aka order n ) will have some number s of contravariant indices and then some number n-s of covariant indices. If s = n, the tensor is pure contravariant, and if s = 0, it is pure covariant, otherwise it is "mixed" (as opposed to " pure"). The transformation of the tensor under F will show a factor Ra a' for each contravariant index, and a factor Sa' a ( = Raa' by (7.5.13) ) for each covariant index, as illustrated by this example : T 'abc de = Ra a' Rb b' Rc c' Sd' d Se' e Ta'b'c' d'e' T 'abc de = Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' . (7.10.1) Note that for R a a' and Raa' the second index is the summation index, but for Sa' a it is the first index. A rank-n tensor always transforms the way an outer product of n vectors transforms if those vectors have indices which type-match those of the tensor. In the above case, an object that would transform the same as Tabc de would be AaBbCcDdEe . ( 7 . 1 0 . 2 ) A tensor of rank-n has 2 n tensor objects in its family since each index can be up or down. For example, the tensor T above is one of 25 = 32 tensors one can form. Of these, one is pure covariant and one is pure contravariant and 30 are mixed. If any of these tensors is a tensor field, such as T abc de(x), then of course all family members are tensor fields. Chapter 7: Standard Notation 115 Direct Products . Consider again the outer product of vectors AaBbCcDdEe. The transformation A'a = Ra a'Aa' occurs in an N-dimensional contravariant vector space we shall call R . In this space one could establish a set of basis vectors, and of course ther e are rules for adding vectors and so on. Transformation B'a = Ra cBc occurs in an identi cal copy of the space R, but transformation D' d = Sd' dDd' = Rdd'Dd' occurs in a covariant version of R we call R¯ . Since dot products (inner products) have been established for vectors in these spaces, they can be regarded as full blown Hilber t Spaces with the caveats of Section 5.10. The transformation of the outer product object, as already noted, is given by A' aB'bC'cD'dE'e = Ra a' Rb b' Rc c' Rdd' Ree' Aa'Bb'Cc'Dd'Ee' (7.10.3) and one can consider the operator R a a' Rb b' Rc c' Rdd' Ree' as a transformation element in a so-called direct product space which in this case would be written Rdp = R ⊗ R ⊗ R ⊗ R¯ ⊗ R¯ ( 7 . 1 0 . 4 ) One could then define ( Rdp)abc de ; a'b'c'd'e' ≡ Ra a' Rb b' Rc c' Rdd' Ree' (7.10.5) so that A'aB'bC'cD'dE'e = ( Rdp)abc de ; a'b'c'd'e' Aa'Bb'Cc'Dd'Ee' (7.10.6) and of course this would apply to any tensor of the same index configuration, such as T 'abc de = Rdpabc de ; a'b'c'd'e' Ta'b'c' d'e' . (7.10.7) This suggests a definition of "tensor" as follows" : tensors are those objects that are transformed by all possible direct product representations formable from the two fundamental vector representations R and R¯. To this set of spaces one would add the identity space 1 to handle tensorial scalars. Appendix E continues this direct product discussion in terms of the basis vectors that form a complete set for a direct product space such as Rdp and shows how to expand tensors on such bases. Lie Groups. The direct product notion is just a formalism, but the formalism has some implications when the space R is associated with a "representation" of a Lie group. In this case, a direct product Rdp = R ⊗ R can be written as a sum of "irreducible" representati ons of that group. What this means is that the transformation elements of Rdp and the objects Tab can be shuffled around with linear combinations so that ( Rdp)ab a'b', when thought of as a matrix with columns labeled by N2 ab possibilities and rows labeled by the N2 a'b' possibilities, appears in "block diagonal form" with all zeros outside the blocks. In this case, the shuffled components of tensor Tab can be regarded as a non-interacting assembly of pieces each of which transforms according to one of those blocks of the shuffled ( Rdp)ab a'b'. The most famous example occurs with N= 3 and the rotation group SU(2) in which case R(1) ≡ R can be decomposed according to R(1) ⊗ R(1) = R(2) ⊕ R(1) ⊕ R(0) where the ⊕ symbols indicate this Chapter 7: Standard Notation 116 block diagonal form. In this case the blocks are 5x5, 3x3 and 1x1, fitting onto the diagonal of the 9x9 matrix area. The numbers L = 0,1,2 here label th e rotation group representations and that label is associated with angular momentum. The elements of the 5x5 block are called D(2) M,M'(φ,θ,ψ) where M,M' = 2,1,0,-1.-2, and where φ,θ,ψ are the "Euler angles" which serve to label a particular rotation. This D(2)object is the L=2 matrix representation of the rotation group. Taking two vectors A and B, one can identify A•B as the combination transforming according to R(0) ("scalar") and AxB ( linearly combined) as that transforming as R(1) ("vector"). The traceless symmetric matrix A iBj - δi,jA•B has 5 independent elements associated with R(2) ( "quadrupole"). This whole reduction idea can be applie d to larger direct products such as R(1) ⊗ R(1) ⊗ R(1) and tensor components Tabc. The Standard Model of elementary particle physics is chock full of direct products of this nature, where the idea of rotational symmetry is extended to ot her kinds of "internal" symmetry, spin and isospin being two examples. Representations of the Lie symm etry group SU(3) are associated with quarks which are among of the fundamental building blocks of the Standard Model. The group discussion above can be applied generally to quantum physics. The basic idea is that if "the physics" (the Hamiltonian or Lagrangian) describi ng some quantum object is invariant under a certain symmetry group (such as rotational symmetry or perhaps some discrete crystal symmetry), then the quantum states of that object can be classified accord ing to the representations of that group. The Bohr hydrogen atom "physics" H ~ ∇ 2-1/|r| has perfect rotation group symmetry and is also symmetric about the axis (angle ψ) from center to electron (no spin). The repres entation functions then must have M' = 0, and then D(L) M,0(φ,θ,ψ) ~ YLM (θ,φ), the famous spherical harmonics th at describe the "orbitals" which have mystified first-year chemistry students for the last 100 years. Historical Note: Ricci and Levi-Civita (see Refs) referred to ra nk-n tensors as "systems of order n" and did not include mixed tensors in their 1900 paper. Nor did they use the Einstein summation convention, since Einstein thought of that later on. They did use the up and down index notation pretty much as it is used today, though the up indices are enclosed in parenthesis. Here is a direct quote from the paper where the nature of the contravariant a nd covariant tensors is described (with crude translation below for non- French readers). Equation (6) had typos which some thoughtful reader corrected: the y subscripts should be r's and the x subscripts should be s's. In our notation ∂x s/∂yr → ∂xs/∂x'r = Ss r = Rrs. Chapter 7: Standard Notation 117 ( 7 . 1 0 . 8 ) We will say that a system of order m is covariant (a nd in this case we will desi gnate its elements by the symbol X r1,r2.... ) (r1,r2.... can each take all the values 1...n), if the elements Y r1,r2.... of the transformed system are given by the formulas (6) . We will designate on the contrary by the symbols X(r1,r2....) the elements of a contravariant system, which is to say of a system where the transformation is represented by the formulas (7) , the elements X and Y being related re spectively to (presumably "are functions of") the variables x and y. Their y is our x', and their n is our N. Notice that the i ndices on the coordinates themselves are taken down, contrary to current usage. They do not explai n why the words contravariant and covariant are used. In our notation, the two equa tions above would be written M'abc... = Raa'Rbb'....... Ma'b'c'... = Σprimed Ma'b'c'... Raa'Rbb'....... (6) M' abc... = Ra a' Rb b'....... Ma'b'c'... = Σprimed Ma'b'c'... Ra a' Rb b''....... (7) (7.10.9) 7.11 The Contraction Tilt-Reversal Rule In some complicated combination of multiple tensor s, imagine there is somewhere a pair of summed indices where one is up and the other is down. As noted above (7.4.15), such a sum is called a contraction . The contracted indices could be on the same obj ect or they could be on different objects. We depict this situation with the following symbolic notation, [-----a---------a----] (7.11.1) where the dashes indicate indices that we don't care about and which won't ch ange -- each one could be up or down. We know we can reverse the tilt this way, as in (7.4.11), Chapter 7: Standard Notation 118 [-----a---------a----] = gab gac [-----b---------c----] (7.11.2) where the first g raises the index b to a, and the se cond g lowers the index c to a. But the two g's are inverses as in (7.4.1), gab gac = gba gac = δb,c, which at once gives the desired result (index b →a) [----- a---------a----] = [----- a---------a----] . // the Contraction Tilt-Reversal Rule (7.11.3) A notable example of course is this: A aYa = AaYa = " A • Y " // or perhaps " A .Y " as noted in Section 5.10 (7.11.4) When is index tilt-reversal allo wed and when is it not allowed? It is always allowed when both indices of the tilted contraction are valid tensor indices. Consider these four examples to be discussed below: R abAb = RabAb Proof: R abAb = Racgcb gbdAd = gcbgbdRacAd = δc dRacAd = RacAc A aRab ≠ AaRab Proof: AaRab = gacAc Rdb g'da = gac g'ad Ac Rdb ≠ AaRab A a∂a = Aa∂a Proof: Aa∂af = gacAc gad∂df = gacgadAc∂df = δc dAc∂df = Ac∂cf (7.11.5) ∂ aAa ≠ ∂aAa Proof: ∂aAa = (gac∂c)(gadAd) = gacgad(∂cAd) + gac(∂cgad)Ad = δcd(∂cAd) + gac(∂cgad)Ad = ∂cAc + (∂agad)Ad ≠ ∂cAc In the first example, since R ab is not a tensor, one is on dangerous ground doing the tilt reversal, but it happens to work because the second index is associated with metric tensor g ij which is the same metric tensor that raises and lowers indices of A b. In the second example, the tilt-reversal fails because the first index of R ab is associated with the x'-space metric tensor g' ij. In the third example, both indices are valid tensor indices (with the same metric tensor). The fourth example shows a failure of the tilt-reversal rule and this example is very important. The inequality becomes an equality only if the underlying tr ansformation F is linear so that R and S and g are then constants independent of positi on. For general F, such as the F involved in curvilinear coordinate transformations, the object ∂ aAb is not a rank-2 tensor and so the object ∂aAa does not represent contraction of two true tensor indices and therefore the "contraction tilt-reversal rule" does not apply. The neutralization rule of Section 7.12 below al so does not apply for this same reason, so ∂aAa does not transform as a scalar under general F. Here is one more example along the lines of the fourth example above that shows the potential danger of reversing a tilt when it is not justified. In this exam ple, we use notation to be introduced in Section 7.16 below. Consider the equation, V a = εabcBb;c ( 1 ) / / v a l i d where Chapter 7: Standard Notation 119 B b = a tensorial vector B b;c = ∂cBb – Γn bcBn = the covariant derivative of vector B b = a rank-2 tensor εabc = a rank-3 tensor density (weight -1) (the Levi-Civita tensor) Va = a vector density (weight -1) (7.11.6) With regard to B b;c : (a) in comma notation one writes ∂cBb = Bb,c ; (b) Γn bc = Γn cb . Since all indices on equation (1) are tensor indices, on e can lower index a and reverse the b and c tilts to get V a = εabcBb;c . (2) // valid (7.11.7) Now go back to equation (1). Because ε abc is antisymmetric on b and c, whereas Γn bc is symmetric on b and c, one can write εabcBb;c = εabcBb,c since the Γ term vanishes by symmetry. Thus one gets V a = εabcBb,c . (3) // valid (7.11.8) Were one to blindly lower a and reve rse the b and c tilts, one would get V a = εabcBb,c . (4) // NOT valid (7.11.9) The reason for "not valid" is that the reversal of the b tilt is not justified. That is, V a = εabcBb,c = εabc∂cBb = gbb' εa b'c ∂c(gbb"Bb") ⇒ Va = gbb' εab'c ∂c(gbb"Bb") = gbb' εab'c ∂c(gbb"Bb") // c tilt reversal is OK = g bb'gbb" εab'c(∂c Bb") + gbb' εab'c (∂c gbb") Bb" = δ b' b" εab'c(∂c Bb") + gbb' εab'c (∂c gbb") Bb" = εabc(∂cBb) + gbb' εab'c (∂c gbb") Bb" = εabcBb,c + gbb' εab'c (∂c gbb") Bb" = εabcBb,c + extra term! (7.11.10) It is due to this extra term that (4) is not valid. Basi cally this is the same as the fourth example above, but the situation is embedded in a more complicated e nvironment (extra tensors, tensor densities, comma notation, covariant derivatives, other tilted indices, et c). One way to summarize the example is this: (Va = εabcBb,c) ⇔ (Va = εabcBb;c) ⇔ (Va = εabcBb;c) ⇔ / (Va = εabcBb,c) (7.11.11) 7.12 The Contraction Neutralization Rule A contracted index pair plays no role in how a te nsor object transforms; the two contracted indices neutralize each other, as we now show. (7.12.1) Chapter 7: Standard Notation 120 First, recall that the indices on a general rank-n tens or (perhaps formed from several tensors) transform the same way that an outer product of n vectors tran sforms, where the vector index types match those of the tensor. The vectors transform this way, V'a = Ra bVb V ' a = Sb aVb . (7.5.3) and (7.5.4) We take the same "big object" [----- a---------a----] appearing in (7.11.1) and now ask how it transforms. Below the X's represent either R or S factors for the dash indices (each of which might be up or down): [-----a---------a----]' = XXXXX Ra b XXXXXXXXX Sc a XXXX [-----b---------c----] = S c a Ra b XXXXX XXXXXXXXX XXXX [-----b---------c----] = δc b XXXXX XXXXXXXXX XXXX [-----b---------c----] // RS = 1 = XXXXX XXXXXXXXX XXXX [-----a---------a----] (7.12.2) where now the only X's left are for the other indices. Again we look at our canonical example, AaYa = AaYa = A•Y . (7.4.14) The contracted vector indices neutralize each other and the resulting object transforms as a scalar. Here are some examples of tensor transforma tions with 0,1 and 2 index pairs contracted: T 'abc de = Ra a' Rb b' Rc c' Sd' d Se' e Ta'b'c' d'e' // no pairs contracted T 'abc ae = Rb b' Rc c' Se' e Ta'b'c' a'e' // index a contracted T 'abc ab = Rc c' Ta'b'c' a'e' // index a and index b contracted Q' = Q where Q = AaYa and Q' = A'aY'a // index a contracted (7.12.3) This shows the idea that one can take a larger tensor like T abc de and form from it smaller (lower rank) tensors by contracting tilted pairs of indices. In the above example list we really have D bc e ≡ Tabc ae = a mixed rank-3 tensor E c = Tabc ab = a contravariant vector (rank-1 tensor) Q = A aYa = a scalar (rank-0 tensor) . (7.12.4) It is similarly possible to build larger tensors from smaller ones, for example Z abc de = Va We gab Lc ( 7 . 1 2 . 5 ) which goes under the same rubric "outer product" mentioned earlier. Chapter 7: Standard Notation 121 7.13 The tangent and reciprocal base vectors and expansions on same Tangent and reciprocal base vectors Here are some basic translations. In the first four lines, n is a label, i is a tensor index: ( en)i → ( en)i // contravariant index i ( e¯n)i → ( en)i // covariant index i (En)i → ( en)i // contravariant index i (E¯n)i → ( en)i // covariant index i ( en)i = Sin (3.2.5) → ( en)i = Si n = Rni // contravariant index i ( E¯n)i = Rni (6.3.2) → ( en)i = Sin = Rn i // covariant index i ( En)i = Rnkgki (6.1.4) → ( en)i = Rn kgki = Rni // contravariant index i Σn(E¯n)a(en)b = δb,a (6.2.16) → Σn(en)a(en)b = δb,a // completeness Σ n E¯n enT = 1 (6.2.24) → Σn en enT = 1 // completeness (matrix form) (7.13.1) As noted earlier, writing a vector in bold such as en is not enough to say whether the vector is contravariant or covariant. Here we use an index position marker * on the right. For example, ( e1)* is a column vector of contravariant components, matching Si n = (en)i from (7.13.1). S = [ e1, e2, e3 .... eN ] (3.2.7) → S* * = [(e1)*, (e2)*, (e3)* .... (eN)*] (7.13.2) R = [ E¯1, E¯2, E¯3 .... E¯N ]T (6.3.3) → R* * = [( e1)*, (e2)*, (e3)* .... (eN)*]T (7.13.3) The relationship between en and en is very simple. Start with (6.1.2) and use (7.4.1) g' ab → g'ab to get the first result below. Then apply g' mn to the first result so g' mnen = g'mn g'ni ei = δmi ei = en which is the second result: En ≡ g'ni ei → en = g'ni ei and en = g'ni ei . (7.13.4) For either contravariant or covarian t indices (indices are not shown!), g' ni raises the label on ei, and inverting one finds that g' ni lowers the label on ei. This fact makes things easy to remember. Using the fact (3.2.6) that ei = ∂'ix , one has g'ni ei = g'ni ∂'ix = ∂'nx so the above line can be expressed as En ≡ g'ni ei → en = ∂'nx and en = ∂'nx . (7.13.5) Chapter 7: Standard Notation 122 The dot products (6.2.4) translate this way: en • em = g¯'nm → en • em = g'nm = ∂'nx • ∂'mx | en| = g'nn = h'n En • em = δn,m → en • em = δn m = ∂'nx • ∂'mx En • Em = g'nm → en • em = g'nm = ∂'nx • ∂'mx |en| = g'nn (7.13.6) The "labels" on the base vectors behave in this dot product structure the same way that up and down "indices" behave. This is the motivation for En → en . Thus, the three final equations can be regarded as the same equation en • em = g'nm where we can raise either or bot h indices/labels to get the other equations. For example, en • em = g'n m = δn m . The dot product (6.1.7) becomes En • um = Rnm → en • um = Rn m // = < en | um> in bra-ket notation (App E (g)) (7.13.7) The matrix Rn m is a "basis change matrix" between basis uk and basis ek . Inverse tangent and reciprocal base vectors Using the rules given above in (6.5.2), g'↔ g R ↔ S e n → u'n e'n → un En → U'n E'n → Un (6.5.2) we can obtain from (7.13.1) and (7.13.6) the corres ponding results for the inverse tangent and reciprocal base vectors: Chapter 7: Standard Notation 123 ( u'n)i → ( u'n)i // contravariant index i ( u¯'n)i → ( u'n)i // covariant index i (U'n)i → ( u'n)i // contravariant index i (U¯'n)i → ( u'n)i // covariant index i ( u'n)i = Rin → ( u'n)i = Ri n = Sni // contravariant index i ( U¯'n)i = Sni → ( u'n)i = Rin = Sn i // covariant index i ( U'n)i = Snkg'ki → ( u'n)i = Sn kg'ki = Sni // contravariant index i Σn(U'¯n)a(u'n)b = δb,a (6.2.16) → Σn(u'n)a(u'n)b = δb,a // completeness R = [ u'1, u'2, u'3 .... u'N ] → R* * = [(u'1)*, (u'2)*, (u'3)* .... (u'N)*] S = [ U¯'1, U¯'2, U¯'3 .... U¯'N ]T → S* * = [( U¯'1)*, (U¯'2)*, (U¯'3)* .... (U¯'N)*]T U'n ≡ gni u'n → u'n = gni u'i and u'n = gni u'i u'n • u'm = g¯nm → u'n • u'm = gnm U'n • u'm = δn,m → u'n • u'm = δn m U'n • U'm = gnm → u'n • u'm = gnm ( 7 . 1 3 . 8 ) Summary table. The summary table given in (6.5.9) was this x'-space x-space (6.5.9) axis-aligned basis vectors e'n un ( e'n)i= δn,i ( un)i= δn,i dual partners to the above E'n Un (E'n)i = g'ni ( Un)i = gni tangent base vectors u'n en ( u'n)i= Rin ( en)i = Sin reciprocal base vectors U'n En ( U'n)i = g'ia Sna ( En)i = gia Rna = g naRia = g' naSia We translate this entire table into Standard Notation: x'-space x-space (7.13.9) axis-aligned basis vectors e'n un ( e'n)i= δni ( un)i = δni dual partners to the above e'n un (e'n)i = g'ni ( un)i = gni tangent base vectors u'n en ( u'n)i = Ri n ( en)i = Si n= Rni reciprocal base vectors u'n en ( u'n)i = g'ia Sn a ( en)i = gia Rn a ( u'n)i = Sn i ( en)i = Rn i Chapter 7: Standard Notation 124 x-space expansions The x-space expansions of (6.6.9) V = V1 u1 + V2 u2 +... = ΣnVn un where Un • V = Vn Un = gni ui V = V¯1 U1 + V¯2 U2 +... = ΣnV¯n Un where un • V = V¯n V = V'1 e1 + V'2 e2 +... = Σn V'n en where En • V = V'n En = g'ni ei V = V¯'1 E1 + V¯'2 E2 +... = Σn V¯'n En where en • V = V¯'n (6.6.9) translate into the following : V = V1 u1 + V2 u2 +... = ΣnVn un where un • V = Vn un = gni ui V = V1 u1 + V2 u2 +... = ΣnVn un where un • V = Vn V = V'1e1 + V'2e2 +... = Σn V'n en where en • V = V'n en = g'ni ei V = V'1e1 + V'2 e2 +... = Σn V'n en where en • V = V'n (7.13.10) x'-space expansions Similarly, the x'-space e xpansions of (6.6.15) V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where E'n • V' = V'n E'n = g'ni e'i V ' = V¯'1 E'1 + V¯'2 E'2 +... = ΣnV¯'n E'n where e'n • V' = V¯'n V' = V1u'1 + V2u'2 +... = Σn Vn u'n where U'n • V' = Vn U'n = gni u'i V' = V¯1U'1 + V¯2U'2 +... = Σn V¯n U'n where u'n • V' = V¯n (6.6.15) translate into the following : V' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where e'n • V' = V'n e'n = g'ni e'i V ' = V'1 e'1 + V'2 e'2 +... = ΣnV'n e'n where e'n • V' = V'n V' = V1u'1 + V2u'2 +... = Σn Vn u'n where u'n • V' = Vn u'n = gni u'i V' = V1u'1 + V2u'2 +... = Σn Vn u'n where u'n • V' = Vn (7.13.11) Summary of all expansions: Using implied sum notation, we can now summarize the eight expansions above, plus the unit vector expansion onto e^n, on just two lines : V = Vn un = Vn un = V'n en = V'n en = V'n e^n // x-space expansions, ⇒ V'n = h'nV'n V' = V'n e'n = V'n e'n = Vn u'n = Vn u'n . // x'-space e xpansions (7.13.12) In all cases one sees a tilted index summation where one index is a vector index and the other is a basis vector label. Half the forms shown above can be obtained from the others by just "reversing the tilt". The power of the Standard Notation makes it self felt in relations like these. Chapter 7: Standard Notation 125 Due to this tilt situation, sometimes a basis like { en} appearing in V = V'n en is called a "covariant basis" while the basis { en} appearing in V = V'n en is called a "contravariant basis". Corresponding expansions of higher rank tens ors are presented in Section 7.17 below. If V is a tensor density of weight W (see Appendix D and E) the rule for adjusting the above expansions is to make the replacement V'n → JW V'n and V'n → JW V'n where J is the Jacobian of (5.12.6). 7.14 Comment on Covariant versus Contravariant Consider this expansion for a vector V in x-space, V = Vnbn Vn = V • bn (7.14.1) where bn is some basis having dual basis bn where as usual bn • bm = δnm, see Section 6.2. Imagine taking Vn → V'n = Rn m Vm and bi → bi' = Qij bj. What Q would cause the following to be true? V = Vnbn = V'nb'n . ( 7 . 1 4 . 2 ) In other words, how does one transform that basis bn such that the vector V remains unchanged if Vn is transformed contravariantly? The answer to this question is that Q ij = Rij since then (using R orthogonality rule #2 of (7.6.4) ) V'nb'n = [Rn m Vm][ Rnj bj] = (Rn m Rnj) Vm bj = δmj Vm bj = Vj bj = Vnbn . (7.14.3) Compare then the transformation of V n with that of the basis bn: V 'n = Rn m Vm bn' = Rnm bm ( 7 . 1 4 . 4 ) The V m vector components transform with Rn m but the basis vectors have to transform with R nm to maintain the invariance of the vector V. One varies with the down-tilt R, while the other varies with the up-tilt R, so the two objects are varying against each other in this tilt sense. They are "contra-varying", so one refers to the components V m as contravariant components with respect to the basis bm . If one starts over with V n components and the bn "dual" (reciprocal) expansion vectors and asks for a solution to this corresponding problem, V = Vnbn = V'nb'n ( 7 . 1 4 . 5 ) one finds not surprisingly that the dual basis must vary as bn' = Rn m bm and then one has Chapter 7: Standard Notation 126 V' n = Rnm Vm bn' = Rn m bm ( 7 . 1 4 . 6 ) which is the previous result with all indices up ↔down. Comparing the tilts, one would say that the V m again "contra vary" with the way the bm vary to maintain invariance of V. But one does not care about the dual basis, one cares about the basis , so relative to the basis bn one has V' n = Rnm Vm bn' = Rnm bm . ( 7 . 1 4 . 7 ) If the basis bm is varied as shown in (7.14.7), then the dual basis bm varies as in (7.14.6) and V of (7.14.5) remains invariant. Comparing now the way the V m transform with the way the basis vectors bm transform in (7.14.7), one sees that both equations have the same tilted R nm. They are "co-varying", so one refers to the components V m as covariant components with respect to the basis bm . 7.15 The Significance of Tensor Analysis "Why is tensor analysis important?", the reader might ask in the midst of this storm of index shuffling. Now is a good time to answer the question. Consider the following sample equation in x-space, where the fields Q, H, T and B may or may not be tensor fields: Q ad c(x) = Hab(x)Tb c(x) Bd(x) . ( 7 . 1 5 . 1 ) Notice that when contracted indices are ignored, the re maining indices have the same type on both sides. If the various objects really were tensors, one would say this was a "valid tensor equation" based on the index structure just described. One says that an equation is "covariant with respect to transformation x' = F(x)" if the equation has exactly the same form in x'-space that it has in x-space , which for our example would be Q'ad c(x') = H'ab(x')T 'b c(x') B'd(x' ) . ( 7 . 1 5 . 2 ) Here the word "covariant" has a new meaning, differ ent from its being a type of vector or index. The meaning is related in the sense that, comparing th e above two equations, everything has "moved" in the same manner ("co-varied") under the transformation. (S ome authors think the word "invariant" is more appropriate; Ricci and Levi-Civita used the term "absolute"; continuum mechanics uses the term "frame-indifferent".) If the objects Q, H, T and B are tensors under F, then covariance of any valid tensor equation like the one shown above is guaranteed !! The reason is that, once the contracted indices on the two sides are ignored according to the "contraction neutralization rule" (7.12.1), the object s on the two sides of the equation have the same indices which are of the same type, so both sides ar e tensors of the same type, and therefore both sides transform from x-space to x'-space in the same way. If one starts, for example, with the primed equation and installs the known transformati ons for all the pieces, one ends up with the unprimed equation. Chapter 7: Standard Notation 127 If this explanation is not convincing, a brute fo rce demonstration can perhaps help out. The following is also a good exercise is using the two tilt fo rms of the R matrix. Recall from (7.5.13) that Sb a = Rab and that SR = 1 is replaced by the va rious orthogonality rules (7.6.4). We shall process the primed equation into the unprimed one : Q'ad c(x') = H'ab(x')T 'b c(x') B'd(x') // x'-space equation (7.15.3) [R aa'Rd d'Rcc'Qa'd' c'(x)] = [Raa'Rbb' Ha'b'(x)] [Rb b"Rcc' Tb" c'(x) ] [Rd d'Bd'(x)] = R aa' Rd d' Rcc'(Rbb' Rb b") Ha'b'(x) Tb" c'(x)Bd'(x) . (7.15.4) Using orthogonality rule #1 of (7.6.4) we continue = R aa' Rd d' Rcc'(δb' b") Ha'b'(x) Tb" c'(x) Bd'(x) = R aa' Rd d' Rcc' Ha'b'(x) Tb' c'(x) Bd'(x) (7.15.5) so that, using the fact that Q is a tensor to replace Q' on the left side of (7.15.3), (R aa'Rd d'Rcc') Qa'd' c'(x) = (Raa' Rd d' Rcc') Ha'b'(x) Tb' c'(x) Bd'(x) . (7.15.6) Now apply the Cancellation Rule (7.6.1 2) three times to conclude that Q a'd' c'(x) = Ha'b'(x) Tb' c'(x) Bd'(x) ( 7 . 1 5 . 7 ) and then remove all primes on indices to get Q ad c(x) = Hab(x)Tb c(x) Bd(x) . // x-space equation (7.15.8) Thus it has been shown that, if all the objects tran sform as tensors, the equation is covariant. Tensor density equations are also covariant. As discussed in Appendix D, a tensor density of weight W is a generalization of a tensor which has the same transf ormation rule as a regular tensor, but there is an extra factor of J-W on the right hand side of the rule, where J is the Jacobian J = detS. For example, Q'ad c(x') = J-WQ Raa'Rd d'Rcc'Qa'd' c'(x) ( 7 . 1 5 . 9 ) would indicate that Q was a tensor density of weight W Q. If WQ = 0, then Q is a regular tensor. With this definition in mind, it is easy to generalize the notion of a "covariant equation" to include tensor densities. Consider some arbitrary tensor equation which we represent by our example above, Q ad c(x) = Hab(x)Tb c(x) Bd(x) . (7.15.8) Suppose all four objects Q, H, T, B are tensor densities with weights W Q, WH, WT, WB. If the four objects Q, H, T, B are tensor densities, and if the up/down free indices match on both sides (the non-contracted Chapter 7: Standard Notation 128 indices), and if WQ = WH + WT + WB, then this is a "valid tensor density equation" and covariance is guaranteed, so it follows that Q'ad c(x') = H'ab(x')T 'b c(x') B'd(x' ) . (7.15.3) It is trivial to edit the above proof by just adding weig ht factors in the right places and then of course they cancel out on the two sides. Examples of covariant tensor equations: In special relativity, which happens to involve linear Lorentz transformations, a fundamental principle is that any "equation of motion" describing anything at all (particles, EM fields, etc) must be covariant with respect to Lorentz transformations, or it cannot be a valid equation of motion (ignoring general relativity). An equation of motion must look the same in a reference frame which is rotated, boosted, or related by any combination of boosts and rotations to some original frame of reference (see Section 5.14)). As was noted earlier, the tradition is to write 4-v ector indices as Greek letters and 3-vector spatial indices as Latin letters. For example, we can define the "electromagnetic field-strength tensor" (rank-2) this way in terms of the 4-vector "vector potential" A μ: Fμν ≡ ∂μAν - ∂νAμ with Aμ = ( 1 c φ, A) (7.15.10) where ∂μ means gμα∂α, the contravariant form of the gradie nt operator. The components are then (7.15.11) where c is the speed of light and of course E and B are the electric and magnetic fields. Maxwell's two inhomogeneous equations (that is, the two with sources) are, in SI units where ε0μ0= 1/c2, ∂νFμν = μ0 Jμ with Jμ = (cρ,J) (7.15.12) while the two homogeneous equations become ∂αFμν + ∂μFνα + ∂νFαμ = 0 or ∂αFμν + cyclic = 0 . (7.15.13) Comment : With some effort, one can show that (7.15.12) says curl B - μ0ε0∂tE = μ0J and div E = ρ/ε0 . Similarly, (7.15.13) says that curl E + ∂tB = 0 and div B = 0, where B = curl A. A is the vector potential, φ is the scalar potential, J is current density, ρ is charge density, all in SI units. One can see that each of the above equations involves only tensors and we expect that in x'-space these equations will take the form Chapter 7: Standard Notation 129 ∂'νF 'μν = μ0 J'μ with J'μ = (cρ',J') and A'μ = ( 1 c φ', A') ∂'αF 'μν + ∂'μF 'να + ∂'νF 'αμ = 0 or ∂'αF 'μν + cyclic = 0 . (7.15.14) Objects like ∂νFμν and ∂αFμν are true rank-3 tensors because the transformation F is linear. Notice that both sides of (7.15.12) transform as a contravariant vector, while both sides of (7.15.13) transform as a covariant rank-3 tensor (the right si de is the all-zero rank-3 tensor). Covariance of tensor equations involving derivatives with non-linear F. A tensor equation which involves derivatives of tensors is non-covariant under transf ormations F which are non-linear. The reason is that the derivative of a tensor is, in that case, not a tensor , as shown in the next Section. Such tensor equations can be made covariant by replacement of all derivatives by covariant derivatives (which are indicated by a semicolon). In general relativity, this is known as the Principle of General Covariance (Weinberg p 106). A simple example is the tensor equation g ab;c = 0 shown in (F.9.13). Examples relating to the transformation from Cartesian to curvilin ear coordinates appear in Chapter 15. 7.16 The Christoffel Business: covariant derivatives This subject is treated in full detail in Appendix F, but here we provide some motivation. It should be noted that a normal derivative is sometimes written ∂aVb = Vb,a with a comma, whereas the covariant derivative discussed below is written V b;a with a semicolon. When a transformation F is non-linear, the matrix Ra b is a function of x. Thus one gets the following transformation for a lower index derivative of a covariant vector field component ∂aVb(x), where a "second term" quite logically appears, (∂'aV'b) = (Rad∂d) (RbcVc) = Rad Rbc (∂dVc) + Rad(∂d Rbc)Vc . (7.16.1) This second term did not arise in (2.4.3) where we looked at ∂ a on a scalar field φ'(x') = φ(x) , (∂' aφ') = (Rad∂d) φ = Rad (∂d φ) . ( 7 . 1 6 . 2 ) In special relativity, for example, where transformations are linear, ∂ d Rbc = 0, there is no second term, and the object ∂aVb transforms as a covariant rank-2 tensor, (∂' aV'b) = Rad Rbc (∂dVc) . // F is a linear transformation (7.16.3) But in the general case the second term is present, so ∂ dVc fails to transform as a rank-2 covariant tensor. In this case, one defines a certain "covarian t derivative" which itself has an extra piece Vb;a ≡ ∂aVb – Γk ab Vk ⇒ V d;c ≡ ∂cVd – Γk cd Vk (7.16.4) where Γc ab is a certain function of the metric tensor g. One then finds that Chapter 7: Standard Notation 130 V'b;a = Rad Rbc V d;c // the notation ∇cVd ≡ V d;c is also commonly used or [∂'aV'b – Γ 'k ab V'k] = Rad Rbc [∂dVc – Γk cd Vk] (7.16.5) so that this covariant derivative of a covariant vector field V c transforms as a covariant rank-2 tensor even with non-linear transformation F (see Christoffel Ref., 1869). This issue arises in general relativity and elsewhere. The object Γc ab (sometimes called the "Christo ffel connection") is given by Γc ab ≡ {ab,c} ≡ ⎩⎨⎧ ⎭⎬⎫c ab ≡ gcd [ab,d] = ½ gcd( ∂agbd + ∂bgad – ∂dgab ) // Christoffel 2nd kind ( 7 . 1 6 . 6 ) Γ dab ≡ [ab,d] ≡ ½ ( ∂agbd + ∂bgad – ∂dgab ) // Christoffel 1st kind and this is where the various "Christoffel symbols" come into play. In general relativity and elsewhere, Γc ab is known as the "affine connection" which represen ts the effect of "curved space" appearing as a force which acts on a mass (that is to say, a gravitational force), see Section 5.15. Warning : There is a differently defined version of Γc ab floating around in the literature. The version used above and everywhere in this document is th at of Weinberg and is the most common form. The derivative of any tensor field other than a scalar fiel d shows this same complication when the underlying transformation F is non-linear. For example, ∂agbd(x) does not transform as a rank-3 tensor, ∂' ag'bd(x') = (Rad∂d)(Rbb'Rdd'gb'd') = RadRbb'Rdd'(∂d gb'd') + other terms (7.16.7) and therefore neither of the Christoffel symbols Γ dab or Γc ab transforms as a tensor in this case. See Appendix F for more detail. 7.17 Expansions of higher order tensors Appendix E clarifies the use of direct product and polyadic notations for describing the basis vector combinations onto which higher order tensors can be expanded in a simple generalization of the vector expansions presented in Section 7.13 above, summarized in (7.13.12). There it was shown that a vector A can be expanded in two interesting ways : A = Σn An un An are the contravariant components of A in x-space A = Σn A'n en . A'n are the contravariant components of A in x'-space (7.17.1) In the first, un are axis-aligned basis vectors, and in the second en are the tangent base vectors. If A is instead a tensor of rank n, thes e expansions are replaced by A = Σijk... Aijk... (ui⊗uj⊗uk...) Aijk... are the contravariant components of A in x-space A = Σijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... are the contravariant components of A in x'-space ( 7 . 1 7 . 2 ) Chapter 7: Standard Notation 131 where there are n indices in each sum, n factors in th e direct products, and n contravariant indices on the components of tensors A in x-space and in x'-space. In the polyadic notation the direct-product crosses are eliminated giving A = Σ ijk... Aijk... uiujuk... Aijk... are the contravariant components of A in x-space A = Σijk... A'ijk... eiejek... A'ijk... are the contravariant components of A in x'-space . (7.17.3) In the case of rank-2 tensors, a product like ui⊗uj = uiuj is called a dyadic (see Section E.4). In this case (only) the product can be visualized as uiuT j which is a matrix constructed from a column vector to the left of a row vector. Thus one can write A = Σij Aij uiuT j Aij are the contravariant components of A in x-space A = Σij A'ij eieT j . A'ij are the contravariant components of A in x'-space . (7.17.4) Section E.7 promotes the interpretation of a rank-2 te nsor A as an operator in a Hilbert space, where the matrices Aij and A'ij are matrices associated with the operator A in different bases, Anm = <un | A | um > = the x-space components of tensor A A'nm = <en | A | em > = the x'-space components of tensor A . (7.17.5) These expansion methods are used in Appendices G and H to derive curvilinear expressions for two objects that play a role in continuum mechanics, ( ∇v) and div(T) (where T is a tensor). 7.18 Collection of Facts about basis vectors e n , u'n and bn. Recall this Developmental Notation bl ock of information regarding the en family of basis vectors, (6.4.1), ( e¯n)i = g¯ij (en)j (E¯n)i = g¯ij (En)j [ e¯n = g¯ en E¯n = g¯ En] (5.8.4) (e'n)i = Rij(en)j (E'n)i = Rij(En)j [ e'n = R en E'n = R En ] (e¯'n)i = Sji(e¯n)j (E¯'n)i = Sji(E¯n)j [ e¯'n = ST e¯n E¯'n = ST E¯n ] (2.5.1) ( en)i = Sin ( En)i = gijRnj = g'njSij (en')i = δi,n ( En')i = g'ni (e¯n)i = g¯ijSjn = Rjig¯'jn (E¯n)i = Rni ( e¯n')i = g¯'ni ( E¯n')i = δn,i (6.3.3) (6.3.9) en • em = g¯'nm ⇒ | en| = g¯'nn = h'n (scale factor) En = g'ni ei En • em = δn,m en = g¯'ni Ei En • Em = g'nm ⇒ | En| = g'nn . (6.2.4) Chapter 7: Standard Notation 132 e'n • e'm = g¯'nm ⇒ | e'n| = g¯'nn = h'n (scale factor) E'n = g'ni e'i E'n • e'm = δn,m e'n = g¯'ni E'i E'n • E'm = g'nm ⇒ | E'n| = g'nn . (6.2.7) ( E¯n)i(en)j = δi,j [ Σ n E¯n enT = 1 ] (6.2.16) and (6.2.24) DN (6.4.1) We shall now manually translate this entire block into Standard Notation. In doing so, we use gab → gab // the contravariant rank-2 metr ic tensor g (and similarly for g') g¯ab → gab // the covariant rank-2 metric tensor g ¯ (7.4.1) R ij → Ri j (7.5.2) Sik → Si k (7.5.4) Si j = Rji (7.5.13) and (7.13.1) for everything else. Matrix equati ons are ignored and only their component forms are translated. Here is the result, ( en)i = gij (en)j ( en)i = gij (en)j (5.8.4) (e'n)i = Ri j(en)j ( e'n)i = Ri j(en)j (e'n)i = Sj i(en)j = Rij(en)j ( e'n)i = Sj i(en)j = Rij(en)j (2.5.1) ( en)i = Si n = Rni ( en)i = gijRn j = g'njSi j (e'n)i = δni ( e'n)i = g'ni (en)i = gijSj n = Rj ig'jn (en)i = Rn i ( e'n)i = g'ni ( e'n)i = δn i (6.3.3) (6.3.9) en • em = g'nm ⇒ | en| = g'nn = h'n (scale factor) en = g'ni ei en • em = δn m en = g'ni ei en • em = g'nm ⇒ | en| = g'nn . (6.2.4) e'n • e'm = g'nm ⇒ | e'n| = g'nn = h'n (scale factor) e'n = g'ni e'i e'n • e'm = δn m e'n = g'ni e'i e'n • e'm = g'nm ⇒ | e'n| = g'nn . (6.2.7) ( en)i(en)j = δij (6.2.16) // completeness SN (7.18.1) Recall that { En, en} are a dual pair in the sense of Section 6.2, For the inverse transformation, we also had {Un', u'n} as a dual pair in Section 6.5, and the block of data about the u'n family of vectors was stated in (6.5.3) in developmental notation. We could translat e that into standard notation as we did the above, but it is easier to translate (7.18.1) using the rules (6.5.2) g'↔ g R ↔ S e n → u'n e'n → un En → U'n E'n → Un (6.5.2 ) Chapter 7: Standard Notation 133 translated to Standard Notation g'↔ g R ↔ S e n → u'n e'n → un en → u'n e'n → un (7.18.2) Applying these rules one at a time to (7.18.1) we obtain this data block for the u'n family, ( u'n)i = g'ij (u'n)j ( u'n)i = g'ij (u'n)j (5.8.4) (un)i = Si j(u'n)j ( un)i = Si j(u'n)j (un)i = Rj i(u'n)j = Sij(u'n)j ( un)i = Rj i(u'n)j = Sij(u'n)j (2.5.1) ( u'n)i = Ri n = Sni ( u'n)i = g'ijSn j = gnjRi j (un)i = δni ( un)i = gni (u'n)i = g'ijRj n = Sj igjn (u'n)i = Sn i= Rin (un)i = gni ( un)i = δn i (6.3.3) (6.3.9) u'n • u'm = gnm ⇒ | u'n| = gnn = hn (scale factor) u'n = gni u'i u'n • u'm = δn m u'n = gni u'i u'n • u'm = gnm ⇒ | u'n| = gnn . (6.2.4) un • um = gnm ⇒ | un| = gnn = hn (scale factor) un = gni ui un • um = δn m un = gni un un • um = gnm ⇒ | un| = gnn . (6.2.7) ( u'n)i(u'n)j = δij (6.2.16) // completeness SN (7.18.3) Finally, we shall restate (7.18.1) for the generic dual pair { Bn, bn} of Section 6.2. Without detailed proof, we claim that one can in gene ral construct a transformation x' = Fb(x) for which some generic complete set of vectors bn(x) are the tangent base vectors like en(x) are for x' = F(x). For this transformation Fb, one would have ( bn)i = (Sb(x))in from (3.2.6), so matrix S b(x) would be known. Then R b = (Sb)-1 is also known, and one would then integrate (2.1.6) d x' = Rb(x) dx to obtain a viable Fb(x). As discussed in Section 6.2, for this given set of { Bn, bn}, we defined W' nk ≡ bn•bk in developmental notation which is analogous to g ¯'nk = en • ek for F(x). Thus suggests we rename W' →w¯' and then we have g ¯'nk = en • ek for F(x) and w ¯'nk = bn • bk for Fb(x), where w ¯' = W. In Standard Notation these equations become g' nk = en • ek and w'nk = bn • bk since W'nk = w¯'nk → w'nk like g ¯'nk → g'nk w'nk → w'nk g ' nk → g'nk . (7.18.4) So the rules for translating (7.18.1) from en to generic bn are are very simple : Chapter 7: Standard Notation 134 e → b // that is, en → bn, en → bn and the same for primed basis vectors g' → w' // that is, g' ij → w'ij and g'ij→ w'ij (7.18.5) Note that the arrows → in (7.18.4) are for DN →SN, whereas those in (7.18.5) are SN e → SN b. The result is ( bn)i = gij (bn)j ( bn)i = gij (bn)j (5.8.4) (b'n)i = Ri j(bn)j ( b'n)i = Ri j(bn)j (b'n)i = Sj i(bn)j = Rij(bn)j ( b'n)i = Sj i(bn)j = Rij(bn)j (2.5.1) ( bn)i = Si n = Rni ( bn)i = gijRn j = w'njSi j (bn')i = δni ( b'n)i = w'ni (bn)i = gijSj n = Rj iw'jn (bn)i = Rn i ( bn')i = w'ni ( b'n)i = δn i (6.3.3) (6.3.9) bn • bm = w'nm ⇒ | bn| = w'nn = h'n (scale factor) bn = w'ni bi bn • bm = δn m bn = w'ni bi bn • bm = w'nm ⇒ | bn| = w'nn . (6.2.4) b'n • b'm = w'nm ⇒ | b'n| = w'nn = h'n (scale factor) b'n = w'ni b'i b'n • b'm = δn m b'n = w'ni b'i b'n • b'm = w'nm ⇒ | b'n| = w'nn . (6.2.7) ( bn)i(bn)j = δij (6.2.16) // completeness SN (7.18.6) 7.19 More on basis vectors and matrix elements of R and S Ambiguity In the summary tables of Section 7.18 we see various equations like these, ( un)i = δni ( en)i = Rni ( e'n)i = δni ( u'n)i = Ri n . (7.19.1) There is a certain ambiguity in writing, for example, ( en)i : it is not clearly stated in what basis component i is evaluated! The basis depends on the expansion in which a component appears. For example, from (7.13.10) we write for a generic vector V, V = ΣiVi ui ui • V = Vi (7.13.10) and we then know that Vi is a component in the u-basis. Applying this to V = en (an x-space vector) we conclude that Chapter 7: Standard Notation 135 en = Σi (en)i ui ui • en = ( en)i . (7.19.2) Thus, ( en)i is in fact a component in the u-basis, not for example the e-basis. We would then write ( en)i = ( en)(u)i = Rni . ( 7 . 1 9 . 3 ) This is the "natural basis" to use for the components of en and for any x-space vector V where we use the natural expansion V = ΣiVi ui shown above. One could however expand en on the en basis to get en = Σi (en)(e)i ei ei • en = ( en)(e)i = δi n . (7.19.4) Then with no ambiguity one could write ( en)(u)i = Rni // natural ( en)(e)i = δni ( 7 . 1 9 . 5 ) Thus, when we write ( en)i, we imply that we are using the u-basis for the components. Since this applies to any x-space vector, it is also true for ( un)i, so ( un)i = (un)(u)i = ui • un . What about x'-space vectors? From (7.13.11) we write V' = Σi(V')i e'i e'i • V' = (V')i . (7.13.11) Applying this to V' = e'n (an x'-space vector) we conclude that e'n = Σi(e'n)i e'i e'i • e'n = ( e'n)i = δi n . (7.19.6) Thus, ( e'n)i is in fact a component in the e'-basis, not for example the u'-basis. We would then write ( e'n)i = ( e'n)(e')i = δni . ( 7 . 1 9 . 7 ) This is the "natural basis" to use for the components of e'n and for any x'-space vector V' where we use the natural expansion V' = Σi(V')i e'i shown above. One could however expand e'n on the u'n basis to get e'n = Σi (e'n)(u')i u'i u'i • e'n = ( e'n)(u')i = ui • en = ( en)i = Rni . (7.19.8) Then with no ambiguity one could write ( e'n)(e')i = δni // natural ( e'n)(u')i = Rni ( 7 . 1 9 . 9 ) Chapter 7: Standard Notation 136 Thus, when we write ( e'n)i, we imply that we are using the e'-basis for the components. Since this applies to any x'-space vector, it is also true for ( u'n)i, so ( u'n)i = (u'n)(e')i = e'i • u'n. We summarize the above discussion as follows: Fact : un and en components are by default presented in the u-basis (7.19.10) e'n and u'n components are by default presented in the e'-basis Table of basis vector dot products Since there are four kinds of basis vectors of interest, un, en, u'n and e'n with lower labels, one might imagine there are 16 scalar products of interest, u e u' e' u u• u u• e u•u' u•e' e e• u e•e e•u' e•e' u' u' •u u'•e u'•u' u'•e' e' e' •u e'• e e'•u' e'•e' However, since a scalar product only ex ists within a Hilbert space (x-sp ace or x'-space), the cross space entries in this table make no sense, so we eliminate them to get u e u' e' u u• u u• e - - e e• u e•e - - u' - - u' •u' u'•e' e' - - e' •u' e'•e' (7.19.11) Since u•e = e• u and u' •e' = e'•u', we see that there are only 6 distinct scalar products of interest. Collecting data from the tables (7.13.1) and (7.1 3.2) we summarize these 6 cases as follows: Table of basis vector dot products: ( un)i = ui • un = < ui | un > = gi n = u'i • u'n = < u'i | u'n > ( en)i = ui • en = < ui | en > = Si n = Rni ( en')i = e'i • e'n = < e'i | e'n > = g'i n = ei • en = < ei | en > ( un')i = e'i • u'n = < e'i | u'n> = Ri n = Sni (7.19.12) Chapter 7: Standard Notation 137 Since a • b = a' • b', the first and third lines each contain 2 of the 6 cases. We are careful to use the correct tensor notation g ni and g'ni for δni as discussed near equation (7.4.17). Doing this, we may state the following : Fact: In any of the above equation lines, one may raise the label n and/or lower the label/index i and end up with another valid equation line. (7.19.13) In this manner, from the 4 lines above one may generate 12 more lines of equations. Note that we have slipped in the Dirac notation < | > described in (E.7.4) as an alternate way to write the dot products. As an example of the above Fact, if we raise the la bel n in the third line we get these valid equations, ( en')i = e'i • en' = < e'i | en' > = g'in = ei • en = < ei | en > and if we then lower index i, we get another set of valid equations, ( en')i = e'i • en' = < e'i | en' > = g'in = ei • en = < ei | en > . To prove the above Fact, one first observes in whic h space a vector lies, so one knows which metric tensor raises and lowers the i ndex. Then one uses these facts from (7.18.1) and (7.18.3), en = g'ni ei e'n = g'ni e'i un = gni ui u'n = gni u'i en = g'ni ei e'n = g'ni e'i un = gni un u'n = gni u'i . (7.19.14) For example, consider the third line in (7.19.12), ( en)i = ui • en = < ui | en > = Rni . Apply g'mn to get g'mn (en)i = g'mn ui • en = g'mn <ui | en > = g'mn Rni or [g'mn en]i = ui • [g'mn en] = < ui | [g'mn] en > = g'mn Rni or ( em)i = ui • [em] = < ui | em > = Rmi // see (7.5.9) about Rmi which shows that the label n can be raised to get an other valid set of equations. Now start with the same third line in (7.19.12) and instead apply g ji to get gji(en)i = gjiui • en = gji<ui | en > = gjiRni or gji(en)i = [gjiui] • en = <gjiui | en > = gjiRni or ( en)j = uj • en = < uj | en > = Rnj // see (7.5.9) about R nj Chapter 7: Standard Notation 138 which shows that the index i can be lowere d to get another valid set of equations. Matrix elements of R and S One can regard the elements R i j and Si j as being certain matrix elements of operators R and S, and this is most easily handled in the Dirac bra-ket notation of (E.7.4). Since V' = R V (vector transformation rule) one sees that a matrix element <a' | R | b> = <a' | R b> must have | b> being an x-space vector, and <a' | being a transposed x'-space vector. We shall now compute matrix elements of R in several basis combinations. As examples of V' = R V we know that, e'n = R en u'n = R un or | e'n> = R |en> |u'n> = R |un> . ( 7 . 1 9 . 1 5 ) We can "close on the left" in various ways. For example, < e'i | e'n> = < e'i | R | en> = g'i n < u'i | e'n> = < u'i | R | en> = Si n = Rni (7.19.16) Here we look up the scalar products on the left of each line using table (7.19.12), making use of Fact (7.19.13), and adjusting the various index names and the up and down index sense. Now start instead with the second equation of (7.19.15) and close two ways, again looking up the basis vector scalar products in (7.19.12), < e'i |u'n> = < e'i | R | un> = Ri n = Sni < u'i |u'n> = < u'i | R | un> = gi n . ( 7 . 1 9 . 1 7 ) Here then are four R matrix elements of interest, < e'i | R | en> = g'i n < u'i | R | en> = Si n = Rni < e'i | R | un> = Ri n = Sni < u'i | R | un> = gi n . ( 7 . 1 9 . 1 8 ) Chapter 7: Standard Notation 139 To get matrix elements of S, we use the fact (7.9.8) that R = ST and the fact (7.9.17) that < a|S|b> = <b|ST|a> = < b|R|a>, so we translate the above four lines: < en | S | e'i> = < e'i | R | en> = g'i n < en | S | u'i> = < u'i | R | en> = Si n = Rni < un | S | e'i> = < e'i | R | un> = Ri n = Sni < un | S | u'i> = < u'i | R | un> = gi n . (7.19.19) and the above then is a full statement of all matrix elements of S and R, for a particular up down sense of the basis vector labels. As usual, g'i n = gi n = δi n but we maintain the true tensor form to allow for up down modifications as per the following claim: Fact : In the above equations, one can raise/lower la bels on either or both sides to get new valid equations. (7.19.20) The proof of this claim is the same as the proof of Fact (7.19.13) above using (7.19.14). For example, < u'i | R | un> = gi n so g ji<u'i | R | un> = gjigi n or < u'j | R | un> = gjn . Similarly < e'i | R | un> = Ri n = Sni ⇒ < e'i | R | un> = Rin = Sni. Interpretation of Ri n and Si n Since V' = R V, one can regard the operator R as a mapping R : X → X' (x-space → x'-space). R may be regarded as a "cross tensor" having one foot in each space. As shown in (7.19.19), R i j = < e'i | R | uj> = [ R(e',u)]i j (7.19.21) so one can regard R as an abstract cross-tensor be ing expanded as follows in the mixed-basis sense of (E.10.4), R = Σ ij [ R(e',u)]i j (e'i ⊗ uj) . ( 7 . 1 9 . 2 2 ) As usual, the coefficients (the cross tens or components) can be projected out using Chapter 7: Standard Notation 140 ( e'a ⊗ ub) • R = Σij [ R(e',u)]i j (e'a ⊗ ub) • (e'i ⊗ uj) = Σij [ R(e',u)]i j (e'a • e'i) ⊗ (ub • uj) = Σij [ R(e',u)]i j δa i δbj = [ R(e',u)]a b . ( 7 . 1 9 . 2 3 ) Similarly, since V = S V' one can regard the operator S as a mapping S : X' → X where, from (7.19.19), Si j = < ui | S | e'j> = [ S(u,e')]i j (7.19.24) and then the cross-tensor expans ion and projection is given by, S = Σij [ S(u,e')]i j (ui ⊗ e'j) ( ua ⊗ e'b) • S = [ S(u,e')]a b . ( 7 . 1 9 . 2 5 ) A Pitfall Example One knows from (7.19.19) that < u'i | R ua> = < u'i | R | ua> = gia = δia . This is also known from, < u'i | R ua> = < u'i | u'a> = δia . Consider now the following slight of hand wherein we get a different result, < u'i | R ua> = [R ua]i = Rij(ua)j = Rijδa j = Ria wrong! Where is the error being made here? Well, the vector u'a = R ua being in x'-space has a natural e'-type basis as discussed below (7.19.7), but here the component [R ua]i is in the u'-basis. When doing something "unnaturally" one must pay more attenti on to labels. The correct version of the above is < u'i | R ua> = [R ua](u') i = [R(u',u)]i j (ua)(u) j = [R(u',u)]i j δa j = [R(u',u)]i a = < u'i | R | ua> = gi a = δi a . (7.19.26) Now consider instead a different example. From (7.19.19) one knows that, < e'i | R ua> = < e'i | R | ua> = Ria . In this case write Chapter 7: Standard Notation 141 < e'i | R ua> = [R ua]i = Rijδa j = Ria and there is no problem because e' is the na tural basis for x'-space. In more detail < e'i | R ua> = [R ua](e') i = [R(e',u)]ij (ua)(u) j = [R(e',u)]ij δa j = [R(e',u)]ia = < e'i | R | ua> = Ria . (7.19.27) We find that the Dirac notation is useful because it provides a bulletproof formalism for avoiding ambiguities such as that of the previous example. The above equation sequence can be written (implied sum on j as usual) < e'i | R ua> = < e'i | R [1] ua> = < e'i | R [ | uj><uj | ] ua> = < e'i | R | uj><uj | ua> = [R(e',u)]ij (ua)(u) j (7.19.28) where we use the (E.7.4) completeness relation 1 = | uj><uj |. In fact x-space completeness: 1 = | uj><uj| = | uj><uj| = | ej><ej| = | ej><ej| x'-space completeness: 1 = | u'j><u'j| = | u'j><u'j| = | e'j><e'j| = | e'j><e'j| (7.19.29) There are just restatements of the (6.2.8) duality idea that bi • bj = < bi | bj > = δi j for any basis. Chapter 8: Length Area and Volume 142 8. Transformation of Differential Length, Area and Volume 8.1 Overview of Chapter 8 This Chapter and all remaining Chapters use th e Standard Notation intr oduced in Chapter 7. The term N-piped is short for N dimensional parallelepiped. The context is Picture B: (8.1.1) Since this Chapter is quite lengthy, a brief overview is in order: The transformation of differential length, area, and vol ume is first framed in terms of the mapping of an orthogonal differential N-pipe d in x'-space to a skewed differential N-piped in x-space. The N-piped in x'- space has axis-aligned edges of length dx' n, while the N-piped in x-space has edges endx'n where en are the tangent base vectors introduced in Chapter 3. We want to learn what happens to the edges, face areas and volume as one differential N-piped is mappe d into the other by the curvilinear coordinate transformation x' = F(x). After solving this problem, we go on to consider the transformations of arbitrary differential vectors, areas and volume. 8.2: The differential N-piped mapping The differential N-piped mapping is descri bed and various symbols are defined. 8.3: Properties of the finite N-piped spanned by the e n in x-space Results from Appendix B concerning finite N-piped geometric properties are quoted. Certain definition changes are made to make the formulas suitable for tensor analysis. The purpose of the lengthy Appendix B is to lend credence to the general formulas for elements of area and volume in N dimensions. 8.4: Back to the differential N-piped mappi ng: how edges, areas and volume transform (a) Setup. The finite N-piped edges en are scaled by curvilinear coordinate variations dx'n to create a differential N-piped in x-space having edges ( endx'n). (b) Edge Transformation. The edges en of the x-space N-piped map to axis-aligned edges e'n in x'-space. (c) Area Transformation. Tensor density methods as presen ted in Appendix D are used here. (d) Volume Transformation. The volume transformation is computed several different ways. (e) Covariant Magnitudes . These are | d x'(n)|, | dA'n | and | dV' | in the Curv ilinear View of x'-space. (f) Two Theorems. (1) g'nn g' = cof(g' nn) and (2) |( Πx i≠nei)| = cof(g'nn) . (g) Cartesian-View Magnitude Ratios. Appropriate for the continuum mechanics application. (h) Nested Cofactor Formulas and STS notation: Description of all "leaf" areas of an N-piped in both descending and ascending orders. Chapter 8: Length Area and Volume 143 (i) Transformation of arbitrary differential vectors, areas and volume. Having built confidence with the general form of vector area and volume expressions in the N-piped case, the N-piped is jettisoned and formulas for the transformation of arbitrary vectors, areas and volume are derived. (j) Concatenation of Transformations. What happens to the transformation of vectors, areas and volumes when two transformations are concat enated? One result is that J = J 1J2. (k) Examples of area magnitude transformation for N = 2,3,4 Example 2: Spherical Coordinates: area patches 8.5: Transformation of Differentia l Volume applied to Integration The volume transformation obtained in Section 8.4 is related to the traditional notion of the Jacobian changing the "measure" in an integration. The "Jacobi an Integration Rule" can then be expressed as a distributional equation. 8.6: Interpretations of the Jacobian 8.7: Volume integration of a tensor field under linear transformations Under suitable conditions, the volume integration of a te nsor field integrand yields a tensor of the same type. 8.2 The differential N-piped mapping Consider these relations involving a differential distance in the n-direction (dx'n > 0): d x'(n) = e'n dx'n x'-space axis-aligned differential vector, and ( e'n)i = δni (3.2.2) d x(n) = en dx'n x-space mapping of the above vector under F-1 or R-1 d x'(n) = R( x) dx(n) relation of the two differential vectors (contravariant rule) (2.1.6) e'n = R( x) en (3.3.2) ( 8 . 2 . 1 ) A superscript (n) on the differentials makes clear there is no implied sum on n. Note that application of R(x) to the second equation yields the first equation, making use of the last two equations, so the second equation really does contain dx' n. In more generality, recall V = Σn V'n en from (7.13.10). The vectors d x(n) span a differential N-piped in x-space, while the d x'(n) span a corresponding differential N-piped in x'-space. The two N-pipeds are related by the mapping x' = F(x). Since the regions are differentially small, this mapping is the same as the linearized mapping d x' = R d x. The metric tensor in x-space will be taken to be g = 1, so it is a Cartesian space. As discussed at the end of Section 5.16, the x'-space N-piped can be viewed in (at least) two ways depending on how the metric tensor g' is set. For a continuum mechanics flow application, one sets g' = 1 and this gives the Cartesian View of the x'-space N-piped. For such flows dot products and magnitudes of vectors like d x(n) are not invariant under the transformation. For our curvilinear coordinates application, however, we set g' = RgRT = RRT and this causes vector dot products and magnitudes to be invariant and we can talk about such objects as being tensorial s calars. This is the Curvilinear View of x'-space. Chapter 8: Length Area and Volume 144 When g' ≠ 1, it is impossible to accurately represent the Curvilinear-View picture of the x'-space N- piped as a drawing in physical space (for N=3). This subject is discussed for a sample 2D system in Section C.5. Although the basis vectors ( e'n)i = δni in x'-space are always axis-aligned, they are only orthogonal for an orthogonal coordinate system, since e'n•e'm = en•em = g'nm . Nevertheless, even for a non-orthogonal system we draw the axes as if they were orthogonal, which at least provides a representation of the notion of "axis aligned" basis vectors. For N > 3 one at least imagine this kind of drawing. Due to these graphical difficulties, in the drawing below the Cartesian View of x'-space is shown. Since g' = 1 for this situation, the axis-aligned basis vectors e'n are in fact unit vectors e^'n and are orthogonal, so the picture becomes at least comprehensible: ( 8 . 2 . 2 ) The orthogonal Cartesian-View N-piped allows visualization of these curvilinear coordinate variations, all dx' k > 0 ( objects below are primed because they exist in x'-space), d L'n ≡ dx'n length n = 1,2....N d A'n ≡ Πi≠ndx'i area d V' ≡ Πidx'i = d A'n dL'n . volume (no implied sum on n) (8.2.3) For example, for N=3 one would have d L'1 ≡ dx'1 d L'2 ≡ dx'2 dL'3 ≡ dx'3 d A'1 = dx'2dx'3 d A'2 = dx'3dx'1 d A'3 = dx'1dx'2 d V' = dx'1dx'2dx'3 = d A'1dL'1 = d A'2dL'2 = dA'3dL'3 . (8.2.4) The Cartesian-View x'-space N-piped is always orthogonal because the ( e'n) are orthonormal axis-aligned unit vectors (since g'=1). In contrast, the x-space N-pi ped is typically rotated a nd possibly skewed as well (if the coordinates x'i describe a non-orthogonal coordinate system). The transformation F and its Chapter 8: Length Area and Volume 145 linearized version R map the skewed x-space N-piped into the orthogonal x'-space N-piped. As one moves around in x-space so that point x changes, the picture on the left above keeps its shape, just translating itself to the new point x', but the picture on the right changes shape and volume because the vectors en(x) are functions of x. It is our goal to write expressions for edges, area s and volumes in these two spaces and to then show how these objects transform between th e two spaces. To this end, we sh all rely on work done in Appendix B which is summarized in the next Section. Followi ng that, we shall add to each edge a differential distance associated with that edge (such as er → erdr in spherical coordinate s), and that will bring us back to the differential N-piped picture above. 8.3 Properties of the finite N-piped spanned by the e n in x-space The finite N-piped spanned by the tangent base vectors en in x-space has the following properties (as shown in Appendix B) : • The N spanning edges are the vectors en which have lengths | en| = h'n (scale factors, (5.11.7) ). (8.3.1) • There are 2 N v e r t i c e s . ( 8 . 3 . 2 ) • There are N pairs of faces. The two faces of each pair are parallel in N dimensions. One face of each pair touches the point where the tails of all the en vectors meet (the near face) while the other does not touch this meeting point (the fa r face). (8.3.3) • Each face of an N-piped is an (N-1)-piped having 2N-1 vertices. The faces are planar surfaces of dimension N-1 embedded in an N dimensional space. (8.3.4) • A face's vector area An is spanned by all the ei except en and is labeled by this missing en vector. ( 8 . 3 . 5 ) • The far face has out-facing vector area An , and the near face has out-facing area vector – An. These vector areas are normal to the faces. (8.3.6) • The vector area An is given by several equivalent expressions: An = |det(Sa b)| en An = σ (-1)n-1 Πx i≠n ei An = σ (-1)n-1 e1 x e2 ... x eN // en missing σ ≡ sign[det(Sa b)] = sign[det(Ra b)] ( An)i = σ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // e n missing (8.3.7) • The volume of the N-piped is given by (see Section 5.12 concerning J) V = | det [ e1, e2, e3 ... eN] | = | det(Sa b) | = g'1/2 = |J| . (8.3.8) Chapter 8: Length Area and Volume 146 The mapping picture above does not apply to a finite N-piped. The finite N-piped just discussed exists in x-space. One might ponder into what shape it maps in x'-space under the transformation F. In the case of spherical coordinates, Fig (1.13), all of x-space maps into a certain orthogonal "office building" in x'- space. A finite N-piped within x-space maps into some very complicated 3D shape within the office building which is bounded by curves wh ich in general are not even coplanar. The point is that a finite N- piped in x-space does not map back into some nice orthogonal N- piped in x'-space and the picture drawn in the previous Section does not apply. However, when differentials are added in the next Section, then, since the mapped regions are very small, the mapping of the x-space differential N-piped is in fact an "orthogonal" N-piped in x'-space. This is because for a tiny region near some point x, the mapping between d x and d x' is described by the linear relation d x' = R( x) dx of (2.1.6). Conventions for defining the area vector and volume . In Appendix B (as reported in (8.3.6)) the area vector An is defined so that the out-facing normal of the x-space N-piped's "far face n" is An, regardless of the sign of det(S). This was done to simplify the computation of the flux of a vector field emerging from the N-piped in the geometric divergence calculation in Chapter 9. That calculation is then valid for either sign of det(S), a sign that we call σ. The following picture illustrates on the left an x-space 3-piped which is obtained by reverse-mapping the x'-space or thogonal 3-piped using an S which has det(S) > 0. On the right is the x-space 3-piped that results for S → -S. These two 3-pipeds are related by a parity inversion of all points through the origin. If the origin lies far away, these two N-pipeds lie far away from each other, a fact not illustrated in the picture: (8.3.9) Notice that A3 for the "far face 3" is outfacing in both cases. This definition of the vector area is not suitable for the vector analysis we are about to undertake. Instead of the above situ ation, we will redefine A3 = e1 x e2 for both pictures, and this will cause the A3 vector in the right picture to point up into the interi or of the N-piped. This new definition allows us to interpret A3 = e1 x e2 as a "valid vector equation" to which we may apply the ideas of covariance and tensor densities. Notice that under a parity transformation, this newly defined A3 is a "pseudovector" which is one which does not reverse direction under a parity transformation, since A3 = (- e1) x (- e2). The subject of parity and handedness and the sign of det(S) is discussed more in Section 6.9. Chapter 8: Length Area and Volume 147 Here then are the expressions for An with this new definition, where the new forms are obtained from (8.3.7) by multiplying by σ = sign ( det(S) ) : An = det(Sa b) en = J en ( 8 . 3 . 1 0 ) An = (-1)n-1 e1 x e2 ... x eN // en missing (8.3.11) ( An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // e n missing (8.3.12) The second line shows that pseudovector areas can on ly exist for an odd number of dimensions (such as N=3). A similar redefinition of the volume will now be do ne. In Appendix B the volume is defined so as to be a positive number regardless of σ with the result V = |det(S)|. We now redefine the volume by multiplication by σ, so that now V = det(S) which is of course a negative number when det(S) < 0, which in turn means the en are forming a left-handed coordinate system as per Section 6.9. So: V = det [ e1, e2, e3 ... eN] = det(Sa b) = J . (8.3.13) 8.4 Back to the differential N-piped mapping : how edges, areas and volume transform (a) The Setup If the edges of the finite N-piped described above are scaled by positive differentials dx' n > 0, the result is a differential N-piped in x-space which has these properties: d x(n) = en dx'n (8.2.1) / / e d g e s d An = J en (Πi≠ndx'i) (8.3.10) // areas dAn = (-1)n-1 (dx'1e1) x (dx'2e2) ... x (dx' NeN) (8.3.11) // en missing from cross product = (-1) n-1 e1 x e2 ... x eN (Πi≠ndx'i) // en missing from cross product = (-1) n-1 (Πx i≠nei) (Πi≠ndx'i) // shorthand notation of (A.10.5) (d An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) // en factor and index missing, (8.3.12) dV = det [dx' e1, dx'e2, dx'e3 ... dx' eN] = det [ e1, e2, e3 ... eN] (Πidx'i) (8.3.13) = det(S a b)(Πidx'i) = J (Πidx'i) // (7.13.2) and (5.12.6) = εabc..x (dx'e1)a(dx'e2)b..... (dx' eN)x // (5.12.8) ( 8 . 4 . a . 1 ) Chapter 8: Length Area and Volume 148 where d An and dV are obtained from An and V according to the new definitions described above. These equations apply to the right side of the N-piped ma pping picture Fig (8.2.2) which is replicated here : (8.2.2) For spherical coordinates, the N-piped on the right would be spanned by these vectors, see (3.4.6), erdr = r^dr, eθdθ= rθ^dθ eφdφ = rsinθ φ^dφ . (8.4.a.2) The reader is reminded that the vectors shown on the right of (8.2.2) exist in x-space, and that this x-space is assumed to be Cartesian for this entire Chapter, meaning g = 1 and V k= Vk for vectors. (b) Edge Transformation The edge d x(n) we know from (2.1.6) transforms as a tens orial vector under transformation F, so d x'(n) = R d x(n) where d x(n) = en dx'n (8.2.1) . (8.4.b.1) Making use of (7.18.1) and RS = 1, evaluation gives [d x'(n)]i = Ri j (en)j dx'n = Ri j Sj n dx'n = (RS)i n dx'n = δi n dx'n ⇒ d x'(n) = e'n dx'n since ( e'n)i = δni (8.4.b.2) so this contravariant edge points along the the n axis di rection in x'-space of Fig (8.2.2). This fact was stated above in (8.2.1), we are just exercising our notation. Chapter 8: Length Area and Volume 149 (c) Area Transformation How does d An transform under F? Looking at the component form stated above in (8.4.a.1), (d An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) // en factor and index missing (8.4.c.1) it is seen that dA n is a combination of tensor objects like εiabc..x and (e2)b. As discussed in Appendix D, a vector density of weight 0 is an ordinary vector such as e2 or d x. The ε tensor (rank-N) is a tensor density of weight -1. In forming more complicated tens or objects, the rule (D.2 .3) is that one adds the weights of the objects being combined. Theref ore, one may conclude that the object d An is a vector density of weight -1. This has the immediate implication that dAn transforms to x'-space under F according to the rule (D.1.4), d A'n = J R d An or (d A'n)i = J Ri j (dAn)j // J-W = J-(-1) = J (8.4.c.2) where J = det(S) is the Jacobian of Sec tion 5.12. Moreover, the above equation for (d An)i is a "valid tensor density equation" as per (7. 15.9) and is therefore covariant. This means that in x'-space the equation has the exact same form, but tensor objects are primed (the dx'i are constants), (d A'n)i = (-1)n-1 ε'iabc..x (e'1)a(e'2)b.... (e'N)x (Πi≠ndx'i) // e'n factor and index missing ( 8 . 4 . c . 3 ) Insertion of (D.5.13) that ε' iabc..x = J2 εiabc..x and (e'n)i = δni then gives (d A'n)i = (-1)n-1 J2 εiabc..x δ1aδ2b...δNx (Πi≠ndx'i) = (-1) n-1 J2 εi123..N (Πi≠ndx'i) // index n missing on ε = δ n i J2 (Πi≠ndx'i) ( 8 . 4 . c . 4 ) The last step follows from the fact that ε i123..N with n missing must vanish if i ≠n, and if i=n then εn123..N = (-1)n ε123..n..N = (-1)n. The conclusion then is that d A'n = J2 (Πi≠ndx'i) e'n since ( e'n)i = δn i (7.18.1) (8.4.c.5) and the covariant vector area d A'n points in the n-axis direction in x'-space. For the covariance-dubious reader, here is an alte rnate derivation of (8.4.c.4) from (8.4.c.2) : (d A'n)i = J Rij (dAn)j = J Rij {[ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) } // en missing = J R ij {[ (-1)n-1 εiabc..x R1a R2b.... RNx (Πi≠ndx'i) } // R nκ missing = J {[ (-1) n-1 εiabc..x Rij R1a R2b.... RNx (Πi≠ndx'i) } // R nκ missing = J {[ (-1) n-1 εiabc..x Sj i Sa 1 Sb 2.... Sx N (Πi≠ndx'i) } . // Sκ n missing (8.4.c.6) Chapter 8: Length Area and Volume 150 where R ij = Sj i by (7.5.13). The εSSS..S object is the determinant of an N-1 dimensional matrix. If i ≠n, then i must be some index like 1 or 2. But then th e determinant has two identical columns, so vanishes. Thus, the result is proportional to δn i. Continuing, = δn i J { (-1)n-1 εnabc..x Sj n Sa 1 Sb 2.... Sx N (Πi≠ndx'i) } // Sκ n missing in group = δn i J { (-1)n-1 εnabc..x Sa 1 Sb 2.... Sj n.... Sx N (Πi≠ndx'i) } = δ n i J { εabc..n...x Sa 1 Sb 2.... Sj n.... Sx N (Πi≠ndx'i) } = δ n i J { det(S) ( Πi≠ndx'i) } = δ n i J2 (Πi≠ndx'i) ( 8 . 4 . c . 7 ) which agrees with the result (8.4.c.4) from the covariant x'-space equation. (d) Volume Transformation What about the volume dV? Assume for the moment that dV is correctly represented this way, where any n will do: dV = d An • dx(n) = (d An)i [dx(n)]i . // no implied sum on n (8.4.d.1) Installing our (8.4.a.1) expressions for d An and d x(n) gives, using (7.18.1) that en • en = 1, dV = J en (Πi≠ndx'i) • (en dx'n) = J (Π idx'i) en • en = J (Πidx'i) (8.4.d.2) which agrees with (8.4.a.1), verifying our assumption (8.4.d.1). Looking at dV = (d An)i[dx(n)]i, dV is seen to be a tensor combination of a vector density of weight -1 with a vector density of weight 0 (an ordinary vector d x(n)), so according to (D.2.3) dV must be scalar density of weight -1. Rule (D.1.4) then says, dV' = J dV . (8.4.d.3) Since dV = J ( Π idx'i) from (8.4.a.1). one gets dV' = J 2 (Πidx'i) . ( 8 . 4 . d . 4 ) Again, one can verify this last result from th e x'-space covariant form of the equation: dV' = d A'n • dx'(n) = { J2(Πi≠ndx'i) e'n } • {e'n dx'n) = J2(Πidx'i) e'n • e'n = J2(Πidx'i) (8.4.c.5) (8.2.1) (8.4.d.5) since e'n • e'n = en • en = 1. Chapter 8: Length Area and Volume 151 An alternate derivation of the dV transform rule dV ' = J dV comes from just staring at (8.4.a.1), dV = ε abc..x (dx'e1)a(dx'e2)b..... (dx' eN)x ( 8 . 4 . d . 6 ) which by the argument above is seen directly to transform as a tensor density of weight -1. To complete the circle, we can verify for a second time the claim (8.4.d.1) that dV = d An • dx(n) : d An • dx(n) = (d An)i [dx(n)]i = {(-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πk≠ndx'k) } (dx'nen)i // en missing ..... = ε abc..i..x (e1)a(e2)b.. (en)κ.. (eN)x (Πkdx'i) // note indices on ε = det(S) ( Πkdx'i) // (5.12.8) where ( eb)a = Sa b from (7.18.1) = dV . // from (8.4.a.1) (8.4.d.7) To summarize the above, by examining the vector dens ity nature of our various objects, we have been able to determine exactly how edges, areas a nd the volume transform under transformation F: edges d x'(n) = R d x(n) or [d x'(n)]i = Ri j [dx(n)]j // ordinary vector areas d A'n = J R d An or (d A'n)i = J Ri j (dAn)j // vector density W = -1 volume dV' = J dV // scalar density W = -1 ( 8 . 4 . d . 8 ) (e) Covariant Magnitudes The x'-space magnitudes here are the "covariant" ones which are associated with the Curvilinear View of x'-space, as discussed above. Since d x(n) is a vector, it follows that | d x'(n)|2 = dx'(n) • dx'(n) = dx(n) • dx(n) = | d x(n)|2 ⇒ | d x'(n)| = | d x(n)| . (8.4.e.1) Since d An is a vector density of weight -1, if follows that, | d A'n |2 = dA'n • dA'n = J2 dAn • dAn = J2| dAn |2 ⇒ | d A'n | = |J| | d An | (8.4.e.2) where we note that d An • dAn, being a combination of two weight -1 vector densities, is a scalar density of weight -2 and thus transforms as shown above. For completeness, we can add from the above table, V ' = J d V ⇒ | dV' | = |J| | dV | . (8.4.e.3) We now gather equations fr om above and take their absolute values on the right. Note from (7.18.1) that |en| = h'n and | en| = g'nn . Chapter 8: Length Area and Volume 152 d x(n) = en dx'n (8.2.1) ⇒ |d x(n)| = | en| dx'n = h'n dx'n d x'(n) = e'n dx'n (8.2.1) ⇒ |d x'(n)| = | e'n| dx'n = h'n dx'n d An = J (Πi≠ndx'i) en (8.4.a.1) ⇒ |d An| = |J| | en| (Πi≠ndx'i) = |J| g'nn (Πi≠ndx'i) d A'n = J2(Πi≠ndx'i) e'n (8.4.c.5) ⇒ |d A'n| = J2 |e'n| (Πi≠ndx'i) = J2g'nn (Πi≠ndx'i) dV = J (Π idx'i) (8.4.a.1) ⇒ |dV| = |J| ( Πidx'i) dV' = J2 (Πidx'i) (8.4.d.4) ⇒ |dV'| = J2 (Πidx'i) . (8.4.e.4) The above can be written more compactly using th e Cartesian-View coordinate variation groupings shown in (8.2.3), d x(n) = en dL'n ⇒ |d x(n)| = h'n dL'n d L'n ≡ dx'n d x'(n) = e'n dL'n ⇒ |d x'(n)| = h'n dL'n ` d An = J d A'n en ⇒ |d An| = (|J| g'nn ) dA'n dA'n ≡ Πi≠ndx'i d A'n = J2 dA'n e'n ⇒ |d A'n| = (J2g'nn ) dA'n dV = J d V' ⇒ |dV| = |J| d V' d V' ≡ Πidx'i dV' = J2 dV' ⇒ |dV'| = J2 dV' . (8.4.e.5) The covariant edge magnitude is of course unchanged by the transforma tion since it is a scalar, while the area and volume magnitudes are magnified by |J| in going from x-space to x'-space. Since dV = d An • dx(n), it is clear that the area's transformation factor of |J| is passed onto the volume. Notice that dV' is always positive regardless of the sign of J, and th is is because x'-space is always a right-handed coordinate system, as in Section 6.9. In contrast , dV can have either sign depending on the sign of J = det(S), and so dV < 0 when the en form a left-handed coordinate system in x-space. (f) Two Theorems : g'nn g' = cof(g' nn) and |(Πx i≠nei)| = cof(g'nn) We now pause to prove two small theo rems which will be used below. Theorem 1 : • g'nn g' = cof(g' nn) where g' = det(g' ij) (8.4.f.1) • g'nn g' = g' / h' n2 (orthog only) For orthogonal coordinates g'nn = 1/h'n2 as shown in (5.11.9), proving the second item The first equality can be shown as follows. First, de fine these two regular matrices, (g'up)ab ≡ g'ab (g'dn)ab ≡ g'ab . Then since these metric tensors are inverses, use fact that A-1 = cof(AT)/detA to get Chapter 8: Length Area and Volume 153 g'up = (g'dn)-1 = cof(g' dnT)/det(g'dn) = cof(g' dn)/det(g'dn) // g' dn symmetric ⇒ (g'up)nn = cof[(g' dn)nn]/det(g'dn) or g'nn = cof[g' nn] / g ' Q E D Theorem 2: |(Πx i≠nei)| = cof(g'nn) ( 8 . 4 . f . 2 ) The quantity on the left is this |(Π x i≠nei)| ≡ | e1 x e2 ... x eN | where en is missing from cross product. (8.4.f.3) We shall give three quick proofs of Theorem 2, the last being valid only for N=3. • First, one form for d An from (8.4.a.1) is this, d An = (-1)n-1 (Πx i≠nei) (Πi≠ndx'i) = (-1)n-1 (Πx i≠nei) dA'n ⇒ |d An| = |(Πx i≠nei)| dA'n . (8.4.f.4) Comparison of this last result with the third line of (8.4.e.5) shows that the following must be true. |(Πx i≠nei)| = |J| g'nn = g'1/2g'nn = cof(g'nn) . QED (8.4.f.5) Note that since g=1 (Cartesian x-space), J2 = g' from (5.12.14), and the last step follows from Theorem 1. • Here is a more direct proof: [ Πx i≠n (ei) is a vector in Cartesian x-space ] | Π x i≠n (ei)|2 = Πx i≠n (ei) • Πx j≠n (ej) = [ Πx i≠n (ei)]k [ Πx j≠n (ej)]k (8.4.f.6) = [ εkabc...x (e1)a(e2)b ...... ( eN)x] [ εka'b'c'...x (e1)a'(e2)b' ...... ( eN)x'] // en missing in both = ε kabc...x εka'b'c'...x {(e1)a(e2)b ...... ( eN)x } (e1)a'(e2)b' ...... ( eN)x' // en missing = e1•e1 e2•e2 .... eN•eN + all signed permutations of the 2nd labels // en missing = g' 11g'22..... g'NN + all signed permutations of the 2nd indices, (7.18.1) // en missing But this is last object is the determinant of the g' ij matrix with g' nn crossed out, which is to say, it is the minor of g' nn. Since g' nn is a diagonal element, the minor and cofactor are the same. Thus, this last object is in fact just cof(g' nn) . Q E D . Chapter 8: Length Area and Volume 154 • A proof for N=3 uses normal vector algebra. Settin g n = 1, for example, one needs to show that | Πx i≠1 ei |2 = | e2 x e3 |2 = cof(g' 11) . To this end, use the vector identity ( A x B) • (A x B) = A2B2 – (A•B)2 and fact (7.18.1) that en • em = g'nm to show that, | e2 x e3 |2 = (e2 x e3) • (e2 x e3 ) = | e2|2 |e3|2 - (e2•e3)2 = g'22 g'33 - (g'23)2 = cof(g' 11). ( 8 . 4 . f . 7 ) and the cases n = 2 and 3 are similar. (g) Cartesian-View Magnitude Ratios In the Cartesian View of x'-space one can write the Cartesian x'-space magnitudes as | d x'(n) |c = dL'n | dA'(n) |c = dA'n | dV' |c = dV' . (8.4.g.1) Then from the three x-space magnitude equations in (8.4.e.5) one obtains the following three ratios of x- space objects divided by their corresponding Cartesian-View x'-space objects: | d x(n)|/ dL'n = h'n = [g'nn]1/2 = the scale factor for edge d x(n) | d A(n)|/ dA'n = g'nn |J| = g'nn g'1/2 = [ g'nn g']1/2 = [cof(g' nn)]1/2 // Theorem 1 above |dV| / d V' = |J| = g'1/2 . / / g ' ≡ det(g'ij) = J2 (8.4.g.2) It is convenient to make the definition dAn ≡ | dA(n)| ( 8 . 4 . g . 3 ) and then the above area magnitude ratio relation may be written dAn = cof(g'nn) dA'n ( 8 . 4 . g . 4 ) and d A'n = Πi≠ndx'i is just a product of the appropriate curvilinear coordinate variations. Since g' nn is a diagonal element of the g' matrix, one can reexpress th e cofactor in the above equation in this manner: cof(g' nn) = minor(g' nn) = det (g' with row n and column n crossed out) (8.4.g.5) On the other hand, one can regard the number cof(g' nn) as one element of the cofactor matrix cof(g') whose elements are given by [cof(g')] ij ≡ cof(g'ij). Then, cof(g' nn) = [cof(g')] nn . (8.4.g.6) Chapter 8: Length Area and Volume 155 (h) Nested Cofactor Formulas and STS notation Descending Leaf Hierarchy The object d An is the "area" of a face on a differential N-piped spanned by the full set of vectors ei but not including en. This face, which is itself an (N-1)-piped, in turn has its own "areas" which are (N-2)- pipeds, and so on, so there is a hierarchy of "areas" of dimensions N-1 all the way down. The area ratios of corresponding areas under transformation F are determin ed by equations similar to (8.4.g.4). For this purpose we define Cof to be a matrix and cof to be a number, Cof(M ij) ≡ the submatrix of M obtained by crossing out row i and column j of M cof(M ij) = (-1)i-j minor(M ij) = (-1)i-j det[Cof(M ij)] = the usual "cofactor" . (8.4.h.1) Then Cof(g' nn) = the submatrix of g' obtained by crossing out row n and column n of g' cof(g' nn) = minor(g' nn) = det[Cof(g' nn)] . (8.4.h.2) Then for example the mth face of face n of an N-piped has area ratio cof[Cof(g' nn)]mm . The object inside the radical is the cofactor of the m,m element of the N-1 x N-1 matrix Cof(g' nn). If we refer to this area as dAn,m we can go down the hierarchy in this manner dA n = cof(g'nn) Πi≠ndx'i face n dA n,m = cof[Cof(g' nn)]mm Πi≠n,mdx'i face m of face n dA n,m,k =cof(Cof[Cof(g' nn)]mm)kk Πi≠n,m,k dx'i face k of face m of face n (8.4.h.3) and so on. As a very simple example, for N = 3 the faces dAn,m are line segments dx'i and one has Cof(g' 33) = ⎝⎛ ⎠⎞ g'11 g'12 g'21 g'22 ⇒ cof[Cof(g' 33)]22 = g'11 = h'12 ⇒ cof[Cof(g' 33)]22 = h'1 ( 8 . 4 . h . 4 ) and dA3,2 = h'1 is in fact the edge length ra tio given above in (8.4.g.2). There is another way to write these area magnitudes if we assume that x-space is Cartesian, but it is difficult to express in standard notation, so we moment arily revert to our developmental notation. In that notation, if x-space is Cartesian we can write from (7.5.6) and (3.2.7) that g ¯' = STS, Chapter 8: Length Area and Volume 156 (8.4.h.5) as was written earlier in (5.11.3). Since the covariant metric tensor g ¯'ij in developmental notation is equal to g' ij in standard notation, we shall use the sta ndard notation below in expressing these tensor elements. The area (magnitude) dA n of face n of a differential N-piped is, from (8.4.g.4) and (8.4.e.5), dAn = cof(g'nn) Πi≠ndx'i . (8.4.h.6) The object cof (g' nn) is the minor of the above bracketed matr ix with row n and column n crossed out. That reduced matrix can in fact be written Cof(g' nn) = S(n)TS(n)) = ⎝⎜⎛ ⎠⎟⎞ e1 e2 .. eN (e1, e2, ...eN) where en is missing from both vectors . (8.4.h.7) so cof(g' nn) = det[S(n)TS(n)] = det [ ⎝⎜⎛ ⎠⎟⎞ e1 e2 .. eN (e1, e2, ...eN) ] . (8.4.h.8) Therefore, dAn = cof(g'nn) Πi≠ndx'i = det[S(n)TS(n)] Πi≠ndx'i where S(n) = (e1,e2....eN) with en missing . (8.4.h.9) The same argument results in dAn,m = cof[Cof(g' nn)]mm Πi≠n,mdx'i = det[S(n,m)TS(n,m)] Πi≠n,mdx'i where S(n,m) = ( e1,e2....eN) with en and em missing . (8.4.h.10) Continuing down one more level, Chapter 8: Length Area and Volume 157 dAn,m,k = cof(Cof[Cof(g' nn)]mm)kk Πi≠n,m,k dx'i = det[S(n,m,k)TS(n,m,k)] Πi≠n,m,k dx'i where S(n,m,k) = ( e1,e2....eN) with en, em, ek all missing . (8.4.h.11) In this descending hierarchy of n-piped "leaf" areas, we eliminate one tangent base vector ei at a time, and the eliminated vectors' labels are used to label the "leaf" whose area is indicated, as in S(n,m). Ascending Leaf Hierarchy One can alternatively build up the hierarchy of areas from the bottom, for example using S[n.m] to indicate the area of the 2-piped spanned by en and em in RN where the top level N-piped lives. One then has dA[n] = det(S[n]TS[n]) dx'n where S[n] = en . (8.4.h.12) In this case S[n]T S[n] = en•en = h'n2 so dA[n] = h'ndx'n as expected. Next, for 2-pipeds, dA[n.m] = det(S[n.m]TS[n.m]) dx'ndx'm where S[n.m] = (en, em) . (8.4.h.13) In this case det[S[n.m]TS[n.m]] = det [ ⎝⎜⎛ ⎠⎟⎞en em (en, em)] = det ⎝⎜⎛ ⎠⎟⎞ en•en en•em em•en em•em = det ⎝⎛ ⎠⎞ g'nn g'nm g'mn g'mm = g ' nng'mm - (g'nm)2 = h'n2 h'm2 - (en•em)2 ( 8 . 4 . h . 1 4 ) so dA[n.m] = h'n2 h'm2 - (en•em)2 dx'ndx'm . ( 8 . 4 . h . 1 5 ) Doing one more step, at the level of 3-pipeds in the hierarchy buildup one would then have dA[n.m.k] = det[S[n.m.k]TS[n.m.k]] dx'ndx'mdx'k where S[n.m.k] = (en, em, ek) (8.4.h.16) and now det[S[n.m.k]TS[n.m.k]] = det [ ⎝⎜⎜⎛ ⎠⎟⎟⎞en em ek (en, em, ek)] = det ⎝⎜⎜⎛ ⎠⎟⎟⎞ g'nn g'nm g'nk g'mn g'mm g'mk g'kn g'km g'kk . (8.4.h.17) Comment 1: Notice that the matrices S appearing above in the above STS structures are in general not square matrices because the number of components N of the ei vectors in RN generally does not match the number of ei vectors in the list which defines S. On the other hand, STS is always a square matrix and therefore always has a determinant. Chapter 8: Length Area and Volume 158 Comment 2: We have used the term "leaf" to refer to any n-piped appearing in the top level N-piped, with n then ranging from N down to 1 (which leaf is just point). In a more precise discussion of this subject, these leaves are called "boundaries" and in addition to having "area", these boundaries have "orientation". We have only discussed the orientation of the top level leaves by using the vector area d An . Usually the precise discussion is couched in the language of "differential forms", and by careful consideration of boundaries and their orientations one is able to derive the generalized Stokes's theorem which reads ∫M dα = ∫∂M α ( 8 . 4 . h . 1 8 ) where α is a differential n-form, d α is an (n+1)-form which is the "exterior derivative" of α, M is a smooth type of surface called a manifold, and ∂M is the boundary of that manifold. This abstract theorem then encompasses all the well-known integral theorems of analysis in any number of dimensions n. See Sjamaar [2015] Sections 5.2 - 5.4 , Section 8.2, and Theorem 9.9 on page 117. The fact that det(STS) is the area scaling factor appears as Theorem 8.4 on Sjamaar page 101 where it is called det(ATA) . See also Appendix F (The Volume of an n-piped embedded in Rm) of Lucht Tensor Products . (i) Transformation of arbitrary differential vectors, areas and volume The above discussion is geared to the N-piped transform picture and reveals how d x(n), dA(n) and dV transform where all these quantities are directly associ ated with the particular differential N-piped in x- space spanned by the ( endx'n) vectors. But suppose d x is an arbitrary differential vector in x-space. Certainly d x' = R d x, so we know how this d x transforms under F. But what about area and volume? Based on the work in Appendix B as carried through into the N-piped transform discussion above, it seems clear that an arbitrary differential area in x-space d A can be represented as follows : (g=1) d A = (d x[1]) x (d x[2]) ... x (d x[N-1]) ( 8 . 4 . i . 1 ) (d A)i = εiabc..x (dx[1])a(dx[2])b.... (d x[N-1])x , (8.4.i.2) where the d x[i] are an arbitrary set of N-1 linearly independe nt differential vectors in x-space. For N=3 one would write d A = dx[1] x dx[2]. Since ε has weight -1 and all other objects are true vectors (weight 0), one again concludes that d A transforms as a tensor density of weight -1, so d A' = J R d A or dA 'i = J Rij dAj . (8.4.i.3) If any of these d x[k] were a linear combination of the N-2 others, one could say d x[k] = Σjαk j dx[j] and then the above (d A)i expression would give a sum of terms each of which vanishes by symmetry, resulting in (d A)i = 0. Finally, given the set of d x[i] linearly independent vectors shown above for i = 1,2..N-1, we can certainly find one more such that all N are then linearly independent, so then we have a set of N arbitrary differential vectors d x[i](arbitrary as long as they are linearly independent), and these will form a volume in x-space, Chapter 8: Length Area and Volume 159 dV = det (d x[1], dx[2], ..... d x[N] ) = εabc..y (dx[1])a(dx[2])b.... (d x[N])y . (8.4.i.4) By the argument just given, inspection shows that this dV transforms as a tensor density of weight -1 so dV' = JdV. These then are the transformation rules for ar bitrary differential vectors, areas and volumes transforming under F : d x' = R d x |dx'| = |d x| dA' = J R d A |dA'| = |J| |dA| dV' = JdV |dV'| = |J| |dV| . J = det(S) J2 = g' (8.4.i.5) Review and covariant form of the d A and dV equations To review, in Cartesian x-space we have these expressions for area and volume d A = (d x[1]) x (d x[2]) ... x (d x[N-1]) dV = det [d x[1], dx[2], ... d x[N]] . (8.4.i.6) Written out in terms of contravariant vector components (d x[i])j these expressions appear as, (d A)i = εiabc..x (dx[1])a(dx[2])b.... (d x[N-1])x dV = εabc..y (dx[1])a(dx[2])b.... (d x[N])y , (8.4.i.7) where ε is the usual permutation tensor involved in cross products and determinants. As noted earlier, these are both "valid tensor densit y equations" as in (7.15.9), so th ey can be written in x'-space as (d A')i = ε'iabc..x (dx'[1])a(dx'[2])b.... (d x'[N-1])x dV' = ε'abc..y (dx'[1])a(dx'[2])b.... (d x'[N])y . (8.4.i.8) where d x'[i] = Rd x[i] and where ε' is the x'-space Levi-Civita tensor. From (D.5.13), each ε' can be written as ε' = J2ε = (g'/g)ε = g'ε, so (d A')i = g' εiabc..x (dx'[1])a(dx'[2])b.... (d x'[N-1])x dV' = g' εabc..y (dx'[1])a(dx'[2])b.... (d x'[N])y . (8.4.i.9) Since the permutation tensor now appears, thes e can be written in terms of cross products, d A' = g' (d x'[1]) x (d x'[2]) ... x (d x'[N-1]) dV' = g' det [d x'[1], dx'[2], ... d x'[N]] . (8.4.i.10) Using contravariant components [d x'[1]]i, this cross product and determinant are formed just as they are in x-space. These two equations then give at least some feel for "the meaning of d A' and dV' in x'-space" . Going back to x-space, we could have wr itten the equations there using g = det(g' ij) = det(δij) = 1 : Chapter 8: Length Area and Volume 160 d A = g (d x[1]) x (d x[2]) ... x (d x[N-1]) dV = g det (d x[1], dx[2], ... d x[N]) . (8.4.i.11) This then is a useful interpretation of the covariant form of these equations. Adding primes to everything in the above two equations yields the previous two equations and only the permutation tensor ε is involved in both sets of equations. With this understanding, the above transformation rule s for differential vectors, areas and volumes can be extended from Picture B to the more general Pictur e A, where g is an arbitrary metric tensor, How things look in developmental notation. Recall that covariant tensor objects get overbars and all indices are down in the developmental notation used in Chapters 1-6 of this document. Here then are some of the above equations expressed in this notation: d A¯ = (d x[1]) x (d x[2]) ... x (d x[N-1]) dV = det [d x[1], dx[2], ... d x[N]] ( 8 . 4 . i . 1 2 ) (d A¯)i = ε¯iabc..x (dx[1])a(dx[2])b.... (d x[N-1])x dV = ε¯abc..y (dx[1])a(dx[2])b.... (d x[N])y (8.4.i.13) (d A¯')i = ε¯'iabc..x (dx'[1])a(dx'[2])b.... (d x'[N-1])x dV' = ε¯'abc..y (dx'[1])a(dx'[2])b.... (d x'[N])y (8.4.i.14) d x' = R d x |dx'| = |d x| dA¯' = J ST dA¯ |d A¯'| = |J| |d A¯| dV' = JdV |dV'| = |J| |dV| J = det(S) J2 = g' (8.4.i.15) Recall from (2.5.1) that a covariant vector transforms as V¯' = ST V¯, and a covariant vector density of weight W will then transform as V¯' = J-W ST V¯ and this explains the middle line of the above three. The contravariant differential area would transform as d A' = J R d A. (j) Concatenation (Composition) of Transformations Consider x" = F2(x') and x' = F1(x) so that x" = F2[F1(x)] ≡ F(x). At some point x, the linearization will yield d x" = R2R1dx as the rule for vector transformation, so the R matrix associated with transformation Chapter 8: Length Area and Volume 161 F is R = R 2R1, and then S = R-1 = R1-1R2-1 = S1S2. Each transformation will have an associated Jacobian: J 1 = det(S1) and J2 = det(S2). The concatenated transformation then has J = det(S) = det(S 1S2) = det(S1)det(S2) = J1J2. The implication is that when one concatenates (composes) two transformations in this manner, the area and volume transformations shown above still apply, where J is taken to be the product of the two underlying Jacobians. For example, one could consider the mapping between two different skewed N-pipeds, each representing a different curvilinear coordinate system , with our orthogonal N-pi ped as an intermediary object, (8.4.j.1) In this case one has F = F2-1F1, so R = R 2-1R1 = S2R1 and then S = S 1R2 so J = J 1/J2. This J then would be used in the above area and volume transformation rules, for example, d Aleft = J R d Aright . In the continuum mechanics flow application, g = 1 on both left and right as well as center, time t 0 is on the right, time t on the left, and the volume transformation is given by dV left = J dVright where J is associated with the combined F. Th is J then characterizes the volume change between an initial and final flow particle where each is skewed in some arbitrary manner. (k) Examples of area magnitude transformation for N = 2,3,4 In (8.4.g.4) it was shown that dAn = cof(g'nn) dA'n. Since this is a somewhat strange result, some examples are in order. Recall that the dAn are the areas of the faces of the differential N-piped in x-space, while the d A'n are the curvilinear coordinate variations one can visualize in the Cartesian-View picture (8.2.2) shown above. For N=2 the area magnitude transformation results ar e (for a general non-orthogonal x'-space system) dA 1 = g'22 dA'1 = h'2 dA'1 d A'1 = dx'1 = dL'1 dA2 = g'11 dA'2 = h'1 dA'2 d A'2 = dx'2 = dL'2 . (8.4.k.1) These equations are simple because the area of a parallelogram "face" is the length of an edge and so these equations just coincide with the length transforma tion results stated above . Remember that a face is labeled by the index of the vector which does not span the face, so h 2' appears in the face 1 equation. For N=3 the area magnitude transformation results are dA 1 = g'22 g'33 - (g'23)2 dA'1 d A'1 = dx'2dx'3 dA2 = g'33 g'11 - (g'31)2 dA'2 dA'2 = dx'3dx'1 dA3 = g'11 g'22 - (g'12)2 dA'3 dA'3 = dx'1dx'2 . (8.4.k.2) Chapter 8: Length Area and Volume 162 For an orthogonal N=3 system the metric tensor g' ab is diagonal, and then the above simplifies to dA1 = g'22 g'33 dA'1 = h'2 h'3 dA'1 d A'1 = dx'2dx'3 dA2 = g'33 g'11 dA'2 = h'3 h'1 dA'2 dA'2 = dx'3dx'1 dA3 = g'11 g'22 dA'3 = h'1 h'2 dA'3 dA'3 = dx'1dx'2 . (8.4.k.3) For an N=4 orthogonal system (g' ij is diagonal with g' ii = h'i2), dA1 = cof(g'11) dA'1 = h'2 h'3 h'4 dA'1 d A'1 = dx'2dx'3dx'4 dA2 = cof(g'22) dA'2 = h'1 h'3 h'4 dA'2 d A'2 = dx'3dx'4dx'1 dA3 = cof(g'33) dA'3 = h'1 h'2 h'4 dA'3 d A'3 = dx'4dx'1dx'2 dA4 = cof(g'44) dA'4 = h'1 h'2 h'3 dA'4 d A'4 = dx'1dx'2dx'3 . (8.4.k.4) Example 2: Spherical Coordinates: area patches Consider again dA n = cof(g'nn) dA'n. Since spherical coordinates are orthogonal, the orthogonal N=3 example above may be used. Example 2 of Chapter 5 showed in (5.13.15) that [ 1,2,3 = r, θ,φ ] , h'1 = h'r = 1 d A'1 = dx'2dx'3 = dθdφ h'2 = h'θ = r d A'2 = dx'3dx'1 = drdφ h'3 = h'φ = rsinθ d A'3 = dx'1dx'2 = drdθ ( 8 . 4 . k . 5 ) Therefore d A1 = dA1e^1 ⇒ d Ar = dAr e^r = dAr r^ with dAr = h'2 h'3 dA'1 = r2sinθ dθdφ d A2 = dA2e^2 ⇒ d Aθ = dAθ e^θ = dAθ θ^ with dAθ = h'3 h'1 dA'2 = rsinθ drdφ d A3 = dA3e^3 ⇒ d Aφ = dAφ e^φ = dAφ φ^ with dAφ = h'1 h'2 dA'3 = rdrdθ (8.4.k.6) so that d Ar = r2sinθ dθdφ r^ ρdφ rdθ ρ = rsinθ d Aθ = rsinθ drdφ θ^ ρdφ dr d Aφ = rdrdθ φ^ r d θ dr (8.4.k.7) where all three vectors are seen to have the correct dimensions length2. As an exercise in staring, the reader is invited to verify these results from the picture below using the hints shown above on the right, Chapter 8: Length Area and Volume 163 (8.4.k.8) 8.5 Transformation of Differential Volume applied to Integration As discussed in Section C.8, the integral ∫D dV h( x) is the same regardless of the way the dV elements are chosen, as long as those elements ex actly fill the integration region D. In the discussion above, |dV| (call it dV a) refers to a positive differen tial volume element in x-space which is typically not aligned with the axes and for a general transformati on F is not in general orthogonal, as shown on the right side of Fig (8.2.2 ). Moreover, the shape of the differential volume N- piped varies over the region of integr ation. Nevertheless, this "rag-tag band" of differential volumes, as mentioned below (C.8.3) for the 2D case, fills the integration region perfectly. Alternatively one could consider |dV| (call it dV b) to be the usual dx 1dx2.....dxN differential volume elements in x-space, and of course this set of differe ntial volume elements also fills the integration space perfectly. Thinking of these two different differential volumes as dV a and dVb, one can see from the definition of the Riemann integral as the limit of a sum, lim Σi dVa(xi) f(xi) = lim Σi dVb(xi) f(xi) (8.5.1) that ∫D dVa h(x) = ∫D dVb h(x) . ( 8 . 5 . 2 ) There would be little meaning to the statement dV a = dVb, since no one is claiming there is some particular skewed N-piped of volume dV a which matches some axis-aligned N-piped of volume dV b . Nevertheless, one could write dV a = dVb as a distributional symbolic equality where the meaning of that symbolic equality is precisely the equivalence of th e two integrals above for any domain D and for any reasonable function h( x). [ Formally h( x) might have to be a "test function" φ(x). Certainly one would require that both integrals converge. ] It was shown in the second last line of (8.4.e.4) that the volume of the skewed N-piped on the right side of Fig (8.2.2) was given by dV a = |J( x')| dV' = |J( x')| ( Πi=1N dx'i) . |J( x')| = g'(x) // g = +1 (8.5.3) Chapter 8: Length Area and Volume 164 Combining this with the distributional symbolic equation dV a = dVb gives d V a = dVb or | J ( x')| ( Πi=1N dx'i) = ( Πi=1N dxi) or | J ( x')| dV' = dVb . ( 8 . 5 . 4 ) Now overriding our previous notation, we can make these new commonly used definitions dV ≡ ( Πi=1N dxi) dV' ≡ ( Πi=1N dx'i) ( 8 . 5 . 5 ) and express the distributional result (8.5.4) as |J( x')| dV' = dV . // |J( θ,ρ)| dρdθ = dxdy in (C.8.6) (8.5.6) We refer to this distributional equality in the example of (C.8.6) as the "Jacobian Integration Rule". The symbolic equation is a s horthand for this equation ∫D dV h( x) = ∫D' dV' |J( x')| h( x) ( 8 . 5 . 7 ) where on the right h( x) = h( x(x')) and region D' is the same region as D expressed in terms of the x' coordinates. Writing out the volume elements this says ∫D ( Πi=1N dxi) h(x) = ∫D' ( Πi=1N dx'i) |J(x')| h( x(x')) (8.5.8) and finally, using J2 = g'/g = g' = det(g ¯') → det(g'ab) from (5.12.14), ∫D ( Πi=1N dxi) h(x) = ∫D' ( Πi=1N dx'i) [ det(g'ab) ] h( x(x')) . (8.5.9) For example, when applied to polar and spherical coordinates, one gets ∫D dxdy h( x) = ∫D' drdθ [r] h( x(r,θ)) det(g'ab) = Πih'i = r (8.5.10) ∫D dxdydz h( x) = ∫D' drdθdφ [ r2sinθ ] h(x(r,θ,φ)) det(g'ab) = Πih'i = r2 sinθ (8.5.11) In the first case h( x) = h(x,y) and h( x(r,θ)) = h(rcos θ,rsinθ). In the second case h( x) = h(x,y,z) and h( x(r,θ,φ)) = h(rsin θcosφ,rsinθsinφ,rcosθ). Chapter 8: Length Area and Volume 165 8.6 Interpretations of the Jacobian Using the facts of Section 5.12 (in Standard Notation) and of the above Sections, one can produce various expressions and interpretations for the J acobian J and its absolute value |J| : J( x') ≡ det(Si j(x')) = det( ∂xi/∂x'k) = 1/det(Ri j(x(x')) = 1/ det(∂ x'i/∂xk) (5.12.6) |J(x')| = det(g'ab(x')) = g'(x') / / g = 1 (5.12.14) |J( x')| = the volume of the N-piped in x-space spanned by the en(x), where x = F-1(x') (8.3.8) |J( x')| = dVN-piped /dV' = ratio of differential x-space N-piped volume / ( Πi=1N dx'i) (8.4.e.5) |J( x')| = dV/dV' = ( Πi=1N dxi)/ ( Πi=1N dx'i) // distributional Jacobian Integration Rule (8.5.6) ( 8 . 6 . 1 ) As discussed in Section 6.9, if the curvi linear coordinates are ordered so that the en form a right-handed coordinate system, then det(S)>0, σ = sign(det(S)) = +1, and |J| = J. Some authors define their Jacobians as the inverse of ours, see (5.12.3). 8.7 Volume integration of a tensor field under linear transformations Recall the distributional statement of (8.5.6), dV' = J -1 dV . (8.7.1) In the language of Appendix D, this says that the vol ume element transforms from x-space to x'-space as a scalar density of weight +1. Since by (D.1.6) g1/2 transforms as a scalar density of weight -1, according to (D.2.3) the object g1/2dV then transforms as an ordinary scal ar (weight 0) (see for example Weinberg p 99 (4.4.6) ). Then if one were to define T ijk... ≡ ∫D g1/2dV Aijk..(x) ( 8 . 7 . 2 ) one might expect that, if Aijk..(x) transforms as a tensor field, then Tijk... might transform as tensor. To investigate this conjecture, consider the above integral in x'-space, T'ijk... ≡ ∫D' g'1/2dV' A'ijk..(x') = ∫D g1/2dV A'ijk..(x'(x) ) // g1/2dV = scalar = ∫D g1/2dV Ri i'Rj j'..... Ai'j'k'...(x) . Chapter 8: Length Area and Volume 166 If the underlying transformation x' = F(x) is linear (see Section 2.8) then the Ra b are independent of position and we continue the above, = Ri i'Rj j'..... ∫D g1/2dV Ai'j'k'...(x) = Ri i'Rj j'..... Ti'j'k'... ( 8 . 7 . 3 ) Therefore, for linear transformations x' = F(x), an integral of the form (8.7.2) of a tensor field is itself a tensor of the same type. Examples: Suppose g = 1 and x' = Rx where R is a global rotation, so we have a linear transformation. For a rotation in developmental notation one has R-1 = RT. But (5.7.6) says g' = R g RT = R 1 R–1 = 1, so g' = 1 as well. We may conclude that the volume integr al of a tensor field of any type is a tensor of the same type. Here are two simple examples: J i = ∫D dV xi Jij = ∫D dV [ r2δij - xixj] . (8.7.4) Under rotations, xi is a true vector, and r2δij - xixj is a true rank-2 tensor (traceless). It follows that Ji and Jij are also tensors. Under rotations, mass density ρ transforms as a scalar, so the following objects are tensors as well, Ii = ∫D dV ρ(x) xi / / v e c t o r Iij = ∫D dV ρ(x) [ r2δij - xixj] . // rank-2 tensor, r2 = Σi(xi)2 (8.7.5) Vector components Ii are the first moments of a mass distribution, while Iij is the usual inertia tensor. Since Iij is real and symmetric, it can be diagonalized by a certain rotation R. We can think of this as a transformation from x-space to x'-space where the resulting tensor I'ij is diagonal. In x'-space, the diagonal elements of the tensor I'ij are then its eigenvalues ( λi = I'ii), while the eigenvectors are the axis-aligned e'i of Chapter 3. Back in x-space, the eigenvectors of the non-diagonal Iij are then en ≡ Se'n where S = R-1. This can be verified as follows (developmental notation) I' = R I R-1 ⇒ I'ij = Rii'Ii'j'(R-1)j'j = Rii'Rjj'Ii'j // contravariant tensor I' e'n = λn e'n // I' is diagonal so I en = [ R-1I' R ] [S e'n] = R-1 I' (RS) e'n = S I' e'n = S λne'n = λn Se'n = λnen . (8.7.6) Chapter 9: Divergence 167 9. The Divergence in curvilinear coordinates Note: Covariant derivations of all the curvilinear differe ntial operator expressions appear in Chapter 15. In Chapters 9 through 13, we provide more "physical" or "brute force" derivations which are, of necessity, much less compact. That compactness is a testament to the power of the covariant derivative formalism, which might be called "semicolon technology". The fo rmalism is not used in Chapters 9 through 13. 9.1 Geometric Derivation of the Curvilinear Divergence Formula In Cartesian coordinates div B = ∇ • B = ∂nBn, but expressed in curvilinear coordinates the right side has a more complicated form. We provide here a geometric derivation (in N dime nsions) of the formula for the divergence of a contravariant vector field expressed in curvilinear c oordinates, which means x'-space coordinates with Picture B. (9.1.1) This derivation is an exercise in using the transformation results obtained in Chapter 8 above, and in understanding the meaning of the components of a vector, as discussed in Section C.5. The divergence of a vector field can be computed in a Cartesian x-space by taking the limit of the flux emerging from a closed volume divided by the size of the volume, in the limit that the volume shrinks down around some point x. Being a scalar field, the divergence is a property of the vector field at some point x and therefore cannot depend on the shape of the closed volume used for the calculation†. If the shape of the volume is taken to be a standard-issue axis-aligned N-piped, the divergence obtained will be expressed in terms of the Cartesia n coordinates and in terms of the Cartesian components of the vector field: [div B](x) = ∂nBn(x) where B = Bnn^. However, if the N-piped shape is the one below, evaluation of this same [div B](x) produces an expression which involves only the curvilinear coordinates and the curvilinear components of the vector field, as will now be demonstrated. † For example, the total amount of water per sec ond flowing through a mathematical closed boundary surrounding a point-source sprinkler head will not depe nd on the shape of that boundary. This fact is part of the Divergence Theorem, see e.g. wiki. We start by considering again our differential non-or thogonal N-piped sitting in x-space, which has edges endx'n, faces d An and volume dV, as discussed in Section 8.4 above: Chapter 9: Divergence 168 (9.1.2) In order to avoid confusion with volume V or area A, we name the vector field B. As just noted, the divergence of a vector field B is the total flux flowing out through the faces of the N-piped divided by the volume of the N-piped, in the limit that all diffe rentials go to 0. Thus one writes symbolically, [div B](x) = (1/dV) ∫ dA•B = (1/dV) ∫ dA(x)•B(x) (9.1.3) where the surface integral is over all the faces of the above x-space differential N-piped. Recall that as we move around in space, the shape (and size) of the above N-piped changes, so the d A of a face changes, hence d A(x). Comment on the div B as a scalar. If B is a tensorial vector field, then we claim that div B is a tensorial scalar field, and one can write [div B]'(x') = [div B](x) . ( 9 . 1 . 4 ) The operator object {(1/dV) ∫dA(x)•} acts as a tensorial vector operator, so that the result of its action on B in (9.1.3) is a tensorial scalar. In (8.4.c.2) and (8.4.d.3) it was shown that d A and dV are vector and scalar densities of weight -1. This means that dV' = J dV and d A' = J Rd A so the ratio d A/dV in our operator is a tensorial vector. The fact that div B is a tensorial scalar is much more obvious from the alternative divergence derivation given in Section 15.3. There we find that div B = Bi ;i which evaluated in Cartesian space becomes div B = Bi ,i = ∂iBi. But in this Chapter we are avoi ding Appendix F covariant derivatives! The task is now to compute the integral ∫dA(x)•B(x) over the N-piped in (9.1.2). Appendix B shows that the N-piped faces come in parallel pairs, so we start by considering pair n. We now restore the original Appendix B convention that An points outward from the N-piped far face regardless of the sign of det(S). Then adding the differential distances ( Πi≠n dx'i) to the first line of (8.3.7) one gets, Chapter 9: Divergence 169 d An(x) = |det(Si j(x'))| en(x) ( Πi≠n dx'i) = g'(x') en(x) ( Πi≠n dx'i) . (9.1.5) From (5.12.6) and (5.12.14) with g = 1 one has |det(S)| = |J| = g'1/2, while en are the reciprocal base vectors called En Chapter 6. Quantities d An, S, en and g' are explicitly shown as functions of space, and as usual x' = F(x). Again , the out-facing vector area for the far face of pair n is d An, while the out-facing vector area for the near face is - d An . From (9.1.5) the contribution to the above diverg ence integral from "far face n" is, approximately, [d An]•B(x) ≈ [g'(x'far) en(xfar) ( Πi≠n dx'i) ]•B(xfar) = g'(x'far) ( Πi≠n dx'i) en(xfar) • B(xfar) , (9.1.6) where xfar is taken to be a point at the center of far face n. Recall from the 2nd last line of (7.13.10) that B(x) = B'n(x')en where B'n(x') = en(x) • B(x) (9.1.7) which says that, when B is expanded on the en, the coefficients B'n of the expansion are the contravariant components of vector B transformed into B' in x'-space (the curvilinear coordinate space) by B' = R B. Thus we write, B'n(x'far) = en(xfar) • B(xfar) . ( 9 . 1 . 8 ) Inserting this into (9.1.6) gives d An•B(xfar) ≈ g'(x'far) ( Πi≠n dx'i) B'n(x'far) . (9.1.9) This far-face n contribution to the flux integral is no w expressed entirely in terms of x'-space objects and coordinates. A similar expression obtains for the near-face n, but the sign of d An is reversed. Adding the contributions of these two faces of pair n gives ∫two faces n dA•B(x) = { g'(x'far) B'n(x'far) – g'(x'near) B'n(x'near) } ( Πi≠n dx'i) . ( 9 . 1 . 1 0 ) In x'-space, if x'near is a point at the center of the near face of face pair n, then x 'far = x'near + e'n dx'n where e 'n = axis-aligned basis vector in x'-space (9.1.11) since these two points map into the near and far f ace-n centers in x-space. For any function f, f( x'far) - f(x'near) ≈ (∂'n f(x'near)) dx'n // no implied sum on n (9.1.12) where a change is made only in coordinate x'n by amount dx'n. Applying to f = J B'n yields { g'(x'far) B'n(x'far) – g'(x'near) B'n(x'near) } ≈ ∂ 'n [g'(x'near) B'n(x'near)] dx'n . (9.1.13) Chapter 9: Divergence 170 In the limit dx'n → 0 one has xnear→ x. Then (9.1.10) with (9.1.13) gives ∫two faces n dA•B(x) = ∂'n [g'(x') B'n(x')] dx'n ( Πi≠n dx'i) = ∂ 'n [g'(x') B'n(x')] ( Πi dx'i) . (9.1.14) Now all N differentials are present in ( Πidx'i). The total flux flowing out through all N pairs of faces of the N-piped in x-space is this same result with an implicit sum on n, so total flux = ∫ dA•B(x) = ∂ 'n [g'(x') B'n(x')] ( Πi dx'i) = ∂ 'n [g'(x') B'n(x')] dV ' (9.1.15) where d V' = Πi dx'i is the volume of the differential N-piped in Cartesian-view x'-space as in (8.2.3). The divergence of B from the defining symbolic expression is then [div B](x) = ∫ dA•B(x) / dV = ∂ 'n [g'(x') B'n(x')] (d V'/dV) (9.1.16) where dV is the volume of the x-space N-piped shown in Fig (8.2.2). In (8.4.e.5) it is shown that dV = (g')1/2 dV' ⇒ (d V'/dV) = 1/ g'(x') // (g')1/2 = J (9.1.17) so that (9.1.16) becomes [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')] // all x'-space coordinates and objects (9.1.18) [div B](x) = ∂nBn(x) . // all x-space coordinates and objects (9.1.19) The added second line just shows [div B](x) expressed in terms of the Cartesian x-space coordinates and objects, while the first line resulting from our derivation shows the same [div B](x) expressed in terms of only x'-space objects and coordinates. The goa l advertised above has been fulfilled. If B is a tensorial vector, then as noted above div B is a tensorial scalar, [div B](x) = [div B]'(x') ( 9 . 1 . 2 0 ) and thus the left sides of both equa tions above could be replaced by [div B]'(x'). 9.2 Various expressions for div B It is shown above that [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')] . (9.2.1) Chapter 9: Divergence 171 To obtain div B written in terms of covariant components B' n, one sets B'n = g'nm B'm to get [div B](x) = [1/ g'(x') ] ∂'n [g'(x') g'nm(x') B'm(x')] . (9.2.2) Recall that the B' m are the coefficients of B when expanded on the en, B = B'nen. In practical work B is expanded on the unit vectors e^n ≡ en/ |en | = en/h'n so that B = B'nen = B'n h'ne^n = B'ne^n where B'n ≡ B'n h'n (9.2.3) and then [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')/ h'n(x') ] . (9.2.4) For example, spherical coordinate work might use e^1, e^2, e^3 = r^, θ^, φ^. As noted earlier, the components Bn(x) are not contravariant vector components since they don't quite transform properly: B'n = Rn mBm ( B'n / h'n) = Rn m (Bn / 1) B'n = h'n (Rn m Bn) . (9.2.5) Our Picture B results so far are these, assuming B is a tensorial vector, General: B'n(x') = Rn mBm(x) x = F-1(x') ≡ x(x') x' = F(x) ≡ x'(x) [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x' ) ] B = B'nen [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')/ h'n(x') ] B = B'n e^n [div B](x) = [1/ g'(x') ] ∂'n [g'(x') g'nm(x') B'm(x')] B = B'nen [div B](x) = ∂n Bn(x) // Bn = Cartesian components of B B = Bn n^ (9.2.6) [div B](x) = [div B]'(x') For orthogonal curvilinear coordinates, one has g' ij = h'i2 δi,j g'ij = h'i-2 δi,j det(g'ij) = Πih'i2 g' = (Πih'i) = h'1h'2....h'N (9.2.7) so the above expressions ca n be written (the arguments x' of the h' n are now suppressed) Orthogonal: [div B](x) = [1/(Πih'i)] ∂'n [(Πih'i) B'n(x' ) ] B = B'nen [div B](x) = [1/(Πih'i)] ∂'n [(Πih'i) B'n(x')/ h'n ] B = B'n e^n [div B](x) = [1/(Πih'i)] ∂'n [(Πih'i) B'n(x')/h'n2] B = B'nen [div B](x) = ∂n Bn(x) // Bn = Cartesian components of B B = Bn n^ [div B](x) = [div B]'(x') ( 9 . 2 . 8 ) Chapter 9: Divergence 172 Comment: Are the equations (9.2.6) valid if B is not a tensorial vector? For such a B one might try to make it be tensorial "by definition" as discussed in Section 2.9. One would then go ahead and define B'n(x') ≡ Rn mBm(x) and claim success. If such a definition does not result in an inconsistency, then such a B has been moved into the class of tensorial vectors. Example 1 in (2.9.3) shows have such an inconsistency might arise, and it is interesting to see how that plays out here. Suppose F is non-linear so that R and g' = RRT are functions of x' and are not constants. Take B(x) = x (the identity field) and try to make it be contravariant by definition, x'n ≡ Rn mxm. The Cartesian divergence is then div B = ∂nBn(x) = ∂nxn = 3. But the first equation of (9.2.6) says div B = [1/ g'(x') ] ∂'n [g'(x') x'n] = ∂ 'n x'n + [1/ g'(x') ] x'n∂'ng'(x') = 3 + other stuff ( 9 . 2 . 9 ) and thus the two calculations for div B disagree. As noted earlier, x' ≡ Rx conflicts with x' = F(x) in the case of non-linear F. This comment can be applied to the tensorial char acter of the differential operators treated in later Chapters. 9.3 Translation from Picture B to Picture M&S ( 9 . 3 . 1 ) Picture M&S reflects the notation used by Moon & Spen cer. In order to avoid a symbol conflict with the Cartesian tensor components, the Curvilinear (now u-space) components are displayed in italics . The rules for translation are • replace x' by u e v e r y w h e r e ( 9 . 3 . 2 ) • replace ∂'n by ∂n meaning ∂/∂un ( exception: on a "Cartesian" line ∂n means ∂/∂xn) • replace g' by g (both the scalar and the tensor) and h n' by hn • put all primed tensor components (scalar, vector, etc) into unprimed italics (eg, B'n → Bn , f ' → f ) After this translation, all unprimed tensor components are functions of x, while all italicized tensor components are functions of u. Here then are the translations of the tw o blocks above: (implied summation everywhere) Chapter 9: Divergence 173 General: now Bn(u) = Rn mBm(x) x = F-1(u) ≡ x(u) u = F(x) ≡ u(x) [div B](x) = [1/ g ] ∂n [g Bn] B = Bnen [div B](x) = [1/ g ] ∂n [g Bn/ hn ] B = Bne^n // M&S 1.06 [div B](x) = [1/ g ] ∂n [g gnm Bm] B = Bnen [div B](x) = ∂nBn // Bn = Cartesian components of B B = Bn n^= Bn n^ [div B](x) = [div B](u) // transformation (scalar) (9.3.3) Orthogonal: [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn] B = Bnen [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn] B = Bn e^n [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn2] B = Bnen (9.3.4) Notice that the scalar function [div B]'(x') → [div B](u) according to the fourth rule of (9.3.2) above, and that the arguments of all h k(u) are suppressed. As an example, for N=3 the second line of (9.3.4) becomes [div B](x) = [1/(h1h2h3)] { ∂1[h2h3 B1(u) ] + cyclic } B = Bn e^n (9.3.5) where + cyclic means two other terms with 1,2,3 cyclically permuted. With the replacements B → E , Bn→ En hn → gnn the 2nd equation of (9.3.3) agrees with Moon & Spencer p 2 (1.06). Comment: We use the "script MT bold" font Bn for components of vectors expanded onto e^n. It had to be something in upper case distinct from Bn and Bn. In practice one can replace Bn with a different symbol and then Bn is just a formal notation appearing in formulas . For example, in spherical coordinates (1,2,3) = (r,θ,φ) one can make the replacements B1, B2, B3 → Br, Bθ, Bφ and these then do not conflict with Bx, By, Bz or Br, Bθ, Bφ . See Section 14.7 for another example. (9.3.6) 9.4 Comparison of various authors ' notations Different authors use different symbols for curvilin ear coordinates. They usually use x-space as the Cartesian space, and then something like u-space or ξ-space as the curvilinear space: Chapter 9: Divergence 174 Curvilinear coords Cartesian coords Curvilinear space gnn Picture C xn x(0)n x-space h n Picture B x'n xn x'-space h' n Moon & Spencer (M&S) p 2 un xn u-space gnn Morse & Feshbach (M&F) p 115 ξn x n ξ-space h n Margenau & Murphy p 192 qn xn q-space Q n (9.4.1) These authors don't use any special notation to disti nguish Cartesian from curvilinear components, nor is it always clear whether a component is a coefficient of a unit vector or not, so one must be careful. For example, on page 115 Morse & Feshbach simply say (9.4.2) which compare to the 2nd line of (9.3.4) above (sum on n now displayed), [div A](x) = [div A](u) = [1/(h1h2h3)] Σn ∂n [h1h2h3 An(u) / hn] A = An e^n (9.4.3) so one should identify the M&F A n with An, the coefficient of e^n. Chapter 10: Gradient 175 10. The Gradient in curvilinear coordinates 10.1 Expressions for grad f This Chapter is considered in the Picture B context, (10.1.1) The gradient of f is defined in Cartesian space by [grad f] n ≡ Gn ≡ ∂nf(x) so G = grad f = ∂nf(x) n^ . (10.1.2) Comment : To follow our conventions, we should perhaps write [grad f] as [ grad f ] a s i n [ ∇f] with a bolded del operator. We decided against this so that grad, div and curl are all written non-bolded. This seems to be a common usage of other authors. Assuming f is a tensorial scalar field under F, then the G n = ∂nf(x) are covariant vector field components under F. This is because, [∂'nf'(x')] = (Rnm ∂m)f'(x') = (Rnm ∂m)f(x) = Rnm [∂mf(x)] , (10.1.3) where recall from (2.10.1) that f'( x') = f( x) for a scalar field. [ In the technical language of Appendix F, one can write from (F.9.1) that tensorial vector f ;n = f,n = ∂nf. ] Since ∂ nf is a tensorial vector, the covariance of Section 7.15 gives us (10.1.2) e xpressed in x'-space, [grad f]' n = G'n ≡ ∂'nf '(x' ) . ( 1 0 . 1 . 4 ) According to the last line of (7.13.10) vector G can be expanded as G = G'1e1 + G'2 e2 +... = ΣnG'n en where en • G = G 'n (10.1.5) where the coefficients G' n are the covariant components of G' in x'-space, which is the transformed G. One can thus write G(x) = [grad f]( x) = G'n en = ∂'nf '(x') en = en ∂'nf '(x') ≡ ∇'CLf '(x') ∇'CL ≡ en∂ 'n G(x) = [grad f]( x) = ∂nf(x) n^ = n^ ∂nf(x) = ∇ f(x) ∇ ≡ n^ ∂n (10.1.6) where the second line shows the usual Cartesian form of the gradient. The first line shows how one could define a "curvilinear gradient" operator ∇'CL ≡ en∂'n, but this does not seem particularly useful. To restate, Chapter 10: Gradient 176 [grad f]( x) = ∂'nf '(x') en // Curvilinear [grad f](x) = ∂nf(x) n^ . // Cartesian (10.1.7) In the first line, the reciprocal vectors en exist in x-space, but the coefficients are expressed entirely in terms of x'-space coordinates and objects. Since f( x) is a scalar field, f( x) = f '(x'), one could regard the derivative appearing in the first line as ∂' nf '(x') = ∂ 'nf(x(x')) where x(x') = F-1(x') . (10.1.8) The contravariant components of grad f are then easily obtained as G' i(x') = [grad f]'i(x') = ∂'if '(x') = g'ij(x') ∂'jf '(x') G = grad f = G'i(x') ei (10.1.9) where now the expansion G' i(x') ei is that of the 3rd line of (7.13.10). If an expansion on unit vectors is desired, the right equation of (10.1.9) can be written, since ei = h'ie^i , G = [grad f](x) = G'iei = (G'i h'i) e^i = G'i e^i where G'i ≡ h'i G'i (10.1.10) so then (10.1.9) reads, G'i(x')/h'i = [grad f]'i(x') = ∂'if '(x') = g'ij(x') ∂'jf '(x') G = grad f = G'i e^i . (10.1.11) Gathering up these results one gets G' i(x') = [grad f]' i(x') = ∂'if '(x' ) G = grad f = G' i(x') ei G'i(x') = [grad f]'i(x') = ∂'if '(x') = g'ij(x') ∂'jf '(x') G = grad f = G'i(x') ei G i(x') = h'i[grad f]'i(x') = h'i ∂'if '(x') = h'i g'ij(x') ∂'jf '(x') G = grad f = G'i(x') e^i ( 1 0 . 1 . 1 2 ) which can be rewritten [grad f]( x) = (∂ 'if '(x')) ei [grad f]( x) = (∂ 'if '(x')) ei = g'ij(x') (∂'jf '(x')) ei [grad f](x) = h'i (∂'if '(x')) e^i = h'i g'ij(x') (∂'jf '(x')) e^i [grad f]( x) = (∂if(x)) n^ = ∇f(x) // Cartesian [grad f]' i(x') = Rij[grad f]j(x) // transformation f '( x') = f( x) (10.1.13) Again, in each of the first three forms above, G = [grad f]( x) is being expressed as a linear combination of e vectors which are in x-space, but the coefficients ar e given entirely in terms of x'-space coordinates and objects. One can compare the last line of (10.1. 13) to the last line of (9.28) that [div B](x) = [div B]'(x'), Since div B is a scalar, there is no mixture of components as in [grad f]' i(x') = Rij[grad f]j(x) . Chapter 10: Gradient 177 For the orthogonal case g'ij = (1/h'i2) δi,j from (5.11.9) so the block above becomes [grad f]( x) = (∂ 'if '(x')) ei [grad f]( x) = (∂ 'if '(x')) ei = (1/h' i2) (∂'if '(x')) ei [grad f]( x) = h'i (∂'if '(x')) e^i = (1/h' i) (∂'if '(x')) e^i [grad f]( x) = (∂if(x)) n^ // Cartesian [grad f]' i(x') = Rij[grad f]j(x) // transformation f '( x') = f( x) (10.1.14) This above equations can be converted from Picture B to Picture M&S using the same rules given in (9.3.2), which we repeat below ( 1 0 . 1 . 1 5 ) • replace x' by u e v e r y w h e r e • replace ∂'n by ∂n meaning ∂/∂un ( exception: on a "Cartesian" line ∂n means ∂/∂xn) (9.3.2) • replace g' by g (both the scalar and the tensor) and h n' by hn • put all primed tensor components (scalar, vector, etc) into unprimed italics (eg, B'n → Bn , f ' → f ) After this translation, all unprimed tensor components are functions of x, while all italicized tensor components are functions of u. The translated results are then (all implied sums) [grad f]( x) = (∂if ) ei [grad f]( x) = (∂if ) ei = gij (∂jf ) ei [grad f]( x) = hi(∂if ) e^i = hi gij (∂jf ) e^i [grad f]( x) = (∂if) n^ // Cartesian [ grad f ]i(u) = Rij[grad f]j(x) // transformation (vector) f ( u) = f( x) = f( x(u)) (10.1.16) Notice that [grad f]' i(x') → [grad f ]i(u) according to the fourth rule, meaning G' i(x') → Gi(u) . For orthogonal curvilinear coordinates, [grad f]( x) = (∂if ) ei [grad f]( x) = (∂if ) ei = (1/h i2) (∂if ) ei [grad f]( x) = hi(∂if ) e^i = (1/h i) (∂if ) e^i // M&S 1.05 (10.1.17) One can always make the replacement f (u) = f( x(u)) in any of the above equations (f scalar). And one more time: the various e vectors are in x-space, but all the coefficients are expressed in curvilinear u- space coordinates and components. With the replacements Chapter 10: Gradient 178 f → φ e^i→ ai h i → gii the last equation of (10.1.17) agrees with Moon & Spencer p 2 (1.05). 10.2 Expressions for grad f • B Sometimes one is interested in the following quantity (back to Picture B) grad f • B ( 1 0 . 2 . 1 ) where f is a tensorial scalar field and B is a tensorial vector field. In this case, since grad f is a tensorial vector field, the quantity grad f • B is a tensorial scalar, and so (grad f)' • B' = (grad f) • B . This quantity grad f • B can be written severa l ways depending on how B is expanded: B = Σi B'i ei ⇒ grad f • B = (∂'nf ') en • Σi B'i ei = (∂'nf ') B'n B = Σi B'i ei ⇒ grad f • B = (∂'nf ') en • Σi B'i ei = (∂'nf ') B'n . (10.2.2) Here grad f comes from lines 2 and 1 of (10.1.14) and the B expansions from (7.13.10), and there is an implied sum on n. The last line can be written, using B'i ≡ B'i h'i , B = Σi [B'i h'i] e^i ≡ Σi B'i e^i ⇒ grad f • B = (∂'nf ') B'n = (∂'nf ') (B'n/h'n) (10.2.3) To summarize: [grad f]( x) • B(x) = (∂'nf '(x')) B'n(x') for B = Σi B'i ei [grad f]( x) • B(x) = (∂'nf '(x')) B'n(x') for B = Σi B'i ei [grad f]( x) • B(x) = (∂'nf '(x')) B'n(x')/h'n(x') for B = Σi B'i e^i [grad f](x) • B(x) = (∂nf(x)) Bn(x) for B = Bn n^ // Cartesian [grad f](x) • B(x) = [grad f]'( x') • B'(x') // scalar (10.2.4) Since grad f • B is a scalar, these results resemble the dive rgence results more than the gradient ones. Everything on the right side of the first three equations involves only x' -space coordinates and components. The conversion from Picture B to Picture M&F is straightforward using rules (9.3.2), [grad f]( x) • B(x) = (∂nf ) Bn B = Σi Bi ei [grad f]( x) • B(x) = (∂nf ) Bn B = Σi Bi ei [grad f]( x) • B(x) = (∂nf ) Bn/hn B = Σi Bi e^i [grad f]( x) • B(x) = (∂nf) Bn // Cartesian B = Bn n^ [grad f](x) • B(x) = [grad f ](u) • B(u) // transformation (scalar) (10.2.5) Chapter 10: Gradient 179 where once again f ( u) = f(x(u)). Comment: According to the Cartesian line above, one can write grad f • dx = ∂nf(x) dxn = df = f( x+dx) - f(x) . (10.2.6) This equation df = [grad f]( x) • dx is sometimes used as an alternate definition of the gradient. If d x is selected to be in the direction of grad f, the dot product has its maximum value, and therefore the gradient points in the direction of the maximum change of a scalar function f( x). For N=2, in the usual 3D plot of real z = f(x,y) the gradient then points "uphill", and the negative of the gradient then points "downhill". Chapter 11: Laplacian 180 11. The Laplacian in curvilinear coordinates We continue to use Picture B, see (11.9) below. The Laplacian (also known as the Laplace-Beltrami operator) is defined by lap f = div (grad f) = div G where G ≡ grad f . (11.1) Since f is (by assumption) a tensorial scalar field, G = grad f is a tensorial vector, as shown in (10.1.3). Then, as claimed in (9.1.4), div(grad f) = div G is a tensorial scalar, meaning [lap f]( x) = [lap f]'( x'). [In Chapter 15 we shall find that lap f = f;i ;i which in Cartesian space is f,i ,i = ∂i∂if = ∂i2f. ] In Cartesian coordinates one writes lap f = ∇ 2f = ∇•∇f = Σn∂n2f ( 1 1 . 2 ) but this form gets modified when lap f is expressed in curvilinear coordinates. The first line of (9.2.6) showed that, div G = [1/ g' ] ∂ 'm [g' G' m] where G = G'nen g' = det(g' ab) . (11.3) Meanwhile, the second line of (10.1.13) showed that, grad f = G = [g'nm (∂'nf ') ] em = G' mem , G' m = g'nm (∂'nf ') , ∂ 'nf ' = ∂'n f(x(x')) = ∂ 'nf '(x') . ( 1 1 . 4 ) Therefore, lap f = div (grad f) = div G = [1/ g' ] ∂'m [g' G' m] = [1/ g' ] ∂'m [g' g'nm (∂'nf ')] (11.5) so the general results can be concisely stated: [lap f]( x) = [1/ g'(x') ] ∂'m [ g'(x') g'nm(x') (∂'nf '(x')) ] // implied sum on n and m [lap f]( x) = Σn ∂n2f(x) // Cartesian f '(x') = f( x) = f( x(x')) [lap f]( x) = [lap f]'( x') // scalar . (11.6) For an orthogonal coordinate system we know from (5.11.9) that g'nm = h'n2 δn,m g'nm = (1/h'n2) δn,m g' = Πi h'i (11.7) so the first line of (11.6) simplifies to [lap f]( x) = [1/(Π ih'i)] ∂'m [(Πih'i) (1/h'm2) (∂'mf ') ] . (11.8) Chapter 11: Laplacian 181 Converting from Picture B to Picture MS with rules (9.3.2) gives, ( 1 1 . 9 ) [lap f]( x) = [1/ g ] ∂m [g gnm (∂nf ) ] [lap f]( x) = Σn ∂n2f(x) // Cartesian f (u) = f( x) = f( x(u)) [lap f]( x) = [ lap f ](u) // transformation (scalar) (11.10) The first equation simplifies in the orthogonal case to [lap f]( x) = [1/ g ] ∂m [g (1/hm2) (∂mf ) ] // orthogonal // M&S 1.09 (11.11) or [lap f]( x) = [1/(Π ihi)] ∂m [ (Πihi) (1/hm2) (∂mf ) ] . (11.12) For N=3 this says, [lap f]( x) = 1/(h1h2h3) { ∂1 [ (h2h3/h1) ∂1f ] + cyclic } . (11.13) With the replacements f → φ hm2 → gmm equation (11.11) agrees with Moon & Spencer p 3 (1.09). Chapter 12: Curl 182 12. The Curl in curvilinear coordinates 12.1 Definition of curl B The vector curl is defined only in N=3 dimensions (but see Section 12.6 below). Picture B is used. (12.1.1) In Cartesian coordinates one writes [curl B]i(x) = [∇ x B(x)]i = εijk∂jBk(x) , (12.1.2) but when expressed in terms of curvilinear coordinates and components, the form is different. Consider the x-space differential 3-piped show n on the right side of Fig (8.2.2), (12.1.3) This 3-piped has three pairs of parallel faces. Within each pair, the "near" face touches the point x at which the tails of the spanning vectors meet, while the "far" face does not. According to (8.4.a.1) we can write ( all dx' i > 0 ) d An = J en (Πi≠ndx'i) . ( 1 2 . 1 . 4 ) According to (8.3.6) the vector - d An is the out-facing area vector for a near face. Therefore the in-facing area vector for a near face is d An. For example, the area vector d A3 of the bottom face in (12.1.3) points up, because the bottom face is a near face and in-facing for it means up. Chapter 12: Curl 183 Consider now the line integral of a vector field B(x) around the boundary of near face n, where the circulation sense of the integral is determined from the right-hand-rule by the direction of d An. Denote this line integral by ( ∫{B•dx)n . ( 1 2 . 1 . 5 ) Sometimes this line integral is referred to as "the circulation" or "the rotation" of B around near face n (and rot B is another notation used for curl B). This terminology is suggestive of a fluid vortex where the fluid velocity v has a large curl component perpendicular to the vortex. In x-space the quantity C(x) ≡ curl B(x) is a vector field defined in the following manner in the limit that all the differentials dx'n → 0 : C • dAn = ( ∫{B•dx)n C ≡ curl B , (12.1.6) Since d An is given in (12.1.4) in terms of en, C should be expanded on the ek. Since C is a vector density of weight -1, (D.2.9) shows that the proper expansion of C onto the e k is given by C = J-1 Σk=13 C'k ek ( 1 2 . 1 . 7 ) so that, using (12.1.4), C • dAn = J-1 (Σk C'k ek) • ( J en ( Πi≠n dx'i) ) = C'n ( Πi≠n dx'i) . (12.1.8) Combining this with (12.1.6) gives, C' n(x') ( Πi≠n dx'i) = ( ∫{B•dx)n . (12.1.9) Since (Πi≠ndx'i) = dAn is the area of face n of our differential N-piped, (12.1.9) says that the nth component of the curl in x'space C'n equals the line integral of B around the boundary of face n, divided by the area of face n, in the limit that all differentials go to zero. Our task is now to compute this line integral and thereby come up with an expression for C'n(x'), the components of C = curl B when B is expanded onto the ek in x-space. 12.2 Computation of the line integral This shall be done for the bottom face (n=3) of the x-space differential 3-piped. As noted below (12.1.4), the area vector d A3 points up in Fig (12.1.3). Our circulation integral by the right hand rule will be counterclockwise around this bottom face boundary as seen in the figure, replicated here, Chapter 12: Curl 184 (12.2.1) In this picture, it is intended that the upper right face is farthest from the viewer, so it is the "back" face of the 3-piped. It's parallel partner face touching the point x is the "front" face. We could call these faces near and far, but front and back seem better here. Then the bottom face about which we are integrating has edges front, right, back, and left. So, grouping the contributions in pairs, one sees that. ( ∫{B•ds)3 ≈ [B(xfront ) - B(xback)] • (e1 dx'1) + [ B(xright ) - B(xleft)] • (e2 dx'2) (12.2.2) where B(xfront ) refers to the value of B at the center of the front edge of the parallelogram which is the bottom face, and similarly for the other three edges. In the limit that the dx'n → 0, this simple approximation of the line integral is "go od enough" to produce the desired results. Motivated by (7.18.1) that ei • ej = δij, expand vector B as in the last line of (7.13.10), B = B'jej where B' j(x') = B(x) • ej (12.2.3) where the B' j are the covariant components of B in x'-space. Inserting this expansion four times into (12.2.2) gives, ( ∫{B•dx)3 = [B'1(x'front ) - B'1(x'back)] dx'1 + [B'2(x'right ) - B'2(x'left)] dx'2 (12.2.4) where x'front = F(xfront ) and similarly for the other three points. In x-space the figure shows, d xBF ≡ xback - xfront = e2dx'2 d xRL ≡ xright - xleft = e1dx'1 . (12.2.5) Applying matrix R gives the corre sponding x'-space equations (recall d x' = Rd x and e'n = Ren), d x'BF ≡ x'back - x'front = e'2dx'2 e'2 = (0,1,0...) d x'RL ≡ x'right - x'left = e'1dx'1 e'1 = (1,0,0...) . (12.2.6) Chapter 12: Curl 185 Using the fact that f( x'+dx') ≈ f(x') + (∂f/∂x'n) dx'n for a variation d x' = e'ndx'n (with no sum on n), one writes, B'1(x'back) ≈ B'1(x'front ) + (∂ B'1/∂x'2) dx'2 where d x' = e'2dx'2 n=2 B'2(x'right ) ≈ B'2 (x'left) + (∂ B'2/∂x'1) dx'1 where dx' = e'1dx'1 n=1 . (12.2.7) The circulation integral (12.2.4) is then ( ∫{B•dx)3 ≈ [– (∂B'1/∂x'2) dx'2] dx'1 + [(∂B'2/∂x'1) dx'1] dx'2 = [ – ( ∂B'1/∂x'2) + (∂B'2/∂x'1)] dx'1 dx'2 = [– ∂'2B'1 + ∂'1B'2] dx'1 dx'2 = [ ∂' 1B'2 – ∂'2B'1] dx'1 dx'2 = ε 3ab ∂'aB'b ( Πi≠3 dx'i) . (12.2.8) Repeating this calculation for faces 1 and 2 produces cyclic results, and all three face line integrals can be summarized as (where equality holds in the limit dx'i → 0) ( ∫{B•dx)n = εnab ∂'aB'b ( Πi≠n dx'i) . (12.2.9) Appendix D discusses the tensor ε known as the Levi-Civita ε tensor. In Cartesian space, the up and down position of the indices does not matter, as fo r any tensor. In non-Cartesian space up and down does matter, as with any tensor. The only fact needed here is (D.4.8) that ε'abc.. = εabc.. where ε' is the tensor in x'-space. In Cartesian space one can regard εabc.. = εabc.. as a bookkeeping permutation tensor with the properties stated below (7.7.4) Installing then εnab = ε'nab into (12.2.9) yields, ( ∫{B•dx)n = ε'nab ∂'aB'b ( Πi≠n dx'i) (12.2.10) and this integral is then given entirely in terms of x'-space coordinates and objects. 12.3 Solving for the curl Recall the expression (12.1.9) involving the curl component C'n in x'-space, C'n(x') ( Πi≠n dx'i) = ( ∫{B•dx)n . (12.1.9) Insert (12.2.10) to get Chapter 12: Curl 186 C'n(x') ( Πi≠n dx'i) = ε'nab ∂'aB'b ( Πi≠n dx'i) . (12.3.1) The differentials cancel out, so then take dx'i→ 0 and thus shrink the 3-piped down to the point of interest x = F-1(x') so that C'n = ε'nab ∂'aB'b . ( 1 2 . 3 . 2 ) Then from (12.1.7) we find, since J = g' , curl B = C = J-1 C'n en // implied sum on n = [(1/ g' ) ε'nab ∂'aB'b ] en . (12.3.3) The comparison between the curvilinear- and Cartesian-expressed curls is this: [curl B](x) = [(1/ g' ) ε'nab ∂'aB'b(x') ] en = (1/ g' ) { [∂'1B'2 - ∂'2B'1] e3 + cyclic } [curl B](x) = εnab ∂aBb(x) n^ = { [ ∂1B2 - ∂2B1 ] 3^ + cyclic } . (12.3.4) Comment 1. The equation curl B = C = [(1/ g' ) ε'nab ∂'aB'b ] en obtained above assumed that B was a vector and the expansion B = B'nen was used. If B were a vector density of weight -1, the expansion would be B = J-1B'nen and the result would be curl B = C = [(1/ g' ) ε'nab ∂'a(J-1B'b) ]. This situation will arise in consideration of the vector Laplacian in Chapter 13. Once again, J = g' . (12.3.5) Comment 2 . Our "big result" of this Section is really (12.3.2) that C'n = ε'nab ∂'aB'b. We could have obtained this same result starting way back with (12.1.2) that Cn = εabc∂aBb . The incantation is that this latter equation, being a tensor densit y equation, is covariant in the sense of (7.15.9) and therefore in x'- space it has the same form with everything primed. But doing it the hard way provides a good physical intuition for the curl of a vector field. (12.3.6) 12.4 Various forms of the curl In (12.3.2) and (12.3.3) we found that, C' n = ε'nab ∂'aB'b C = J-1C'n en B = B'n en curl B = [(1/ g' ) ε'nab ∂'aB'b ] en . (12.4.1) If it is desired to have contravariant components of B, one gets C'n = ε'nab ∂'a(g'bcB'c ) C = J-1C'n en B = B'n en curl B = [(1/ g' ) ε'nab ∂'a(g'bcB'c )] en . (12.4.2) For practical applications, one usua lly wants both vectors expanded on the e^n unit vectors in this way Chapter 12: Curl 187 C = J-1C'n en = (J-1C'n h'n) e^n ≡ C 'n e^n C 'n = h'n J-1C'n B = B'n en = (B'n h'n) e^n ≡ B'n e^n B'n = h'nB'n . (12.4.3) Using then C 'n = (J-1h'n)C'n and C = (J-1h'n)C'n e^n and J = g' one finds, C'n = [(1/ g' ) h'n ε'nab ∂'a(g'bc B'c/h'c )] C = C 'n e^n B = B 'n e^n curl B = [(1/ g' ) h'n ε'nab ∂'a(g'bc B'c/h'c )] e^n curl B = C . (12.4.4) To summarize: B'c = Rc dBd g' = g'( x') etc. [curl B](x) = ε'nab [(1/ g' ) ∂ 'aB'b ] en B = B'nen (12.4.1) [curl B](x) = ε'nab [(1/ g' ) ∂ 'a(g'bcB'c )] en B = B'nen (12.4.2) [curl B](x) = ε'nab [(1/ g' ) h'n ∂'a(g'bc B'c/h'c )] e^n B = B 'n e^n (12.4.4) [curl B](x) = εnab ∂aBb(x) n^ // Cartesian B = Bn n^ (12.4.5) Converting from Picture B to Picture MS with rules (9.3.2), ( 1 2 . 4 . 6 ) one gets : [curl B](x) = εnab [(1/ g ) ∂aBb ] en B = Bnen [curl B](x) = εnab [(1/ g ) ∂a(gbcBc )] en B = Bnen [curl B](x) = εnab[(1/ g ) hn ∂a(gbc Bc/hc )] e^n B = Bne^n [curl B](x) = εnab ∂aBb(x) n^ // Cartesian B = Bn n^ (12.4.7) Warning : The object εnab in the first three equations is now in u-space which is non-Cartesian, so up and down index positions do matter, but when indices are all up, it continues to be the normal permutation tensor. Determinant Notation Consider the following determinant, det(M) = ε abcXaYbZc = ⎪⎪⎪⎪ ⎪⎪⎪⎪ X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 . Chapter 12: Curl 188 If one regards X n as a vector Xn with components ( Xn)i one could write the above line as three determinant equations, Qi = det(M i) = εabc(Xa)iYbZc = ⎪⎪⎪⎪ ⎪⎪⎪⎪ (X1)i (X2)i (X3)i Y1 Y2 Y3 Z1 Z2 Z3 , i = 1,2,3 . One could then combine these into a single vector equation, Q = ⎪⎪⎪⎪ ⎪⎪⎪⎪ X1 X2 X3 Y1 Y2 Y3 Z1 Z2 Z3 = εabc XaYbZc . As an example, suppose Xn = en , Yn = ∂n (an operator), and Z n = Bc . Then one has Q = ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 = εabc XaYbZc = εnab XnYaZb = εnab en(∂aBb) . But the expression on the right is recognized as g [curl B](x) from the first line of (12.4.7). It is in this notation that we now rewrite (12.4.7) in the traditional determinant notation: [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bn en (12.4.8) [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 g1cBc g2cBc g3cBc B = Bn en (12.4.9) [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ h1 e^1 h 2 e^2 h 3 e^3 ∂1 ∂ 2 ∂ 3 (g1c/hc) Bc (g2c/hc) Bc (g3c/hc) Bc B = Bn e^n (12.4.10) [curl B](x) = ⎪⎪⎪⎪ ⎪⎪⎪⎪ 1^ 2^ 3^ ∂1 ∂2 ∂3 B1 B2 B3 // here ∂n= ∂/∂xn and Bi = Bi(x) B = Bn n^ (12.4.11) 12.5 The curl in orthogonal coordinate systems For such systems g ij = δi,jhi2 and det(g ab) = h12h22h32 so g = h1h2h3 . It is then a simple matter to convert all the above forms and the results are: Chapter 12: Curl 189 Picture B: B'c(x') = Rc dBd(x) h i' = hi'(x') etc. [curl B](x) = ε'nab [(1/(h1'h2'h3') ∂'aB'b ] en B = B'nen [curl B](x) = ε'nab [(1/(h1'h2'h3')) ∂'a(h'b2B'b )] en B = B'nen [curl B](x) = ε'nab [(1/(h1'h2'h3')) h'n ∂'a(h'b B'b) )] e^n B = B 'n e^n [curl B](x) = εnab ∂aBb(x) n^ // Cartesian B = Bn n^ (12.5.1) Picture M&S (12.4.6) : Bc(u) = Rc dBd(x) h i = hi(u) etc. [curl B](x) = εnab [(h1h2h3)-1 ∂aBb ] en B = Bnen [curl B](x) = εnab [(h1h2h3)-1 ∂a(hb2Bb )] en B = Bnen [curl B](x) = εnab[(h1h2h3)-1 hn ∂a(hb Bb)] e^n B = Bne^n (12.5.2) [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bn en (12.5.3) [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 h12B1 h22B2 h32B3 B = Bn en (12.5.4) [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ h1 e^1 h2 e^2 h3 e^3 ∂1 ∂2 ∂3 h1 B1 h2 B2 h3 B3 B = Bn e^n // M&S 1.07a ( 1 2 . 5 . 5 ) With the replacements B → E Bn→ En e^n→ an h i → gii (h 1h2h3)-1 → (1/ g ) equation (12.5.5) agrees with Moon & Spencer p 3 (1.07a). 12.6 The curl in N > 3 dimensions Looking at the basic form (12.4.7) of the curl, [curl B]n(x) = εnab ∂aBb(x) // Cartesian (12.6.1) it is hard to imagine a generalization to N>3 dimensi ons where the curl is still a vector. The only vectors available for construction purposes are ∂n and Bn. For N=4 one might try out various generalizing forms [curl B]n(x) = (1/ g ) εnabc ∂a(∂b Bc) = (1/ g ) εnabc ∂a∂b Bc ? [curl B]n(x) = (1/ g ) εnabc ∂a (BbBc) ) ? Chapter 12: Curl 190 but these two forms vanish because antisymmetric ε is contracted against something symmetric. Thus the idea of using multiple cross products as used in Appendix A does not prove helpful. At this point we call upon the Appendix F notion of covariant derivatives and related semicolons. The rank-2 tensor B b;a – Ba;b = ∂aBb – ∂bBa discussed in (D.8.5) provides the logical extension of the curl to N > 3 dimensions. For N=3 it happens that the object can be associated with a vector curl, [curl B]n = εnab [Bb;a – Ba;b ]/2 = εnab Bb;a = εnab [∂aBb – ∂bBa ]/2 = εnab∂aBb . (12.6.2) In relativity where N=4, there is no vector curl, and one sees B b;a – Ba;b referred to as the covariant curl, and ∂aBb – ∂bBa as the ordinary curl (Weinberg p 106 (4.7.2)). Writing the N-dimensional contravariant curl co mponents in this manner in Cartesian x-space, [curl B]ij = (Bj;i – Bi;j) ( 1 2 . 6 . 3 ) one can then ask how this generalized curl would be expressed in terms of x'-space coordinates and objects. (This curl is a regular rank-2 tensor with weight 0, the vector curl has weight -1 ). The general issue of expanding tensors is addressed in Appendix E where this general result is obtained in (E.2.11), A = Σ ijk... A'ijk... (ei⊗ej⊗ek...) . A'ijk... = contravariant components of A in x'-space ( 1 2 . 6 . 4 ) Applying this to A = [curl B] one gets [curl B] = Σij[curl B]'ijei⊗ej = Σij(B'j;i – B'i;j)ei⊗ej . (12.6.5) The Cartesian components of curl B in x-space can th en be expressed in terms of x'-space components and coordinates, [curl B]ab(x) = Σij [curl B]'ij (ei)a(ej)b = Σij(B'j;i – B'i;j)(ei)a(ej)b (12.6.6) where the tangent base vectors en exist as usual in x-space, and B'i = B'i(x') where x' = F(x). Using (F.9.5) one finds that (B' a;b – B'b;a) = ( ∂'aB'b - ∂'bB'a ) + ( g'acΓ'b cn - g'bcΓ'a cn)B'n (12.6.7) where Γ is the affine connection. As in previous Sections we could write this in various ways involving components B' n, B'n, and B 'n. Since (12.6.7) is covariant, in x-space it takes the same form with no primes (B a;b – Bb;a) = ( ∂aBb - ∂bBa ) + ( gacΓb cn - gbcΓa cn)Bn (12.6.8) = ( ∂aBb - ∂bBa ) . Chapter 12: Curl 191 The second line is due to having Γ = 0 by (F.4.16) for x-space being Cartesian with g = 1. This is also true for a quasi-Cartesian x-space having g = G, as discussed in (1.10), because the x-space basis vectors are then constants and Γ involves their spatial derivatives. Chapter 13: Vector Laplacian 192 13. The Vector Laplacian in curvilinear coordinates 13.1 Derivation of the Vector Laplacian in general curvilinear coordinates The vector Laplacian is defined in terms of the vector curl which is only defined for N=3 dimensions. The context is Picture B, (13.1.1) The definition of the vector Laplacian of a vector field B(x) is ∇2B ≡ grad(div B) – curl (curl B) . (13.1.2) If B is a vector, div B is a scalar, grad(div B) is a vector, so we expect ∇2B to be a vector. An annoying observation is that, according to Section D.8, curl B is a vector density of weight -1 and therefore curl(curl B) a vector density of weight -2, and so appl es are being subtracted from oranges to make ∇2B. This tensor conundrum is resolv ed in Section 15.7 so we ignore it for now and proceed undaunted. In Cartesian coordinates, one finds that, as proven below (15.7.3), [∇2B]i = ∇2(Bi) ≡ Σn ∂n2Bi , (13.1.3) but expressed in general curvilinear c oordinates the form gets modified. To avoid confusion, some authors use different symb ols for the vector Laplacian operator. For example, Moon and Spencer use @ in place of ∇2 and we will honor these authors by using that symbol here, so @B ≡ grad(div B) – curl (curl B) . (13.1.4) In order to make use of the re sults of earlier Sections, define G ≡ grad(f) where f = div B (13.1.5) V ≡ curl C where C ≡ curl B (13.1.6) so that @B = G – V . ( 1 3 . 1 . 7 ) The first line of (10.1.13) gives this expression for G, Chapter 13: Vector Laplacian 193 G = grad(f) = ( ∂'kf ') ek . ( 1 3 . 1 . 8 ) Now replace f 'using the first line of (9.2.6) f ' = f '( x') = f( x) = div B = [1/ g' ] ∂ 'i [g' B'i] (13.1.9) so that G = ∂'k{ (1/ g' ) ∂ 'i (g' B'i)} ek = ∂' n{ (1/ g' ) ∂ 'i (g' B'i)} en . (13.1.10) The second term V is little more complicated. First, from the first line of (12.4.5) write, C = curl B = ε'nab [(1/ g' ) ∂ 'a{ B'b} ] en = J-1C'n en = (1/ g' )C'n en // J = g' V = curl C = ε'ncd [(1/ g' ) ∂ 'c{ J-1C'd} ] en = J-2V'n en = (1/g') V'n en (13.1.11) where recall from (D.4.8) that ε'abc... = εabc.. = the usual permutati on tensor, but written up and primed so as to be in covariant form. The factors of J appearing in the expansions on the far right are due to the fact that C is a vector density of weight - 1 and V a vector density of weight -2, see (D.2.9). From (13.1.11) we extract these facts, C' n = ε'nab ∂'a{ B'b} ( 1 3 . 1 . 1 2 ) V'n = g' ε'ncd ∂'c{ (1/ g' )C'd} . (13.1.13) Now lower the index on C' n in (13.1.12) to get C' d = g'deC'e = g'de ε'eab (∂'aB'b) , (13.1.14) and insert this into (13.1.13) to get V' n = g' ε'ncd ∂'c{ (1/g' ) g'de ε'eab (∂'aB'b) } = g' ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'aB'b) } . (13.1.15) Then from (13.1.11) we have this expression for V, V = curl C = (1/g') V'n en = [ (1/ g' )ε'ncd ε'eab ∂'c { (1/ g' ) g'de(∂'aB'b) }] en . (13.1.16) We can then summarize our results: Chapter 13: Vector Laplacian 194 @B = G – V (13.1.7) G = ∂'n{ (1/ g' ) ∂'i (g' B'i)} en (13.1.10) B = B'nen V = [ (1/ g' ) ε'ncd ε'eab ∂'c { (1/ g' ) g'de(∂'aB'b) }] en (13.1.16) B = B'nen = [ (1/ g' ) ε'ncd ε'eab ∂'c { (1/ g' ) g'de(∂'a[g'bfB'f]) }] en . B = B'nen (13.1.17) Often one wants vectors expanded onto the unit vectors e^n B = B'n en = B'n h'n e^n = B 'ne^n ⇒ B'n = B 'n/h'n (13.1.18) and the above set of equations becomes @B = G – V G = h'n ∂'n{ (1/ g' ) ∂'i (g' B 'i/h'i)} e^n B = B'n e^n V = h'n [ (1/ g' ) ε'ncd ε'eab ∂'c {(1/ g' ) g'de(∂'a[g'bf B 'f/h'f]) }] e^n . (13.1.19) To this list we can add the Cartesian form @B = [∇2Bn ] n^ . // Cartesian, n^ = un B = Bn n^ (13.1.20) Converting from Picture B to Picture MS using rules (9.3.2) gives, ( 1 3 . 1 . 2 1 ) @B = G – V G = ∂n{ (1/ g ) ∂i (g Bi)} en B = Bnen V = [ (1/ g ) εncd εeab ∂c { (1/g ) gde(∂a[gbfBf]) }] en G = hn ∂n{ (1/ g ) ∂i (g B i/hi)} e^n B = Bn e^n V = hn [ (1/ g ) εncd εeab ∂c { (1/g ) gde(∂a[gbfBf/hf]) }] e^n @B = [∇2Bn ] n^ // Cartesian, n^ = un B = Bn n^ (13.1.22) In the equations above, all the ∂i mean ∂/∂ui and the argument u of all functions is suppressed. The Cartesian form will be verified below. Comment : The above general statement of @B in curvilinear coordinates is amazingly complicated. Using (D.10.37-39), one could replace εncd εeab = δ(ncd;eab) = δ n ec ad b = δn,eδc,aδd,b + 5 similar terms, Chapter 13: Vector Laplacian 195 but this just expands the V term into six terms in place of one. The alternate path of starting with [ @B]n = B'n;j ;j as shown in Section 15.8 leads to a result involving the affine connection Γ, as well as ∂iΓ and ΓΓ terms and is also quite unpleasant. We must accept the fact that @B is a complicated object which complicates the study of differential equations (such as Navier-Stokes) in general coordinates. 13.2 The Vector Laplacian in orthogonal curvilinear coordinates We continue in Picture M&S and shall use only the e^n expansion. Setting g ij = hi2δi,j and gij = (1/hi)2δi,j from (5.11.8), our equation (13.1.22) simplifies somewhat, @B = G – V = hn [gnk ∂k{ (1/ g ) ∂i (g Bi(u)/hi)} – (1/ g ) εncd εeab ∂c{ (1/ g ) gde (∂a [gbf Bf(u)/hf]) } ] e^n = hn [δk,n ∂k{ (1/ g ) ∂i (g Bi(u)/hi)} (1/hn)2 – (1/ g ) εncd εeab ∂c{ (1/ g ) hd2δd,e (∂a [hb2δb,f Bf(u)/hf]) } ] e^n = h n [ ∂n{ (1/ g ) ∂i (g Bi(u)/hi)} (1/hn)2 – (1/ g ) εncd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb(u)]) } ] e^n = [ (1/h n) ∂n{ (1/ g ) ∂i (g Bi/hi)} – (hn/g ) εncd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb]) } ] e^n . = [ A n – (hn/g ) Bn ] e^n ( 1 3 . 2 . 1 ) where A n ≡ (1/hn) ∂n{ (1/ g ) ∂i (g Bi/hi)} B n ≡ εncd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb]) } . (13.2.2) Now define, T ≡ (1/ g ) ∂i (g Bi/hi) Γd ≡ (1/ g ) hd2 εdab{ (∂a [hb Bb]) } ) (13.2.3) so that An = (1/hn) ∂nT Bn = εncd ∂cΓd . ( 1 3 . 2 . 4 ) Then (13.2.1) says @B = [ (1/h n) ∂nT – (hn/g ) εncd ∂cΓd ] e^n where T ≡ (1/ g ) ∂i (g Bi/hi) and Γ d ≡ (1/ g ) hd2 εdab{ (∂a [hb Bb]) } ) . (13.2.5) Chapter 13: Vector Laplacian 196 It is conventional to write this out more specifically as @B = [ (1/h 1) ∂1T – (h1/g ) ε1cd ∂cΓd ] e^1 + cyclic But ε1cd ∂cΓd = (∂2Γ3 - ∂3Γ2) so @B = [ (1/h 1) ∂1T – (h1/g ) (∂2Γ3 - ∂3Γ2)] e^1 + cyclic = [ ( 1 / h 1) ∂1T + (h1/g ) (∂3 Γ2 – ∂2 Γ3) ] e^1 + [ ( 1 / h 2) ∂2T + (h2/g ) (∂1 Γ3 – ∂3 Γ1) ] e^2 + [ ( 1 / h 3) ∂3T + (h3/g ) (∂2 Γ1 – ∂1 Γ2) ] e^3 . // M&S 1.11 (13.2.6) Writing out the Γd from (13.2.3), Γ1 = (1/ g ) h12 ( ∂2 [h3 B3] – ∂3 [h2 B2] ) Γ2 = (1/ g ) h22 ( ∂3 [h1 B1] – ∂1 [h3 B3] ) Γ3 = (1/ g ) h32 ( ∂1 [h2 B2] – ∂2 [h1 B1] ) . T = (1/ g ) ∂i (g Bi/hi) ( 1 3 . 2 . 7 ) With the replacements B → E Bn→ En e^n→ an hi → gii T → ϒ (13.2.8) equations (13.2.6) and (13.2.7) agree with Moon and Spencer p 3 (1.11). 13.3 The Vector Laplacian in Cartesian coordinates First we verify that our gene ral orthogonal coordinates fo rm (13.2.5) reduces to the expected result for the case that u-space is the same Cartesia n space as x-space. One then has, hi = 1 g = 1 u = F(x) = x ⇒ R = S = 1 ( e^n)i = Sni = δni ⇒ e^n = n^, B = Bn e^n = Bn n^ ⇒ Bn = Bn = Bn . (13.3.1) Then (13.2.5) becomes @B = [ ∂nT – εnab∂a Γb ] n^ // implied sum on n where T = ∂iBi Γ b = εbcd (∂cBd) ( 1 3 . 3 . 2 ) or Chapter 13: Vector Laplacian 197 @B = [ ∂n{∂iBi} – εnab∂a { εbcd (∂cBd)} ] n^ = [ ∂ n{div B} – εnab∂a { (curl B)b } ] n^ = [ ∂ n{div B} – [curl (curl B)]n } ] n^ = g r a d ( d i v B) – curl (curl B) (13.3.3) which agrees with our starting definition of @B (13.1.4). Second , we verify the claim of (13.1.3) that in Cartesian coordinates [@B]n = ∇2 Bn . (13.1.3) We start with the first line of (13.3.3), [ @B]n = ∂n∂iBi – εnab εbcd ∂a∂cBd = ∂n∂iBi – (δncδad – δndδac) ∂a∂cBd // from (D.10.22) = ∂n∂iBi – ∂n∂dBd + ∂c∂cBn = ∂2 cBn = ∇2(Bn) = ∇2(Bn) . (13.3.4) Comment : This is a proof of the Cartesian-coordinates vector identity [ ∇(∇•B) - ∇x(∇xB)]n = ∇2(Bn) . Thus it has been shown that, in Cartesian coordinates, [@B]n = ∇2 (Bn) = [ grad(div B) – curl (curl B) ]n . (13.3.5) It is this second and rather complicated form whic h allowed us to obtain the expression (13.2.5) for @B expressed in general curvilinear coordinates. Chapter 14: Summary 198 14. Summary of Differential Operators in curvilinear coordinates 14.1 Summary of Conventions and How To The results are given in the Picture M&S cont ext, and are copied from Chapters 9-13. (14.1.1) The Standard Notation of Chapter 7 is used throughout the tables. In all the differential operator equations below, an operator acts either on a tensorial vector field B or on a tensorial scalar field f. On the right side of the dr awing above, objects are said to be in x-space, and f( x) and Bn(x) (components of B) are x-space tensorial objects. On the le ft side of the drawing objects are said to be in u-space. The function f is represented in u-space as f (u), while there are three different ways to represent the components of B, called Bn(u), Bn(u) and Bn(u). There is a big distinction between the x- space objects and the u-space objects. For the scalar, f (u) = f(x) = f( x(u)) and so f has a different functional form than f. For the vector components, the u-space components are linear combinations of the x-space components, for example Bn = Rn mBm ( or B = RB, contravariant vector transformation). See (9.3.6) concerning the font used for Bn(u). On lines marked "Cartesian", ∂n = ∂/∂xn and f( x) and Bn(x) appear (Cartesian components). On other lines, ∂ n = ∂/∂un, and the f and B objects appear in italics and are functions of u. The other functions like h n, gab and g are also functions of u. The vectors en, e^n and en all exist in Cartesian x-space. The en are the tangent base vectors of Chapter 3, and the en are the reciprocal base vectors of Chapter 6. The unit vectors e^n ≡ en/ |en| = en/hn are used as well. The dot product A•B is the covariant one of Section 5.10, namely, A iB¯i → AiBi = gijAiBj . For each differential operator, the object on the LHS of the equations is always the same: it is a differential operator acting on f( x) or B(x) in x-space . In the Cartesian lines, the RHS expresses that LHS object in terms of Cartesian objects and Cartesian coordi nates. On the other lines, the RHS expresses that exact same LHS x-space object in terms of Curvilinear (u-space) objects a nd coordinates. When the LHS is a scalar, the LHS object can be considered to be in either x-space or u-space. When the LHS is a vector, that LHS object is in x-space but can be related to u-space objects by a linear transformation by R. Chapter 14: Summary 199 How to Compute Things for some Arbitrary Coordinate System For some arbitrary (and possibly non-orthogonal) coordinate system in some arbitrary number of dimensions N, one can start with the defining equations such as the following ones for spherical coordinates, x = rsin θcosφ that is: x = F-1(u ) u = (r,θ,φ) y = rsin θsinφ z = rcos θ . In the Picture M&S context and in developmental notation, one uses (2.1.5) S ik ≡ (∂xi/∂uk) to compute the matrix S. Then the covariant me tric tensor comes from (5.3.3), g ¯ = STGS. Normally one has G = 1 for Cartesian x-space, but see (1.10) for qua si-Cartesian. The scale factors are then h i = g¯ii and g = det(g¯), as shown in (5.11.7) and (5.12.20). The contra variant metric tensor is then found as g = g ¯-1 (matrices). The conversion to Standard Notation is then g ¯→ gab and g→gab. These calculations are easily automated as shown for example in the Maple code of (G.6.1) and (G.6.2). If the coordinate system is orthogonal, then g ¯ will be diagonal and one will have g ¯nm = hn2 δn,m and gnm = hn-2 δn,m and finally g = det(g ¯) = Πihi. For the eleven classical orthogonal systems the scale factors h i are easily found online or in othe r reference sources, so one can then avoid the above steps (or verify them). At this point, any formulae appearing in the tables below can be used. Chapter 14: Summary 200 The expressions marked below appear on pages 2 or 3 of Moon & Spencer (M&S). general: g ≡ det(gab) hn2 ≡ gnn ∂i = gij∂j Bn = hnBn en = hne^n orthogonal: g = (Πihi) = h1h2...hN gnm = hn2 δn,m gnm = hn-2 δn,m (14.1.2) ___________________________________________________________________________ 14.2 divergence divergence general: (9.3.3) (14.2.1) [div B](x) = [1/ g ] ∂n [g Bn] B = Bnen [div B](x) = [1/ g ] ∂n [g Bn/ hn ] B = Bne^n // M&S 1.06 [div B](x) = [1/ g ] ∂n [g gnm Bm] B = Bnen [div B](x) = ∂nBn // Bn = Cartesian components of B B = Bn n^= Bn n^ [div B](x) = [div B](u) // transformation (scalar) divergence orthogonal: (9.3.4) (14.2.2) [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn] B = Bnen [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn] B = Bn e^n [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn2] B = Bnen divergence orthogonal N=3: (9.3.5) (14.2.3) [div B](x) = [1/(h1h2h3)] { ∂1[h2h3 B1(u) ] + cyclic } B = Bn e^n ___________________________________________________________________________ 14.3 gradient and gradient dot vector gradient general: (10.1.16) (14.3.1) [grad f]( x) = (∂if ) ei [grad f]( x) = (∂if ) ei = gij (∂jf ) ei [grad f]( x) = hi(∂if ) e^i = hi gij (∂jf ) e^i [grad f]( x) = (∂if) n^ // Cartesian [ grad f ]i(u) = Rij[grad f]j(x) // transformation (vector) f (u) = f( x) = f( x(u)) gradient orthogonal: (10.1.17) (14.3.2) [grad f]( x) = (∂if ) ei [grad f]( x) = (∂if ) ei = (1/h i2) (∂if ) ei [grad f]( x) = hi(∂if ) e^i = (1/h i) (∂if ) e^i // M&S 1.05 Chapter 14: Summary 201 gradient dotted with a vector: (10.2.5) (14.3.3) [grad f]( x) • B(x) = (∂nf ) Bn B = Bnen [grad f]( x) • B(x) = (∂nf ) Bn B = Bnen [grad f]( x) • B(x) = (∂nf ) Bn/hn B = Bn e^n [grad f]( x) • B(x) = (∂nf) Bn // Cartesian B = Bn n^ [grad f]( x) • B(x) = [grad f ](u) • B(u) // transformation (scalar) ___________________________________________________________________________ 14.4 Laplacian Laplacian general: (11.10) (14.4.1) [lap f]( x) = [1/ g ] ∂m[g gnm (∂nf ) ] [lap f]( x) = ∂n2f(x) // Cartesian f (u) = f( x) = f( x(u)) [lap f]( x) = [ lap f ](u) // transformation (scalar) Laplacian orthogonal: (11.12) (14.4.2) [lap f]( x) = [1/(Π ihi)] ∂m[ (Πihi) (1/hm2) (∂mf ) ] // orthogonal // M&S 1.09 Laplacian orthogonal N=3: (11.13) (14.4.3) [lap f]( x) = 1/(h1h2h3) { ∂1 [ (h2h3/h1) ∂1f ] + cyclic } ___________________________________________________________________________ Chapter 14: Summary 202 14.5 curl curl general: (12.4.7-11) // N=3 only (14.5.1) [curl B](x) = εnab [(1/ g ) ∂aBb ] en B = Bnen [curl B](x) = εnab [(1/ g ) ∂a(gbcBc )] en B = Bnen [curl B](x) = εnab[(1/ g ) hn ∂a(gbc Bc/hc )] e^n B = Bne^n [curl B](x) = εnab ∂aBb(x) n^ // Cartesian B = Bn n^ [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bnen [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 g1cBc g2cBc g3cBc B = Bnen [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ h1 e^1 h 2 e^2 h 3 e^3 ∂1 ∂ 2 ∂ 3 (g1c/hc) Bc (g2c/hc) Bc (g3c/hc) Bc B = Bne^n // M&S 1.07 [curl B](x) = ⎪⎪⎪⎪ ⎪⎪⎪⎪ 1^ 2^ 3^ ∂1 ∂2 ∂3 B1 B2 B3 // here ∂n= ∂/∂xn and Bi = Bi(x) B = Bn n^ curl orthogonal: (12.5.2-5) (14.5.2) [curl B](x) = εnab [(h1h2h3)-1 ∂aBb ] en B = Bnen [curl B](x) = εnab [(h1h2h3)-1 ∂a(hb2Bb )] en B = Bnen [curl B](x) = εnab[(h1h2h3)-1 hn ∂a(hb Bb)] e^n B = Bne^n [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bnen [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 h12B1 h22B2 h32B3 B = Bnen [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ h1 e^1 h2 e^2 h3 e^3 ∂1 ∂2 ∂3 h1 B1 h2 B2 h3 B3 B = Bne^n // M&S 1.07a ___________________________________________________________________________ Chapter 14: Summary 203 14.6 vector Laplacian vector Laplacian general: (13.1.22) // N=3 only (14.6.1) [@ B](x) = G – V G = ∂n{ (1/ g ) ∂i (g Bi)} en B = Bnen V = [ (1/ g ) εncd εeab ∂c { (1/g ) gde(∂a[gbfBf]) }] en G = hn ∂n{ (1/ g ) ∂i (g B i/hi)} e^n B = Bn e^n V = hn [ (1/ g ) εncd εeab ∂c { (1/g ) gde(∂a[gbfBf/hf]) }] e^n @ B = [∇2Bn ] n^ // Cartesian, n^ = un B = Bn n^ vector Laplacian orthogonal: (13.2.6-7) (14.6.2) [@ B](x) = [(1/h 1) ∂1T + (h1/g ) (∂3 Γ2 – ∂2 Γ3) ] e^1 // M&S 1.11 + [ ( 1 / h 2) ∂2T + (h2/g ) (∂1 Γ3 – ∂3 Γ1) ] e^2 + [ ( 1 / h 3) ∂3T + (h3/g ) (∂2 Γ1 – ∂1 Γ2) ] e^3 T = (1/ g ) ∂i (g Bi/hi) Γ1 = (1/ g ) h12 ( ∂2 [h3 B3] – ∂3 [h2 B2] ) Γ2 = (1/ g ) h22 ( ∂3 [h1 B1] – ∂1 [h3 B3] ) Γ3 = (1/ g ) h32 ( ∂1 [h2 B2] – ∂2 [h1 B1] ) o r (13.2.5) (14.6.3) [@ B](x) = [ (1/h n) ∂nT – (hn/g ) εnab∂a Γb ] e^n T = (1/ g ) ∂i (g Bi/hi) Γb = (1/ g ) hb2 εbcd ( ∂c [hd Bd]) ___________________________________________________________________________ Chapter 14: Summary 204 14.7 Example 1: Polar coordinates: a practical curvilinear notation From our many visits to this example we know that : general: g ≡ det(gab) hn2 ≡ gnn ∂i = gij∂j Bn = hnBn en = hne^n orthogonal: g = (Πihi) = h1h2...hN gnm = hn2 δn,m gnm = hn-2 δn,m (14.1.2) e1 = r(-sinθ,cosθ) = eθ = r e^θ // = r θ^ e2 = (cosθ,sinθ) = er = e^r // = r^ (3.4.2) (14.7.1) θ r x y gij = ⎝⎛ ⎠⎞ r2 0 0 1 θ r R = ⎝⎛ ⎠⎞-sinθ/r cosθ/r cos(θ) sinθ θ r u 1 = θ u2 = r (!!) (14.7.2) (5.13.11) (3.4.1 ) h1 = hθ = gθθ = r h 2 = hr = grr = 1 . (5.13.11) (14.7.3) Assume one is working with a 2D vector velocity field v(x) -- our first encounter with a "lower case" vector field which we have been careful to suppor t with the general notations above. Since one knows the names of the variables 1= θ and 2=r, one might define the following new variables on the first line to be the officially named variables on the second line v θ vr vθ vr v θ vr vx v y v1 v2 v1 v2 v1 v2 v1=v1 v2=v2 contravariant covariant unit vector Cartesian (italic) (italic) (non-italic ) (non-italic) (14.7.4) One ends up with the comfortable v = vθθ^ + vrr^ notation as shown below. vθ ≡ v1 = h1ν1 = hθ vθ = r vθ // unit vector projection components vr ≡ v2 = h2ν2 = hr vr = vr vθ = Rθ xvx + Rθ yvy = -sinθ /r vx + cosθ/r vy vr = Rr xvx + Rr yvy = cosθ vx +sinθ vy (14.7.5) so vθ = r vθ = -sinθ vx + cosθ vy vr = vr = cosθ vx +sinθ vy v = vnen = vθeθ + vrer v = vne^n = vθe^θ + vre^r = vθθ^ + vrr^ v = vnn^ = vxx^ + vyy^ . ( 1 4 . 7 . 6 ) As an example of a differential operator, consider the divergence for orthogonal coordinates from (14.2.2) Chapter 14: Summary 205 [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn] . B = Bn e^n (14.7.7) Applied to the present situation one gets (h 1 = hθ = r and h 2 = hr = 1), [div v](x) = [1/(h1h2)] { ∂1 [h2 v1] + ∂2[h1 v2] } V = vn e^n = [ 1 / ( h θhr)] { ∂θ [hr vθ ] + ∂r[hθ vr] } = (1/r) { ∂θvθ + ∂r(rvr) } . (14.7.8) Suppose v x and vy are constants. Then the Cartesian expression says [div v](x) = ∂nvn = ∂xvx + ∂yvy = 0 + 0 = 0 . (14.7.9) Equation (14.7.9) agrees : [div v](x) = (1/r) { ∂ θvθ + ∂r(rvr) } = (1/r) { ∂θ[-sinθ vx + cosθ vy] + ∂r(r [cosθ vx + sinθ vy]) } = (1/r) { [-cos θ vx - sinθ vy] + [cosθ vx + sinθ vy] } = 0 . (14.7.10) The notation illustrated here works for any curvilinear coordinates. Chapter 15: Covariant Method 206 15. Covariant derivation of all curvili near differential operator expressions 15.1 Review of Chapters 9 through 13 Let us briefly review the discussion of Chapters 9 through 13 concerning the expression of differential operators div, grad, lap, curl and @ in curvilinear coordinates. The underlying framework was provided by Picture B, (15.1.1) where the coordinates x'n of x'-space were the curvilinear coordinate s of interest, while the coordinates of x-space were the Cartesian coordinates. The transformation x' = F(x) provided the connection between the curvilinear coordinates and the Cartesian ones. In Chapter 9 the object div B was treated as the total B "flux" (∫B•dA) emitting from an N-piped in x- space divided by the volume of that N-piped, in the limit the N-piped shrank to a point. Using Appendix B formulas for the N-piped area, and using the key expansion B = B'nen , // (7.13.10) line 3 where the en are tangent base vectors in x-space and the B'n are components of B in x'-space, we obtained in (9.1.18) a way to write the Cartesian-space div B in terms of curvilinear x' coordinates and x'-space objects, namely, B'n(x'), g'( x') and the gradient operator ∂'i = ∂/∂x'i. In Chapter 10 the vector grad f was expanded using (7.13.10) line 4 that V = V'nen to get grad f = [grad f]' nen = (∂'nf ')en where en are the reciprocal base vectors. This result was expressed in various ways, and gradf •B was also treated such that gradf •B = (∂'nf ')en • B'mem = (∂'nf ') B'n . [ Recall that f '(x') = f( x). ] In Chapter 11 the Laplacian was written as lap f = div (grad f) and then the results of Chapters 9 and 10 for div and grad were used to obtain (11.5) which expresses lap f in terms of x'-space coordinates and objects B'n(x'), g'( x'), g'nm(x') and ∂'i . In traditional notation, one writes ∇2f = ∇ • [ ∇f ] = div (grad f). In Chapter 12 the x'-space curl component C'n = [curl B]'n was treated as the circulation line integral of B around near face n of the same N-piped used in Chapter 9 for divergence, divided by the area of that Chapter 15: Covariant Method 207 face, again in the limit the N-piped shrank to a point. With the expansions B = B'nen and C = J-1C'nen, this led to an expression (12.3.3) for curl B in terms of the curvilinear x' coordinates and the x'-space objects B' n(x'), g'( x'), g'nm(x'), ε'nab and ∂'i . In Chapter 13 the vector Laplacian was treated using the definition @ B ≡ grad(div B) – curl (curl B) and then the results of Chapters 9, 10 and 12 on div, grad and curl were recruited to produce an expression (13.1.17) for @ B in terms of x'-space coordinates and various x'-space objects. 15.2 The Covariant Method We shall now repeat all of the above work using "the covariant method" which, as will be shown, gets most results extremely quickly, but at a cost of requiring knowledge of tens or densities and covariant differentiation, as described in Appendices D and F. Imagine that one has a Cartesian x-space expre ssion for some tensor object of interest Q, Q---- = [ Cartesian form with various up and down indices and various de rivatives ] (15.2.1) where Q---- means that Q has some number of up and down indices. The following four steps should then be carried out: 1. Arrange any summed index pairs within [....] so th ey are "tilted". Then when [...] is "tensorized" those tilted pairs will be true contractions. Note that doing this does not affect the Cartesian va lue of this object [...] since up and down indices are the same in Cartesian space. 2. Replace all derivatives with cova riant derivatives. Thus is done by first writing all derivatives in comma notation, and then by replacing thos e commas with semicolons. For example: ∂ αBa = Ba,α → Ba;α . // B a;α = ∂α Ba – Γn aαBn (F.9.2) (15.2.2) Objects like ∂αBa which are not tensors become objects like B a;α which are tensors according to the theorem of (F.7.1). Note again that doing this does not affect the Cartes ian value of this object [...], because in Cartesian x-space Γ n aα = 0 as per (F.4.16). For a more general case like B abc..x; α shown in (F.7.2), making the replacement B abc..x, α makes no difference as long as B abc..x is a true tensor (W = 0), because all those Γ correction terms seen in (F.7.2) vanish in Cartes ian space. The same is true regardless of index positions. 3. Insert appropriate powers of g as needed to achieve appropriate tensor density weights so that all terms in [...] have the same weight and this weight equals the weight of Q---- . Tensor densities are discussed in Appendix D, and this weight adjustment idea appears in (D.2.3c). Chapter 15: Covariant Method 208 Note again that doing this does not affect the Cart esian value of this object [...] because g = 1 in Cartesian space (g = det(g ij) ). Comment : Since x-space is Cartesian, we know from (5.12.16) that g > 0 and g' > 0, so we don't need to use expressions like |g| and |g'| in our current context. 4. At this point one has Q---- = { tensorized form of the Cartesia n expression of interest } . (15.2.3) Since this is then a "true tensor equation" (it could be a true tensor density equa tion), according to the rule of covariance described in Section 7.15 the above e quation can be converted to x'-space by simply priming Q and all objects within { }. One then has Q' ---- = { tensorized form of the Cartes ian expression of interest }' (15.2.4) where the notation {...}' means that everything inside {...} is primed. Perhaps at this point one does some simplificati ons of the resulting {...}' object. The result then is the expression of Q---- in general curvilinear c oordinates if the underlying transformation x' = F(x) describes the transformation from Cartesian to curvilinear c oordinates. For a vector , recall (7.13.10) that Q = Q'nen so the "curvilinear coordinates" expression of Q involves the the x'-space components Q'a(x') but the tangent base vectors en are in x-space. A similar statement can be made for Q being any tensor, as shown for example in (E.2.9) which says Q = Σijk Q'ijk (ei⊗ej⊗ek) . These four steps then comprise "the covariant method ". The method of course applies to tensor analysis applications other than "curvilinear coordinates". Uniqueness. One might obtain two tensorizations of a Cart esian expression that look different. Both these tensorizations must lead to the same Q'---- in x'-s pace. For example, suppose one finds two tensorizations of a vector Qa , call them {...} 1a and {...} 2a which, when evaluated in Cartesian x-space, are equal {...} 1a = {...}2a when both sides are evaluated in Cartesian x-space . (15.2.5) One can then transform both objects to x'-space in the usual manner, {...}' 1a = Ra b{...}1b and {...}' 2a = Ra b{...}2b . (15.2.6) Therefore, since {...} 1a = {...}2a in x-space, one must have {...}' 1a = {...}'2a in x'-space. Example: In (15.7.6) below we will estab lish the following tensorization of ( @B)n (the vector Laplacian), ( @B)n = {.....} 1n = (Bj ;j);n – g-1/2εnab(g-1/2gbcεcdeBe;d);a . (15.2.7) Chapter 15: Covariant Method 209 Since in Cartesian x-space one can write ( @B)n = ∇2(Bn) = ∂j∂jBn = Bn,j ,j , another tensorization of (@B)n is given by (@B)n = {.....} 2n = Bn;j ;j . (15.2.8) Therefore, the following must be true in x'-space B'n;j ;j = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a (15.2.9) Explicitly verifying (15.2.9) takes some work and we do it below in Section 15.8. Both forms are used in Appendix I to compute @B in spherical and cylindrical coordinates. In the following Sections we shall in short orde r derive the curvilinear expressions for the basic differential operators using the cova riant method outlined above. This me thod is used as well (along with parallel brute force methods) to obtain expressions for objects ∇ v, divT ,∇T in Appendices G, H, J. 15.3 divergence (Chapter 9) In Cartesian coordinates, [div B] = ∂iBi = Bi ,i. The true tensorial tensor which matches this is [div B] ≡ Bi ;i since Γ = 0 for Cartesian coordinates by (F.4.16). That is to say, from (F.9.3), B a ;α = ∂α Ba + Γa αn Bn (F.9.3) so B i ;i = ∂i Bi + Γi in Bn = ∂i Bi = Bi ,i since Γc ab ≡ 0 . Since Bi ;i is a scalar, it is the same in x-space as in x'-space (Section 7.15 covariance), [div B] = [div B]' = B'i ;i = ∂'i B'i + Γ 'i in B'n (15.3.1) where B' n are the contravariant components of vector B in x'-space. But recall (F.4.2) in x'-space, Γ'i in = (1/ g' ) ∂ 'n(g ' ) . (F.4.2) Therefore [div B] = ∂'n B'n + (1/ g' ) ∂'n(g' )B'n = (1/ g' ) ∂'n (g' B'n). (15.3.2) This matches the result (9.2.1) obtained in Chapter 9 by geometric methods. 15.4 gradient and gradient dot vector (Chapter 10) If f is a scalar, then G = grad f transforms as an ordinary vector according to (2.4.2) and (2.5.1), so it is already a tensor. According to line 4 of (7.13.10) the expansion of such a vector may be written Chapter 15: Covariant Method 210 G = [grad f] = G'iei = (∂'if ') ei . (15.4.1) Also, grad f • B is a scalar, again already a tensor. Using this fact and line 3 of (7.13.10) that B = B'nen , [grad f]' • B' = [grad f] • B = [(∂'if ') ei] • (B'nen) = (∂'if ')B'nδi n = (∂ 'if')B'i (15.4.2) These two results match (10.1.7) and the second line of (10.2.4) of Chapter 10. As a reminder, since f is a scalar field, f '(x') = f( x). 15.5 Laplacian (Chapter 11) In Cartesian x-space the Laplacian of a scalar function f can be written as [lap f] Cart = ∂i∂if = f,i ,i . (15.5.1) The tensorized version of the Laplacian is then the following, using the , →; rule, lap f = f;i ;i . ( 1 5 . 5 . 2 ) To verify that this is so (we already know it is so), we call upon (F.9.19) with scalar B → f, f;a ;α = ∂α f;a + Γa αn f;n . (F.9.19) (15.5.3) Then (F.4.16) says Γ = 0 so f ;a ;α = ∂α f;a = ∂α f,a = f,a ,α so finally f;i ;i = f,i ,i . Since (15.5.3) is a true tensor equation, Section 7. 15 covariance says it takes this form in x'-space, f';a ;α = ∂'α f';a + Γ'a αn f';n . (15.5.4) But lap f = f;i ;i is a scalar, so we have [lap f] = [lap f]' = f' ;i ;i = ∂'i f';i + Γ'i in f';n = ∂'i f',i + Γ'i in f',n = ∂' i∂'if' + Γ'i in (∂'n f') . (15.5.5) But identity (F.4.2) says Γ'i in = (1/ g' ) ∂'n(g' ) so then lap f = ∂' i∂'if' + (1/ g' ) ∂ 'n(g' ) (∂'n f') = [ 1 / g' ] ∂'n [g' (∂'nf ')] . (15.5.6) This matches (11.5) found in Chapter 11. Chapter 15: Covariant Method 211 15.6 curl (Chapter 12) In Cartesian space we know that C = curl B and Ci = εijk∂jBk = εijk Bk,j. The obvious tensorized form is Ci = εijk Bk;j which we could verify using (F.9.2) with Γ = 0, but we know this is correct from the general theory (Step 2) presented in Section 15.2 . Since this is a true tensor equation, Section 7.15 covariance tells us that it takes this form in x'-space, C'i = ε'ijk B'k;j . ( 1 5 . 6 . 1 ) Recall from (D.4.8) that ε' ijk = εijk = the permutation tensor. As shown in Section D.8, C ≡ curl B is a vector density of weight -1 because it contains the ε tensor which has that weight. According to (E.2.26) the expansion of such a vector density is given by C = J-1C'iei . Therefore, curl B = C = J-1C'i ei = J-1 [ε'ijkB'k;j]ei = (1/ g' ) ε'ijk B'k;jei (15.6.2) But we now do call upon (F.9.2) in x'-space to show that ε'ijk B'k;j = ε'ijk B'k,j : B' a;α = ∂'α B'a – Γ'n aαB'n covariant rank-2 tensor // 2nd term is sym on a ↔α (F.9.2) so B' k;j = ∂'j B'k – Γ'n kjB'n so ε' ijk B'k;j = ε'ijk [∂'j B'k – Γ'n kjB'n] = ε'ijk ∂'j B'k – ε'ijkΓ'n kjB'n = ε' ijk ∂'j B'k – 0 // because ε' is antisymmetric on jk, but Γ'n kj is symmetric on jk † = ε' ijk B'k,j . ( 1 5 . 6 . 3 ) Inserting this result into (15.6.2) gives curl B = (1/ g' ) ε'ijk B'k,jei = (1/ g' ) ε'ijk(∂'jB'k)ei . This matches the result (12.3.3) found in Chapte r 12 based on geometric circulation integrals. † Q ≡ AijSij = AjiSji = (-Aij)(+Sij) = - AijSij = -Q ⇒ Q = 0 (15.6.4) i↔ j symmetries 15.7 vector Laplacian (Chapter 13) The vector Laplacian operator takes a bit more work. Since this is a major example of carrying out the "tensorization process" of Section 15.2, much detail is provided. In Chapter 13 the vector Laplacian is written @ (instead of ∇2) and is defined by Chapter 15: Covariant Method 212 @B ≡ grad(div B) – curl (curl B) . (13.1.4) (15.7.1) Recall from (13.3.4) that that in Cartesian space, [grad(div B) – curl (curl B)]n = ∇2(Bn) = (@B)n // Cartesian x-space (15.7.2) so we are allowed to use the LHS here to find the curvilinear expression of @B . In components, (@B)n = ∂n(∂jBj) - εnab∂a [curl B] b = ∂ n(∂jBj) - εnab∂a (εbde∂dBe) . (15.7.3) In Cartesian space up and down index position does not ma tter, see (5.9.1), and we are just jockeying the indices in search of a tensorized form with contr acted indices where possible. Writing (15.7.3) in comma notation, ( @B)n = (Bj ,j),n – εnab(εbdeBe,d),a (15.7.4) one is then free to replace commas with semicolons since Γ = 0 in Cartesian coordinates, see (F.4.16), (@B)n = (Bj ;j);n – εnab(εbdeBe;d);a . (15.7.5) There is a "technical difficulty" visible here. The first term (B j ;j);n is a true vector since it is the contraction of a true rank-3 tensor on two tilted indices, but the second te rm is a rank-3 tensor density of weight -2. In the second term, in order to neutralize the weight of each ε tensor density (from (D.4.9) each ε has weight -1), we shall now add a benign factor of g-1/2, as suggested in (D.2.3c), since g-1/2 is a scalar density of weight +1. This in turn is so from (5.12.13) that g' = J2g which says g has a weight of -2. The factor g-1/2 is benign since g = 1 in Cartesian coordina tes. So, with this upgrade installed, ( @B)n = (Bj ;j);n – g-1/2εnab(g-1/2εbdeBe;d);a . (15.7.6) Now both objects T b ≡ (g-1/2εbdeBe;d) and Vn ≡ g-1/2εnab Tb;a are true vectors, so the above equation says that a true vector is the difference between two other true vectors. Anticipating a few steps ahead when our ship lands in x'-space, we know that εbdeBe;d = εbdeBe,d due to the e↔ d symmetry of the Γ term in B e;d = Be,d - Γs edBs of (F.9.24), and this motivates us to perform a "tilt change" in this product. Such a tilt-change also gets the ;d index "down" which is a nicer place for it to be since it will soon become ∂d. The tilt operation is done as follows , ε bdeBe;d = εbd eBe ;d = εbdeBe;d = gbc εcdeBe;d . (15.7.7) Recall that tilt changes like this are only allowed with true tensor indices, (7.11.3), and that is why the ε tilt reversal can be done only after the semicolon in B e;d is installed. Eq. (15.7.6) then reads, Chapter 15: Covariant Method 213 (@B)n = (Bj ;j);n – g-1/2εnab(g-1/2gbcεcdeBe;d);a . (15.7.8) Although εbde is a fine tensor, we really prefer εcde since this is the permutation te nsor, and that is why the gbc factor is added in (15.7.7). At this point every index is a tensor index, so (15.7. 8) is a "true tensor equation" in the sense of Section 7.15. The above equation agrees exactly with the Cartesian expression of ( @B)n since g = 1, g bc = δb,c, and the semicolons are commas since Γ = 0, a fact verified in the Comment above. The point is that we have found a way to write the Cartesian expression of ( @B)n such that both terms are true tensors. The equation is therefore now "covariant" and the equation in x'-space can be obtained simply by priming all objects: (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a . (15.7.9) The semicolons were installed to obtain a true tensor equation, but now that we have successfully arrived in x'-space, we want to remove as many semicolons as possible because they imply "extra terms" since now Γ' ≠ 0. Consider for example, this grouping which appears in the above, ε' nab (g'-1/2g'bcε'cdeB'e;d);a = ε'nab T'b;a T' b ≡ (g'-1/2g'bcε'cdeB'e;d) . (15.7.10) Since T b;a = Tb,a – Γn baTn from (F.9.24), ε'nabT'b;a = ε'nab T'b,a because εnab is antisymmetric on a,b while Γn ba is symmetric, see (A.4.4). Thus, (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d),a . (15.7.11) ↑ But ε'cdeB'e;d = ε'cdeB'e,d for the exact same symmetry reason, so (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e,d),a . (15.7.12) ↑ Meanwhile, (F.9.1) says D;n = D,n so that (B'j ;j);n = (B'j ;j),n , since the object B'j ;j = B;j ;j is a scalar like D. Therefore, (@B)'n = (B'j ;j),n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e,d),a . (15.7.13) ↑ Since ε'cde is a constant (the permutation tensor, see below (D.4.1)), we now pull it through the ∂'a derivative implied by ,a and then show ing all derivatives the above becomes ( @B)'n = ∂'n(B'j ;j) – g'-1/2ε'nabε'cde ∂'a(g'-1/2g'bc∂'dB'e) . (15.7.14) From (15.3.2) and (F.9.1) that B ;j = B,j we know that [div B] = B;j ;j = B,j ;j = (1/ g' ) ∂'j (g' B'j) = (1/ g' ) ∂'i (g' B'i) . (15.7.15) Using this in the first term on right of (15.7.14) gives, Chapter 15: Covariant Method 214 (@B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i)} – g'-1/2ε'nabε'cde ∂'a(g'-1/2g'bc∂'dB'e) . (15.7.16) To compare this result to that of Chapter 13, we shuffle the indices in the second term as follows, g'-1/2ε'nabε'cde ∂'a(g'-1/2g'bc∂'dB'e) // now take a,b,c,d,e → A,B,C,D,E g'-1/2ε'nABε'CDE ∂'A(g'-1/2g'BC∂'DB'E) // now take A →c, B→d, C→e, D→a, E→b g'-1/2ε'ncdε'eab ∂'c(g'-1/2g'de∂'aB'b) . (15.7.17) Then (15.7.16) reads, ( @B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i)} – g'-1/2ε'ncdε'eab ∂'c(g'-1/2g'de∂'aB'b) . (15.7.18) This may now be compared with (13.1.17) which we quote : @ B = G – V G = ∂'n{ (1/ g' ) ∂'i (g' B'i)} en B = B'nen V = [ (1/ g' ) ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'aB'b) }] en B = B'nen (15.7.19) The results agree, and further processing of this result is done in Chapter 13. This concludes our discussion of how the curvilinear-coordinate differential opera tor forms of Chapters 9,10,11,12,and 13 may be obt ained relatively quickly using the noti ons of tensor densities (App. D) and covariant derivatives (App. F), without resorting to the use of N-piped geometric constructs. 15.8 Verification that two tensorizations are the same In Section 15.7 we obtained the x-space @B "tensorization" (15.7.8) which written in x'-space says (@B)'n = B'j ;j;n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a . (15.7.9) (15.8.1) Since in Cartesian coordinates ( @B)n = ∇2(Bn) = ∂j∂j Bn = Bn,j ,j , an obvious alternate tensorization is Bn,j ,j → Bn;j ;j so that in x'-space (@B)'n = B'n;j ;j . ( 1 5 . 8 . 2 ) In Section 15.2 we showed that tensorization is unique, so these two tensorizations must be the same. We know they are the same, but in the time-honored traditi on of "trust but verify", we would like to actually verify their equality. As many readers know, verifyin g things that are supposedly obvious often leads to the discovery of problems. The claim that the above two tensorizations are the same is not general in the abstract. It is specific in that it applies in the "tensor world" built upon a transformation x' = F(x) where x-space is Cartesian. The two forms are not equal if x-space has some metric tensor other than g = 1. Our task then is to show that the follo wing is true when x-space is Cartesian, Chapter 15: Covariant Method 215 B'n;j ;j = B'j ;j;n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a ? (15.8.3) The plan is to rewrite the above equation in a sequence of reducing steps until we get to an equation that obviously is true. Then one then can reverse the sequence to conclude that (15.8.3) is true. We shall mark each step with a question mark to the right, to indicate that this is "som ething we want to show is true". When the final step is reached which is true, the reader then goes backwards and erases the question marks. As a first and rather large step in te rms of clutter reduction, we claim that, (g' -1/2g'bcε'cdeB'e;d);a = g'-1/2g'bcε'cde (B'e;d);a . (15.8.4) This is an application of several facts derived in Appendix F. The main idea is Theorem (F.10.15) concerning what objects are "extractable" from a covariant derivative expression. Eq. (F.10.17) shows that ε' cde is extractable since ε 'cde α; = 0, while (F.10.18) shows that g' bc is extractable since g' bc;α= 0. Finally, (F.10.20) says that, for any real s, g's is extractable since (g's);α = 0, recalling that g' > 0. We then arrive at B' n;j ;j = B'j ;j;n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a = B ' j ;j;n – g'-1/2ε'nabg'-1/2g'bcε'cde (B'e;d);a // now use g' bc and note ε'nab = ε'ban = B 'j ;j;n – g'-1ε'ban ε'bde B'e;d;a . Therefore B'n;j ;j = B'j ;j;n – g'-1ε'bdeε'ban B'e;d;a ? (15.8.5) We would next like to have all the B' tensors be multiples of B' e;d;a which appears in the last term, so : B' n;j ;j = B'n;a ;a = g'ne g'adB'e;d;a B'j ;j;n = B'd ;d;n = g'de g'na B'e;d;a . (15.8.6) Installing these expressions into (15.8.5) gives g' ne g'adB'e;d;a = g'de g'na B'e;d;a – g'-1 ε'bdeε'bna B'e;d;a ? or [ g' ne g'ad - g'de g'na + g'-1 ε'bde ε'bna ] B'e;d;a = 0 ? (15.8.7) At this point, one might hope that [....] = 0 and we ar e done. If this were true, our claim (15.8.3) would be true for any metric tensor g, but we know that is not the case, so we do not expect to have [...] = 0. The next step is to call upon (D.11.11) expressed in x'-space which says Chapter 15: Covariant Method 216 g'-1 ε'sABε'sA'B' = g'AA'g'BB' – g'BA'g'AB' (D.11.11) or g'-1 ε'bdeε'bna = g'dng'ea – g'en g'da . (15.8.8) Putting this into (15.8.7) gives [ g'ne g'ad - g'de g'na + g'dng'ea – g'eng'da ] B'e;d;a = 0 ? or [ - g' de g'na + g'dng'ea ] B'e;d;a = 0 ? or [ g' dng'ea - g'deg'na ] B'e;d;a = 0 ? (15.8.9) As expected, we find [...] ≠ 0. We have to use the fact that x-space is Cartesian. To this end write the covariant transformation rule for rank-3 tensor B e;d;a as shown for example in (7.10.1), B'e;d;a = ReERdDRaA BE;D;A = ReERdDRaA BE,D,A = ReERdDRaA ∂D∂A BE (15.8.10) where the critical Cartesian x-space fact will be that ∂ D∂A BA is symmetric under A↔ D , Inserting the above transformation rule into (15.8.9) gives, [ g'dng'ea - g'deg'na ] ReERdDRaA∂D∂A BE = 0 ? or g' dng'ea ReERdDRaA ∂D∂A BE = g'deg'na ReERdDRaA ∂D∂A BE ? (15.8.11) At this point we recall from (7.5.9) that g' ** raises the first index of R ** , whereas the second index of R** goes up and down for free since g = 1 in x-space, so we will take second i ndices all down. Thus, R aERnDRaA∂D∂A BE = RdERdDRnA ∂D∂A BE ? or R a ERn DRaA∂D∂A BE = Rd ERdDRn A∂D∂A BE ? (15.8.12) Now on the right side do D ↔A on these dummy indices, but then restore the ∂D∂A order so Ra ERn DRaA∂D∂A BE = Rd ERdARn D∂D∂A BE ? (15.8.13) and changing d →a on the right Ra ERaARn D ∂D∂A BE = Ra ERaARn D∂D∂A BE ? (15.8.14) The two sides are exactly the same, so we are done and thus (15.8.3) is verified. Appendix A: Reciprocal Base Vectors 217 Appendix A: Reciprocal Base Vectors the Hard Way A.1 Introduction Note : This Appendix is written in the developmental not ation, not the Standard Notation, though a few equations are translated to the latter form. The rules for translation to Standard Notation are (7.13.1) (7.5.2) (7.5.4) (7.4.1) (7.4.1) En → en Rij → Ri j Sij → Si j g¯'nm → g'nm g'nm → g'nm . The tangent base vectors en are defined in Chapter 3, while the reciprocal base vectors En are defined in Chapter 6. Here are some facts regarding th ese vectors gathered from those chapters: (3.2.5) and (6.1.4) (6.2.4) (6.2.4) and (6.1.2) (3.2.7) and (6.6.3) and (6.1.2) (en)k = Skn en • em = g¯'nm | en| = g¯'nn = h'n S = [ e1, e2, e3 .... eN ] ( En)i ≡ gia Rna En • Em = g'nm |En| = g'nn R = [ E¯1, E¯2, E¯3 .... E¯N ]T = g' na Sia en • Em = δn,m En ≡ g'ni ei en = g¯'ni Ei . (A.1.1) These facts are all applicable in the Picture A c ontext with arbitrary metric tensors g' and g, (A.1.2) This Appendix begins with a differe nt definition of something called Ek. Although the definition is meaningful in the general Picture A context, the object so defined only agrees with the Ek of Chapter 6 if x-space is Cartesian (g = 1). The reason can be traced to the fact that the dot product rule en • Em = δn,m is only valid for the Appendix A definition of Em when g = 1 because only then is a cross product orthogonal to all its component vectors. One application of the reciprocal base vectors is in the study of curvilinear coordinates where one always takes g = 1, and g' is then the curvilinear coordinates metric tensor of interest. Therefore, the reader should think of this Appendix in the context of Picture B (A.1.3) Appendix A: Reciprocal Base Vectors 218 A.2 Definition of E n The reciprocal base vectors are defined in the following very strange looking and clumsy manner, ( Ek)α ≡ det(R) (-1)k-1 εαi1i2i3...ik...iN (e1)i1 (e2)i2 ...... ( ek)ik.......... ( eN)iN (A.2.1) where N is the number of dimensi ons of the Cartesian x-space RN in which the vectors en and En exist. Notice that the ε subscript i k is "crossed out" and the same for factor ( ek)ik . Crossed out means they are simply missing, they are omitted. Thus, in the above expression there are N-1 implied summation indices (α is fixed) and there are N-1 factors of the form ( en)in . The object ε has N subscripts and is the "totally antisymmetric tensor" in N dimensions: ε123...N ≡ +1, and each time any two indices on ε are swapped, ε negates. For example, ε1234 = 1 but ε1432 = -1. If two indices are the same, then ε = 0. A.3 Simpler notation To avoid dealing with subscripts on subscripts, one can rewrite the a bove definition in a less precise but simpler notation ( Ek)α ≡ det(R) (-1)k-1εαabc...x (e1)a(e2)b ...... ( eN)x // κ(k) and ( ek)κ are missing (A.3.1) In this notation, subscript x stands for the N th letter of the alphabet (imagine N ≤ 26). If κ is the kth letter of the alphabet, then κ is missing from the indices on ε, and the factor ( ek)κ is missing from the product of factors. For example, if k = 2, then summation index κ = b is missing from the ε. Now take the ε subscript α and slide it right to the "hole" where κ is missing, picking up a minus sign for each step of this slide. Moving k-1 positions results in (-1)k-1. Thus the above becomes, ( Ek)α ≡ det(R) εabc..α..x (e1)a(e2)b ...... ( eN)x // (ek)κ is missing, α in κ position (the kth) (A.3.2) Example : For N = 3 the above becomes, ( E1)α ≡ det(R) εαbc(e2)b(e3)c ⇒ E1 = det(R) e2 x e3 a is missing ( E2)α ≡ det(R) εaαc(e1)a(e3)c ⇒ E2 = det(R) e3 x e1 b is missing ( E3)α ≡ det(R) εabα(e1)a(e2)b ⇒ E3 = det(R) e1 x e2 c is missing (A.3.3) and the results are cyclic. Here is a detail from the middle line εaαc(e1)a(e3)c = – εαac(e1)a(e3)c = + εαca(e1)a(e3)c = εαca(e3)c (e1)a = [ e3 x e1 ]α (A.3.4) A.4 Generalized Cross Product of N-1 vectors of dimension N One can define a generalized "cross product" of N-1 v ectors, each of dimension N, in this fashion: Qa ≡ εabc...x BbCcDd.....Xx ( A . 4 . 1 ) Appendix A: Reciprocal Base Vectors 219 where x and X represent the Nth letter of the alphabet. The ε object is again the totally antisymmetric tensor with N indices. In vector notation one writes this symbolically as Q = B x C x D x ... x X / N-1 factors, N-2 crosses . (A.4.2) This vector notation is defined by the previous line. The vector Q is orthogonal to all the vectors from which it is constructed! For example (here is the point where Q • C ≡ gabQaCb needs to be QaCa, so g = 1 is required in x-space) Q • C = CaQa = Ca εabc...x BbCcDd.....Xx = BbDd...Xx { εabc...x CaCc } . (A.4.3) But {..} is the contraction of something symmetric under a ↔c (CaCc) with something antisymmetric under a↔c (εαabc...x ) and therefore {..} = 0. In general, S acAac = Sca Aca // relabel both dummy summation indices = S ac (-Aac) // S is Symmetric, A is antisymmetric = - S ac Aac // = the negative of the starting expression = 0 . ( A . 4 . 4 ) Similarly, Q•A = 0, Q•B = 0 and so on. Swapping the position of any two vectors in the generalized cross product causes Q to change sign. For example, swapping B and C, Qa ≡ εabc...x CbBcDd.....Xx = εacb...x CcBbDd.....Xx // b ↔ c = - εabc...x BbCcDd.....Xx = -Q a . // swap indices on ε (A.4.5) Thus, the notions of orthogonality and interc hange are consistent with the regular Q = B x C cross product for N=3. When N=2, one must be a little careful with this notation. The component equation is Q a ≡ εab Bb ⇒ Q 1 = B2 and Q2 = -B1 . (A.4.6) One might be tempted to express the vector equation as Q = B since there are no "no crosses". This vector equation is wrong , while the component equation is correct . One can rescue the vector notation by a simple trick. When N=2 the vector B can be represented of course as B = B11^ + B2 2^. Imagine this 2D space to be embedded in the usual 3D space with a third axis 3^. Then consider this 3D cross product: Q = B x 3^ ⇒ Q a ≡ εabc Bb(3^)c = εabc Bbδ3,c = εab3 Bb = εabBb . (A.4.7) Thus, this trick reproduces the correct component equation, and it makes more obvious the fact that Q is orthogonal to B. Summary : The generalized cross product Q of N-1 vectors each of dimension N can be expressed in both component and vector notation: Appendix A: Reciprocal Base Vectors 220 Qa ≡ εabc...x BbCcDd.....Xx Q = B x C x D x ... x X / N-1 factors, N-2 crosses (A.4.8) Q is orthogonal to all the vectors from which it is composed. Swapping any two vectors negates Q. When N=2, one can rescue the otherwise failing v ector notation by thinking of it as saying Q = B x 3^. Comment: Notice that Q = B x C x D is defined for 4-vectors only. This is a completely different animal from the object Q = B x (C x D) which is defined for 3-vectors onl y. This latter object contains two ε factors, while the former only one. A.5 Missing Man Formation We now make a small variation in the no tation. Start with the above equation, Qa ≡ εabc...x BbCcDd.....Xx , ( A . 5 . 1 ) then change a to α, back up all the Latin letters by one (but leave the last as "unknown" x), and assume that some subscript κ and factor K κ are "missing". The result is, Qα ≡ εαac...x AaBbCc.....Xx . // κ and Kκ are missing (A.5.2) There are still N-1 factors, and one can still write this in vector notation Q = A x B x C x ... x X // K is missing (A.5.3) and of course it is still true that Q•C = 0, etc. For N=2 the vector notation is rescued as in (A.4.7) above. A.6 Apply this Notation to E Compare the above Q α of (A.5.2) to the {...} part of the (A.3.1) definition of ( Ek)α , ( Ek)α ≡ det(R) (-1)k-1{ εαabc...x (e1)a(e2)b ...... ( eN)x } // κ(k) and ( ek)κ are missing; N ≥ 2 (A.3.1) (A.6.1) Therefore, the definition of Ek for N > 2 can be written in this vector notation, Ek ≡ det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing; N > 2 . (A.6.2) The reciprocal base vector Ek is thus orthogonal to all the ta ngent base vectors from which it is constructed (remember ek is missing)! For example, for N=3 the three E vectors are given by E1 = det(R) (-1)1-1 e2 x e3 = det(R) e2 x e3 E2 = det(R) (-1)2-1 e1 x e3 = det(R) e3 x e1 E3 = det(R) (-1)3-1 e1 x e2 = det(R) e1 x e2 ( A . 6 . 3 ) Appendix A: Reciprocal Base Vectors 221 which agrees with the results (A.3.3). For N =2 ( E's label corresponds to the missing e's label ), E1 = det(R) (-1)1-1 e2 x 3^ = det(R) e2 x 3^ or ( E1)k = det(R) εka(e2)a E2 = det(R) (-1)2-1 e1 x 3^ = - det(R) e1 x 3^ or ( E2)k = -det(R) ε ka(e1)a (A.6.4) One can combine these two lines into one as follows ( eg, k = 1, then 3-1 = 2, etc) Ek = det(R) (-1)k-1 e3-k x 3^ = det(R) e3-k x 3^ or ( E1)k = det(R) (-1)k-1εka(e3-k)a . (A.6.5) The vector "trick" notation shows that E1•e2 = 0 and E2•e1 = 0, E1•e2 = det(R) e2 x 3^ • e2 = 0 E2•e1 = -det(R) e1 x 3^ • e1 = 0 ( A . 6 . 6 ) and also E1•e1 = det(R) εka(e2)a (e1)k = det(R)det[ e1, e2] = det(R)det(S) = 1 E2•e2 = -det(R) ε ka(e1)a (e2)k = -det(R)det[ e2, e1] = det(R)det(S) = 1 . (A.6.7) It is shown next that these N=2 results are special cases of a general fact: Em • en = δm,n . Eq. (5.11.5) showed that em • en = g¯'mn . The other two dot products are now considered. A.7 Compute E m • en One can now compute, for general N, Ek • ek = ( Ek)α(ek)α = { det(R) (-1)k-1εαabc...x (e1)a(e2)b ...... ( eN)x } ( ek)α . (A.7.1) k is missing Slide α to the right in the ε subscript field and put it into the hole of the missing subscript κ, picking up (-1)k-1. At the same time, move the ( eκ)α to the left and position it in its proper place in the product of factors, Ek • ek = ( Ek)α(ek)α = { det(R) εabc..α..x (e1)a(e2)b ... (eκ)α ... (eN)x } = det(R) det [ e1, e2, e3 .... eN ] = det(R) det(S) = det(RS) = det(1) = 1 (A.7.2) where S = [ e1, e2, e3 .... eN ] from (3.2.7). We already know that Ek is orthogonal to all the en which form the generalized cross product, therefore Em • en = δm,n ( A . 7 . 3 ) which is the "duality relation" discusse d more generally in Section 6.2. Appendix A: Reciprocal Base Vectors 222 A.8 Compute E n • Em Since the vectors { en } are linearly independent and thus form a basis in RN, Em can be expanded onto the en , Em = Σn An(m) en ( A . 8 . 1 ) δm,k = Em • ek = Σn An(m) en • ek = Σn An(m) g¯'nk . (A.8.2) Multiplying both sides of (A.8.2) by g' ki and summing on k gives LHS = Σk g'ki δm,k = g'mi RHS = Σn An(m) (Σk g¯'nk g'ki) = Σn An(m) (g¯'g')ni = Σn An(m)δn,i = Ai(m) . (A.8.3) Therefore A i(m) = g'mi so, Em = Σn An(m) en = Σn g'mn en ( A . 8 . 4 ) which is to say Ek is this linear combination of the ei, which is the definition used in Chapter 6, (6.1.2), Ek = Σi g'ki ei = g'ki ei // implied sum on i // Std Notation: ek = Σi g'ki ei (A.8.5) This may be compared with the previous result (A.6.2), Ek ≡ det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing; (A.6.2) It seems rather impressive that these two dissimilar ways of writing E are equal. Finally, En • Em = En • (g'mi ei) = g'mi (En • ei) = g'mi δn,i = g'mn = g'nm // g' is symmetric (A.8.6) A.9 Summary of relationship between th e tangent and reciprocal base vectors en • em = g¯'nm En • Em = g'nm en • Em = δn,m En = Σi g'ni ei en = Σi g¯'ni Ei g¯' = g'-1 . (A.9.1) Although these results have just been derived in the Picture B context, they are also valid in the more general Picture A context, as shown in Chapter 6 in which the equation En = Σi g'ni ei is used as the definition of En . As a reminder, the cross product expression for En is only valid in Picture B where g=1. In Standard Notation, the summa ry above can be restated as en • em = g'nm en • em = g'nm en • em = δnm en = Σi g'ni ei en = Σi g'ni ei g'ab = (g'ab) -1 (A.9.2) Appendix A: Reciprocal Base Vectors 223 A.10 Another Cross Product Notation and another expression for E Go back to the general cross product of N-1 vectors each of dimension N, Q = B x C x D x ... x X . // N-1 factors, N-2 crosses (A.4.2) Replace B,C,D ... by vectors A(n), Q = A(1) x A(2) x A(3) x ... x A(N-1) // N-1 factors, N-2 crosses (A.10.1) It is convenient to write this using a product symbol Π x , Q = Πx i=1N-1 A(i) = Πx i A(i) ( A . 1 0 . 2 ) where in the second form it is unders tood that i takes on all values i = 1 to N-1. The superscript x means that this is not a regular product, it is our generalized cross product. This Πx symbol also implies correct handling of the special case N=2 such that Q = Πx i=11 A(i) ≡ A(1) x 3^ // ≠ A(1) (A.10.3) as discussed in (A.4.7) above. This same Πx notation can be applied to the "missing ma n formation" of Section A.5 above. Suppose Q = A(1) x A(2) x A(3) x ... x A(N) // A(n) is missing . (A.10.4) One can write this as Q = Πx i=1..N,i ≠n A(i) ≡ Πx i≠n A(i) . ( A . 1 0 . 5 ) And of course this idea can be applied to the expression for Ek Ek = det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing; (A.6.2) Ek = det(R) (-1)k-1 Πx i≠k ei . ( A . 1 0 . 6 ) Once again, for N=2 the Π x symbol implies that ( ek = "missing", e3-k = the one not missing) Πx i≠k ei = Πx i=1..2,i ≠k ei = e3-k x 3^ ( A . 1 0 . 7 ) Ek = det(R) (-1)k-1 e3-k x 3^ ( A . 1 0 . 8 ) which is the "trick" notation of s(A.4.7) above for the N=2 case. Appendix B: Geometry of N-pipeds 224 Appendix B: The Geometry of Parallelepipeds in N dimensions B.1 Overview This Appendix presents a simple method for construc ting an N dimensional parallelepiped, which name we shorten to "N-piped". It is found that an N-piped has 2N vertices and N pairs of faces for a total of 2N faces, and the locus of points that make up each of these faces is stated. Each face of an N-piped is in fact an (N-1)-piped which has 2N-1 vertices and is planar in N dimensions (meaning it lies on an N-1 dimensional flat surface). The two faces which ma ke up each face pair lie on parallel planes in RN. For example, for N=3 each face is a 2-piped having 23-1 = 4 vertices, and there are N=3 face pairs for a total of 6 faces, and each pair of faces is planar in 3 dimensions. For N=4 there are 4 pairs of faces for a total of 8 faces. Each face is a 3-piped having 2 4-1 = 8 vertices. For example, one would say that each face of a 4-cube is a 3-cube. It is not intuitively obvious that two faces each of which is a regular cube can in fact lie on surfaces which are planar and parallel in 4 dimensions, but we show how this works below. It is then shown that, if the N-piped is spanned by the N tangent base vectors en of Chapter 3, the normal vectors for the pairs of parallel faces are just the reciprocal base vectors En of Chapter 6. Section B.5 focuses on the area and volume of N-pipeds in various dimensions, and simple expressions for the volume and vector areas of the faces of an N-piped are obtained. Rather than just state the results in N dimensi ons, we attempt an inductive approach to provide motivation for the N dimensional results. In this appro ach, cases N = 2,3.. are treated with nearly identical boilerplate templates to build up the inductive case. A reader interested in the details here should r ead Appendix A first since use is made of various Appendix A results including the generalized cross product idea. On the other hand, a reader not interested in details might just read the Appe ndix B Summary presented below in Section B.6. B.2 Preliminary: Equation of a plane in N dimensions Consider an arbitrary plane drawn in N space which does not pass through the origin. There is some point on that plane which lies closer to the orig in than all other points on the plane. Let p be a vector from the origin to that closest point, and let r represent a point lying on the plane, ( B . 2 . 1 ) Since p is normal to the plane, and since r-p is a vector lying in the plane, it follows that p•(r-p) = 0 ⇒ r•p = p2 ⇒ r•p^ = p . (B.2.2) Therefore, one way to write the equation of a plane in N dimensions is Appendix B: Geometry of N-pipeds 225 r•p^ = p r = (x1, x2, .....xN) . (B.2.3) where p^ is the unit vector normal to the plane which poin ts "away from the origin", and where p > 0 is the distance of closest approach of the plane to the origin. In the limit p →0, the plane passes through the origin and the equation is then r•p^ = 0 where p^ is either normal to the plane. B.3 N-pipeds and their Faces in Various Dimensions (a) The 1-piped Start with N = 1 where the piped is some arbitrary line segment e1 in direction e^1 having length e 1, with one end affixed to the origin of the real axis : (B.3.a.1) This piped has two vertices located at v1 = 0 and v2 = e1. These two vertices are also the "faces" of this 1-piped, so there are two faces (one pair of faces) . These faces are 0 dimensi onal and therefore don't point in any direction (they are the endpoints of the vector e1). The 1-piped is a piece of a plane in 1 dimension (a line). One can think of the vertex at the origin as the "generator 0- piped" and the other vertex as the partner face of the generator, in the sense of the generator idea described below. The loci of points in the 1-piped's interior "volume" is given by rvolume1 = α1 e1 0 ≤ α1 ≤ 1 . (B.3.a.2) The volume of this 1-piped is e 1. (b) The 2-piped Now add another dimension, going to N=2. Introduce a unit vector e^2 in some arbitrary direction in R2 other than e1 so that e1 and e2 are linearly independent. Take the 1- piped described above (line segment) and translate it by e2 to create a new copy of the line segment. The original 1-piped we call the generator piped, and the copy is the partner of the generator piped which, it will be shown, lies on a plane (a 1-plane = line) which is parallel to the plane of the genera tor piped, but its plane does not pass through the origin. In N=2 dimensions, the generator 1-piped and its partner are now "faces" of a 2-dimensional object, a parallelogram = a 2-piped. Draw line segments from all the vertices of the generator piped to matching vertices of its partner piped (add 2 line segments) to make 2 additional "side" faces. One of these faces necessarily touches the origin, and the other face does not. Faces always occur in parallel pairs, one face of which touches the origin, and one of which does not , the latter we will call the "partner" face. For our 2-piped, each face is a 1-piped. There are now four faces, each is a line segment. Appendix B: Geometry of N-pipeds 226 (B.3.b.1) The loci of the 2-piped's volume and of its four 1-piped faces are given by rvolume2 = α1e1 + α2 e2 0 ≤ α1,α2 ≤ 1 rface2 = α1e1 0 ≤ α1 ≤ 1 // the generator face rface2p = α1e1 + e2 0 ≤ α1 ≤ 1 // partner of the generator face rface1 = α2e2 0 ≤ α2 ≤ 1 // side face touching the origin rface1p = α2e2 + e1 0 ≤ α2 ≤ 1 // partner of the above side face (B.3.b.2) The origin-touching faces are numbered using the index of the en vector that does not appear in the locus for the face. This seems strange but for N > 2 it will be clear why this is done. The above results can be summarized as rvolume3 = Σnαnen 0 ≤ αn ≤ 1 rface(i) = Σn≠iαnen 0 ≤ αn ≤ 1 i = 1,2,3 rface(ip) = Σn≠iαne + ei 0 ≤ αn ≤ 1 i = 1,2,3 (B.3.b.3) According to Note 1 following (6.2.7), it is possible to construct vectors E1 and E2 as linear combinations of e1 and e2 such that the following is true: Ei• ej = δi,j // Ek = Σi=12 g'ki ei, see Note 4 following (6.2.7) (B.3.b.4) If one interprets the en vectors as tangent base vectors for some transformation F, then the two vectors En are the corresponding reciprocal base vectors which are discussed in Chapter 6 and Appendix A. Consider now these dot products: E2 • rface2 = E2 • α1e1 = 0 ⇒ E^2 • rface2 = 0 E2 • rface2p = E2 • [ α1e1+ e2] = 1 ⇒ E^2 • rface2p = 1/E2 . (B.3.b.5) According to (B.2.3), the first line says that face 2 lies on a plane which passes through the origin and which has normal vector E^2. Also according to (B.2.3), the second line says that face 2p has the same normal and its plane is therefore parallel to face 1 but misses the origin by distance 1/|E 2|. Similarly, E1 • rface1 = E1 • α2e2 = 0 ⇒ E^1 • rface1 = 0 E1 • rface1p = E1 • [ α2e2+ e1] = 1 ⇒ E^1 • rface1p = 1/E1 . (B.3.b.6) Appendix B: Geometry of N-pipeds 227 These two faces are also parallel, both having normal E^1. The first touches the origin while the partner's plane misses the origin by distance 1/|E 1| . The conclusions that En is normal to face n, and that the pair of faces n and np are parallel, do not depend on the specific upper endpoints of the ranges of α1 and α2 which happen to be given as 1 above. This seems pretty obvious since rescaling the edges of a parallelogram does not affect its normal vector. (c) The 3-piped Now add another dimension, going to N=3. Introduce a unit vector e^3 in some arbitrary direction in R3 so that ( e^1,e^2,e^3) are linearly independent. Take the 2-piped described above (parallelogram) and translate it by distance e 3 in the e^3 direction to create a new copy of the 2-piped. The original 2-piped we call the generator piped, and the copy is the partner of the generator piped which, as will now be shown, lies on a plane which is parallel to that of the generator pipe d, but which does not pass through the origin. In N=3 dimensions, the generator 2-pipe d and its partner are now "faces" of a 3-dimensional object, a parallelepiped = a 3-piped. Draw line segments from all 22 vertices of the generator piped to the corresponding vertices of its partne r piped (add 4 line segments), to get 4 additional side faces. Two of these faces necessarily touch the origin, and the other two do not. For the 3-piped, each face is a 2-piped. There are now 2*3 = 6 faces, each is a 2-piped. (B.3.c.1) The loci of the 3-piped's volume and of its six 2-piped faces are given by rvolume3 = α1e1 + α2 e2 + α3 e3 0 ≤ α1,α2,α3 ≤ 1 rface3 = α1e1 + α2 e2 0 ≤ α1,α2 ≤ 1 // the generator face rface3p = α1e1 + α2 e2 + e3 0 ≤ α1,α2 ≤ 1 // partner face to the above rface2 = α1e1 + α3 e3 0 ≤ α1,α3 ≤ 1 // the generator face rface2p = α1e1 + α3 e3 + e2 0 ≤ α1,α3 ≤ 1 // partner face to the above rface1 = α2e2 + α3 e3 0 ≤ α2,α3 ≤ 1 // the generator face rface1p = α2e2 + α3 e3 + e1 0 ≤ α2,α3 ≤ 1 // partner face to the above (B.3.c.2) Notice that the partner face locus is created from the non-partner face by adding "the other" base vector. For example, face 3 is "spanned" by base vectors e1 and e2 so e3 is added to get the partner face face3p. A Appendix B: Geometry of N-pipeds 228 partner is just a copy of the non-partner which is tran slated by a constant vector. The above results can be summarized in this concise manner: rvolume3 = Σnαnen 0 ≤ αn ≤ 1 rface(i) = Σn≠iαnen 0 ≤ αn ≤ 1 i = 1,2,3 rface(ip) = Σn≠iαne + ei 0 ≤ αn ≤ 1 i = 1,2,3 (B.3.c.3) According to Note 1 following (6.2.7), it is possible to construct vectors E1, E2, E3 as linear combinations of e1, e2, e3 such that the following is true, Ei• ej = δi,j // Ek = Σi=13 g'ki ei, see Note 4 following (6.2.7) (B.3.c.4) If one interprets the en vectors as tangent base vectors, then the three vectors En are the corresponding reciprocal base vectors which are discussed in Chapter 6 and Appendix A. Consider now these dot products: E1• rface1 = E1• [ α2e2 + α3 e3] = 0 ⇒ E^1• rface1 = 0 E1• rface1p = E1• [α2e2 + α3 e3 + e1] = 1 ⇒ E^1• rface1p = 1/E1 . (B.3.c.5) According to (B.2.3), the first line says that face 1 lies on a plane which passes through the origin and which has normal vector E^1. Also according to (B.2.3), the sec ond line says that face 1p has the same normal and its plane is therefore parallel to face 1 but misses the origin by distance 1/|E 1| . A similar pair of equations obtains for each of the other face pairs. (d) The N-piped Now add another dimension, going from N-1 to N. Introduce a unit vector e^N in some arbitrary direction in RN so that ( e^1... e^N ) are linearly independent. Take the (N-1 )-piped described above and translate it by distance e N in the e^N direction to create a new copy of the (N-1)-piped. The original (N-1)-piped we call the generator piped, and the copy is the partner of the generator piped whic h, it will be shown, lies on a plane which is parallel to that of the generator piped, but which does not pass through the origin. The generator (N-1)-piped and its partne r are now "faces" of a N-dimensional object, an N-piped. Adding this partner piped doubles the total vertex count. Draw line segments from all 2N-1 vertices of the generator piped to the corresponding vertices of its partner piped to get 2N-2 additional side faces for a total now of 2N faces. There are N pairs of "faces" because there are N ways to omit a single ei from the list of vectors which span a face, so including the partner faces an N- piped has 2N faces in total. Half of these faces necessarily touch the origin, and the other half do not. Each face is an (N-1)-piped. It is convenient to refer to the partner face of a pa ir as "the far face" and the other one, which touches the origin, as "the near face". We leave the N-piped drawing which would be labele d (B.3.d.1) to the reader's imagination, and we omit a detailed list of the face volumes which we could called (B.3.d.2), giving just the summary below. Appendix B: Geometry of N-pipeds 229 The loci of the N-piped's volume and of its 2N (N-1)-piped faces are given by: rvolumeN = Σnαnen 0 ≤ αn ≤ 1 rface(i) = Σn≠iαnen 0 ≤ αn ≤ 1 i = 1,2...N rface(ip) = Σn≠iαnen + ei 0 ≤ αn ≤ 1 i = 1,2...N (B.3.d.3) Notice that the partner face locus is created from the non-partner face by adding "the other" base vector. For example, face i is "spanned" by base vectors en n ≠ i, so it is ei that one adds to get the partner. A partner is just a copy of the non-partner which is translated by a constant vector. According to Note 1 following (6.2.7), it is possible to construct vectors E1...EN as linear combinations of e1.. eN such that the following is true, Ei• ej = δi,j // Ek = Σi=12 g'ki ei, see Note 4 following (6.2.7) (B.3.d.4) If one interprets the en vectors as tangent base vectors, then the N vectors En are the corresponding reciprocal base vectors which are discussed in Chapter 6 and Appendix A. Consider now these dot products: Ei• rface(i) = Ei• [Σn≠iαnen] = 0 ⇒ E^i• rface(i) = 0 Ei• rface(ip) = Ei• [Σn≠iαne + ei] = 1 ⇒ E^i• rface(ip) = 1/Ei (B.3.d.5) The first line says that face i lies on a plane which passes through the origin and which has normal vector E^i. The second line says that face ip has the same normal and its plane is therefore parallel to face i but misses the origin by distance 1/|E i| . B.4 The question of inward versus outward facing normal vectors It has been shown above that, for an N-piped, the pair of faces i and ip has normal vector Ei. For one of these faces, Ei will be an outward directed normal, while for the other it will be an inward directed normal. One might like to know which is which. Here is one way to find out. First, construct these three vectors (piped) center = Σn(1/2) en // vector from origin to piped center (face i) center = Σn≠i(1/2) en // vector from origin to center of face i (face ip) center = Σn≠i(1/2) en + ei // vector from origin to center of face ip . (B.4.1) Construct vectors from piped center to face centers ( results here are fairly obvious) Appendix B: Geometry of N-pipeds 230 (face i) center - (piped) center = [Σn≠i(1/2) en] - Σn(1/2) en = - (1/2) ei (face ip) center - (piped) center = [Σn≠i(1/2) en+ ei ] - Σn(1/2) en = + (1/2) ei . (B.4.2) Then compute Ei • {(face i) center - (piped) center } = Ei • [- (1/2) ei ] = -(1/2) < 0 Ei • {(face ip) center - (piped) center } = Ei • [+ (1/2) ei ] = +(1/2) > 0 . (B.4.3) One may conclude that Ei is an outward pointing normal for face ip (the far face of the pair). Therefore, - Ei is an outward pointing normal for face i (the near face of the pair), which recall is the face which touches the origin.. Here is an illustration for the case N = 2: (B.4.4) B.5 The Face Area and Volume of N-pipeds in Various Dimensions We embark now on another long march to inductively a rrive at results for the general N case. Tracing the first few cases N = 2,3,4 and then extrapolating to N = N probably gives more insight than a formal induction proof which is not attempted here. Each case below is treated with the same boilerplate template which first treats Face Area and then Volume. (a) The 2-piped For this case N = 2, the reader is encouraged to st are at the above Fig (B.4.4) while reading the text. Face Area. The area of a 2-piped face (a line segment) is just the length of the edge which is the face, A1 = |e2| A2 = |e1| . ( B . 5 . a . 1 ) where here we maintain the plan of labeling an area by the index of the spanning vector which is omitted in making the area. The vector areas can be written, based on the work above, A1 = |e2| E^1 / / Ek = Σi g'ki ei , see Note 4 following (6.2.7) A2 = |e1| E^2 ( B . 5 . a . 2 ) and these vectors are out-facing for faces 1p and 2p. We claim that both these results can be expressed in a single formula, Appendix B: Geometry of N-pipeds 231 An = |det(S)| En . (B.5.a.3) One can see that the direction is correct for n = 1,2, so it is just a question of verifying the magnitude. One must show that |det(S)| | E1| = | e2| and |det(S)| | E2| = | e1| (B.5.a.4) or | Ek| = |det(R)| | e3-k| k=1,2 . // RS = 1 (B.5.a.5) Using the N=2 trick notation from (A.6.5), Ek = det(R) (-1)k-1 e3-k x 3^ (A.6.5) (B.5.a.6) so that | Ek | = | det(R)| | e3-k x 3^| = |det(R)| | e3-k| k = 1,2 (B.5.a.7) since e3-k and 3^ are perpendicular. QED. We stress the formula An = |det(S)| En because it will turn out that this is valid for all N ≥ 2 . One can restate An = |det(S)| En using the cross product notation presented in (A.10.6): An = |det(S)| En = |det(S)| det(R) (-1)n-1 Πx i≠n ei = σ (-1) n-1 Πx i≠n ei σ ≡ sign(det(S)) = sign(det(R)) (B.5.a.8) with the understanding that for the N=2 case Π x i≠n ei has the special meaning of (A.10.7), so that in fact An = σ (-1)n-1 [ Πx i≠n ei ] = σ (-1)n-1 [ e3-n x 3^ ] // for N = 2, n = 1,2 (B.5.a.9) Volume . The volume of a 2-piped is the base times the he ight of a parallelogram, familiarly given by the cross product of the edges, volume(2) = | e1 x e2 | = | εab (e1)a(e2)b | = | det [ e1, e2 ] | = | det(S) | // see (3.2.7) (B.5.a.10) where S is the linearized transformation matrix for N= 2, see Chapter 2. Of course strictly in N=2 the notation e1 x e2 has no meaning, so one has to imagine a 3^ dimension to give it meaning. The second form does have a meaning for N=2, and that meaning is |( e1)1 (e2)2 – (e1)2 (e2)1|. (b) The 3-piped Face Area: The faces of a 3-piped are 2-pipeds. For N=2, the 2-piped volume is this from (B.5.a.9), volume(2) = | εab (e1)a(e2)b | , (B.5.a.9) (B.5.b.1) Appendix B: Geometry of N-pipeds 232 where e1 and e2 are 2D vectors. For the 2-piped which is "face 3" of the 3-piped -- a "near" face which touches the origin of the 3D skewed en coordinate system -- vectors e1 and e2 are 3D vectors. This face 3 is the top face shown in Fig (B.3.c.1) above. The firs t 2 components of each of these 3D vectors are the same as the components of the 2D ei vectors, while the 3rd components are both 0. This is so because face 3 lies in a plane defined by this 3rd component be ing 0. The volume(2) formula expressed in terms of these new 3D vectors is therefore |ε ab3 (e1)a(e2)b|, where ε is now a 3D ε tensor. The conclusion is that the area of the top face in Fig (B.3.c.1) is, A3 = |εab3 (e1)a(e2)b| ( B . 5 . b . 2 ) and this then is the scalar area of both face 3 and its partner face 3p, the far face. Similar arguments would then support these other area expressions A 1 = |ε1ab (e2)a(e3)b| A2 = |εa2b (e3)a(e1)b| . (B.5.b.3) Since indices on ε can be swapped for free due to the absolu te value, the non-summed index can be put first in all three cases and one may then conclude that A 1 = |e2 x e3| face 1 and face 1p A2 = |e3 x e1| face 2 and face 2p A3 = |e1 x e2| . face 3 and face 3p (B.5.b.4) In (A.3.3) it is shown that E1 = det(R) e2 x e3 so that e2 x e3 lines up with E1 if det(R) > 0. Regardless of the sign of det(R), we define the vector areas to point in the + E^n directions. Thus, A1 ≡ |e2 x e3| E^1 face 1p out-facing A2 ≡ |e3 x e1| E^2 face 2p out-facing A3 ≡ |e1 x e2| E^3 . face 3p out-facing (B.5.b.5) These three equations can be combin ed into the following single formula An = |e1 x ... x e3| E^n . // en missing (B.5.b.6) where the ei are reordered for free due to the absolute value signs. But (A.6.2) says En = det(R) (-1)n-1 e1 x ... x e3 // en missing (A.6.2) (B.5.b.7) so | En| = | det(R) | | e1 x ... x e3 | . // en missing (B.5.b.8) Thus, Appendix B: Geometry of N-pipeds 233 An = E^n |En| / |det(R)| = |det(S)| En = |det(S)| det(R) (-1)n-1e1 x ... x e3 // en missing = σ (-1)n-1e1 x ... x e3 / / en missing w h e r e σ ≡ sign[det(S)] = sign[det(R)] (B.5.b.9) To summarize, for N=3 one has An = |det(S)| En = σ (-1)n-1e1 x ... x e3 // en missing = σ (-1)n-1 Πx i≠n ei ( B . 5 . b . 1 0 ) where the last line uses the shorthand notation of (A.10.5). This An result has the same formal form as that of the 2-piped case (B.5.a.8). Volume. The volume of a 3-piped (with each of the above An in turn treated as the base) is base times height, so volume(3) = | A1 • e1 | = | A2 • e2 | = | A3 • e3 | or volume(3) = | e2 x e3 • e1 | = | e3 x e1 • e2 | = | e1 x e2 • e3 | . (B.5.b.11) Here is a drawing showing the last case ( σ = +1), where "base" is A 3 = | e1 x e2 | and "height" is e 3 cosθ , (B.5.b.12) Using ε notation one can write e3 • e1 x e2 = (e3)i εijk(e1)j(e2)k = εijk (e1)j(e2)k (e3)i = εjki (e1)j(e2)k(e3)i = det [ e1, e2, e3] = det(S) // see (3.2.7) (B.5.b.13) so that volume(3) = | e3 • e1 x e2 | = | εabc (e1)a(e2)b(e3)c | = | det [ e1, e2, e3] | = | det(S) | . (B.5.b.14) This volume expression has the same form as that for the 2-piped, (B.5.a.10). Appendix B: Geometry of N-pipeds 234 (c) The 4-piped Face Area: The faces of a 4-piped are 3-pipeds. For N=3, the 3-piped volume is stated in (B.5.b.14), volume(3) = | εabc (e1)a(e2)b(e3)c | (B.5.c.1) where e1,e2,e3 are 3D vectors. For the 3-piped which is "face 4" of the 4-piped -- a "near" face which touches the origin of the 4D skewed en coordinate system -- vectors e1,e2,e3 are 4D vectors. The first 3 components of each of these 4D vectors ar e the same as the components of the 3D ei vectors, while the 4th components are all 0. This is so because face 4 li es in a plane defined by this 4th component being 0. The volume(3) formula expressed in terms of these new 4D vectors is therefore | ε abc4 (e1)a(e2)b(e3)c |, where ε is now a 4D ε tensor. The conclusion is that A 4 = | εabc4 (e1)a(e2)b(e3)c | (B.5.c.2) and this then is the scalar area of both face 4 and its partner face 4p, the far face. Similar arguments would then support these other area expressions A 1 = | ε1abc (e2)a(e3)b(e4)c | A2 = | εa2bc (e3)a(e4)b(e1)c | A3 = | εab3c (e4)a(e1)b(e2)c | . (B.5.c.3) Since indices on ε can be swapped for free due to the absolu te value, the non-summed index can be put first in all three cases and one may then conclude that A 1 = |e2 x e3 x e4| face 1 and face 1p A2 = |e3 x e4 x e1| face 2 and face 2p A3 = |e4 x e1 x e2| face 3 and face 3p A4 = |e1 x e2 x e3| face 4 and face 4p . (B.5.c.4) where, as discussed in (A.4.1) and (A.4.2), Q = A x B x C is defined by Q k = εkabc AaBbCc . (B.5.c.5) In (A.6.2) it was shown that E1 = det(R) e2 x e3 x e4 so that e2 x e3 x e4 lines up with E1. Regardless of the sign of det(R), we define the vector areas to point in the + E^n directions. Thus An = |e1 x ... x e3| E^n // en missing n = 1,2,3,4 (B.5.c.6) where the ei are reordered for free due to the absolute value signs. But (A.6.2) says En = det(R) (-1)n-1 e1 x ... x e4 // en missing (B.5.c.7) so |En| = | det(R) | | e1 x ... x e4 | . // en missing (B.5.c.8) Appendix B: Geometry of N-pipeds 235 Thus, An = |En| E^n / |det(R)| = |det(S)| En = |det(S)| det(R) (-1)n-1e1 x ... x e4 // en missing = σ (-1)n-1e1 x ... x e4 // en missing σ ≡ sign[det(S)] = sign[det(R)] (B.5.c.9) To summarize, for N=4 one has An = |det(S)| En = σ (-1)n-1e1 x ... x e4 // en missing = σ (-1)n-1 Πx i≠n ei σ ≡ sign[det(S)] = sign[det(R)] (B.5.c.10) This An expression has the same form as those of the 2-piped and the 3-piped found earlier. Volume . The volume of a 4-piped (with each of the above An in turn treated as the base) is base times height, so volume(4) = | A1 • e1 | = | A2 • e2 | = | A3 • e3 | = | A4 • e4 | (B.5.c.11) or volume(4) = | e2 x e3 x e4 • e1 | = | e3 x e4 x e1 • e2 | = | e4 x e4 x e1 • e3 | = | e1 x e4 x e2 • e4 | . Using ε notation one can write the first case as e2 x e3 x e4 • e1 = (e1)a εabcd(e2)b(e3)c(e4)d = εabcd(e1)a(e2)b(e3)c(e4)d = d e t [ e1, e2, e3, e4] = det(S) // see (3.2.7) (B.5.c.12) so that volume(4) = | εabcd(e1)a(e2)b(e3)c(e4)d | = | det [ e1, e2, e3, e4] | = | det(S) | (B.5.c.13) This volume expression has the same form as those of the 2-piped and the 3-piped. (d) The N-piped Face Area: The faces of an N-piped are (N-1)-pipeds. If th ere had been an (N-1)-piped section prior to this one, the volume formula there would have been volume(N-1) = | ε abc...x (e1)a(e2)b... (eN-1)x | (B.5.d.1) where e1,e2,...eN-1 were (N-1)D vectors (that is, 3D vectors if N=4). For the N-piped which is "face N" of the N-piped -- a "near" face which touches the origin of the ND skewed en coordinate system -- vectors e1,e2,...eN-1 are ND vectors. The first N-1 components of each of these ND vectors are the same as the components of the (N-1)D ei vectors, while the Nth components are a ll 0. This is so because face N lies in a plane defined by this Nth component being 0. The volume(N-1) formula expressed in terms of these Appendix B: Geometry of N-pipeds 236 new ND vectors is therefore | εabc...xN (e1)a(e2)b... (eN-1)x |, where ε i s n o w a n N D ε tensor. The conclusion is that ( compare this to the N=4 expression (B.5.c.2) ), AN = | εabc...xN (e1)a(e2)b... (eN-1)x | (B.5.d.2) and this then is the scalar area of both face N and its partner face Np, the far face. Similar arguments would then support similar expressions for the other faces, for example, A 1 = | ε1abc...x (e2)a(e3)b... (eN-1)w (eN)x | A2 = | εa2bc...x (e3)a(e4)b... (eN)w (e1)x | . (B.5.d.3) Since indices on ε can be swapped for free due to the absolu te value, the non-summed index can be put first in all three cases and one may then conclude that A 1 = |e2 x e3 x e4 x e5... x eN| face 1 and face 1p A2 = |e3 x e4 x e5 x e6... x e1| face 2 and face 2p A3 = |e4 x e5 x e6 x e7... x e2| face 3 and face 3p ... AN = |e5 x e6 x e7 x e8.. x e3| face N and face Np (B.5.d.4) where, as discussed in (A.4.1) and (A.4.2), Q = A x B x C.... X is defined by Q k = εkabc...x AaBbCc .....Xx . (B.5.d.5) In (A.6.2) it was shown that E1 = det(R) e2 x e3 ... eN so that e2 x e3 ... eN lines up with E1. Regardless of the sign of det(R), we define the vector areas to point in the + E^n directions. Thus An = |e1 x ... x eN| E^n // en missing n = 1,2,3...N (B.5.d.6) where the ei are reordered for free due to the absolute value signs. But (A.6.2) says En = det(R) (-1)n-1 e1 x ... x eN // en missing (B.5.d.7) so | En| = | det(R) | | e1 x ... x eN | // en missing (B.5.d.8) Thus, An = |En| E^n / |det(R)| = |det(S)| En = |det(S)| det(R) (-1)n-1e1 x ... x eN // en missing = σ (-1)n-1e1 x ... x eN // en missing σ ≡ sign[det(S)] = sign[det(R)] (B.5.d.9) To summarize, for N=N one has An = |det(S)| En = σ (-1)n-1e1 x ... x eN // en missing = σ (-1)n-1 Πx i≠n ei σ ≡ sign[det(S)] = sign[det(R)] (B.5.d.10) Appendix B: Geometry of N-pipeds 237 This An expression has the same form as those of the 2-piped, the 3-piped and the 4-piped. Volume . The volume of a N-piped (with each of the above An in turn treated as the base) is base times height, so volume(N) = | A1 • e1 | = | A2 • e2 | = ... = | AN • eN | or volume(N) = | e2 x e3 x e4....eN • e1 | = ... (B.5.d.11) Using ε notation one can write the first case as e2 x e3 x e4...eN • e1 = ( e1)a εabc...x (e2)b(e3)c.....(eN)x = εabc...x (e1)a(e2)b(e3)c....(eN)x = d e t [ e1, e2, e3, ....eN] = det(S) // see (3.2.7) (B.5.d.12) where x is the Nth letter of the alphabet, so that volume(N) = | εabc...x (e1)a(e2)b(e3)c....(eN)x | = | det [ e1, e2, e3, ....eN] | = | det(S) | . (B.5.d.13) This volume expression has the same form as those of the 2-piped, the 3-piped and the 4-piped. This result is also consistent with the volume( N-1) expression stated in (B.5.d.1). B.6 Summary of Main Results of this Appendix 1. One way to write the equation of a plane in N dimensions is r•p^ = p r = (x1, x2, .....xn) (B.6.1) where p^ is the unit vector normal to the plane which poin ts "away from the origin", and where p > 0 is the distance of closest approach of the plane to the origin. In the limit p →0, the plane passes through the origin and the equation is then r•p^ = 0 where p^ is either normal to the plane. 2. An N-piped has 2N vertices as shown by the inductive c onstruction method presented above. (B.6.2) 3. The locus of points making up the (closed) interior of an N-piped spanned by e1...eN is given by rvolumeN = Σn=1Nαnen 0 ≤ αn ≤ 1 (B.6.3) The tails of all the vectors e1...eN meet at the origin of RN space. 4. There are N pairs of faces on an N-piped, and each face is an (N-1)-piped having 2 N-1 vertices. The total face count is 2N. Each face is spanned by a subset of N-1 of the base vectors en, so each face is Appendix B: Geometry of N-pipeds 238 "missing" one of the en and the face is labeled using the index of this missing base vector. The loci of points making up the faces of an N-piped are given by rface(i) = Σn≠iαnen 0 ≤ αn ≤ 1 i = 1,2...N rface(ip) = Σn≠iαnen + ei 0 ≤ αn ≤ 1 i = 1,2...N (B.6.4) where "face i" has a corner touching the origin of th e N-piped (near face), while its parallel partner face "ip" does not touch the origin (far face). 5. If the N-piped spanning vectors en are the tangent base vectors associated with some transformation F, then Ei• ej = δi,j where Ei are the reciprocal base vectors. In th is case, the equations of the planes of the faces of the N-piped can be written in the form shown in item 1 above, E^i• r(face i) = 0 E^i• r(face ip) = 1/Ei i = 1,2...N (B.6.5) so that both faces of a pair i are planar (in N di mensional space) and they have the same normal vector E^i so the faces of a pair lie on parallel planes. 6. The vector Ei is an outward-facing normal for face ip, while - Ei is an outward-facing normal vector for face i (which touches the origin). (B.6.6) 7. The out-facing vector area of face ip of an N-piped can be expressed as Ai = |det(S)| Ei Ai = σ (-1)i-1 Πx j≠i ej σ ≡ sign[det(S)] = sign[det(R)] Ai = σ (-1)i-1 e1 x e2 ... x eN // ei m i s s i n g ( B . 6 . 7 ) where ei is the vector missing from the face's spanning set. The outfacing area for face i is - Ai. The last two lines are shorthands for the follo wing, as discussed in (A.5.2), ( Ai )α = σ (-1)i-1 εαabc..x (e1)a (e2)b ... (eN)x // where (e i)i and i are missing (B.6.8) For N=2 the last two expressions for Ai are interpreted as shown in (B.5.a.9), Ai = σ (-1)i-1 e3-i x 3^ i = 1 , 2 ( B . 6 . 9 ) 8. The volume of an N-piped spanned by e1...eN is given by volume(N) = | det(S) | = | det [ e1, e2, e3, ....eN] | = | εabc...x (e1)a(e2)b(e3)c....(eN)x | (B.6.10) where one can regard the tangent base vectors as the columns of the linearized transformation matrix S as shown in (3.2.7). Appendix C: Elliptic Polar Coordinates 239 Appendix C: Elliptical Polar Coords, View s of x'-space, Jacobian Integration Rule This Appendix is written in the develo pmental notation of Chapters 1-6. C.1 Elliptical polar coordinates The 2D "elliptic" coordinate system has coordinate lines which are orthogonal ellipses and hyperbolas. When rotated about its two symmetry axes, this sy stem generates 3D prolate or oblate spheroidal coordinates. This is not the 2D coordinate system described in this Appendix. For "elliptical polar" coordinates, the coordinate lines are taken instead as the ellipses from elliptic coordinates, and the rays from polar coordinates. This non-orthogonal system is perhaps not very useful, but provides a good "sandbox" in which to study general aspects of coordinate systems. The transformation x' = F(x) is given by x ' - s p a c e x-space (Cartesian) ρ2 = x2/a2 + y2/b2 x2 + y2 = r2 still x' 1 = θ x 1= x tanθ = y / x x ' 2 = ρ x 2 = y (C.1.1) Writing the first equation above as 1 = x 2/(ρa)2 + y2/(ρb)2 ( C . 1 . 2 ) it should be clear that ρ serves to label an ellipse of semi-major axis ρa, and semi-minor axis ρb, while θ labels the ray at angle θ , as in polar coordinates. The inverse transform x = F-1(x') is given by x = aρcosθ x/a = ρ cosθ ⇒ x2/a2 + y2/b2 = ρ2 y = bρsinθ y/b = ρsinθ ⇒ tan θ = y/x . (C.1.3) The matrix S is given by S11 = (∂ x/∂θ) = -aρsinθ S12 = (∂ x/∂ρ) = acosθ S ik ≡ ( ∂xi/∂x'k) (2.1.5) S21 = (∂ y/∂θ) = bρcosθ S22 = (∂ y/∂ρ) = bsinθ ( C . 1 . 4 ) S = ⎝⎛ ⎠⎞-aρsinθ acosθ bρcosθ bsinθ ⇒ det(S) = -abρ and R = S-1 = ⎝⎛ ⎠⎞ -sinθ /(aρ) cosθ/(bρ) cosθ/a sinθ/b . The tangent base vectors en can be read off as the columns of S, see (3.2.7) : e1 = ρ(-asinθ,bcosθ) = eθ |eθ| = ρ a2sin2θ + b2cos2θ ≡ hθ eθ = |eθ| e^θ e2 = (acosθ,bsinθ) = eρ | eρ| = a2cos2θ + b2sin2θ ≡ hρ eρ = |eρ| e^ρ . (C.1.5) The covariant metric tensor from (5.7.9) or (5.11.3) is, Appendix C: Elliptic Polar Coordinates 240 g¯' = STS = ⎝⎛ ⎠⎞ρ2{a2sin2(θ) + b2cos2(θ)} [b2-a2]ρ sin(θ)cos(θ ) [b2-a2]ρ sin(θ)cos(θ ) a2cos2(θ) + b2sin2(θ) = ⎝⎜⎛ ⎠⎟⎞e1•e1 e1•e2 e2•e1 e2•e2 (C.1.6) which is clearly non-diagonal (but symmetric) as expected. When a = b = 1 it reduces to the polar coordinates system metric tensor where then ρ = r. The coordinate system is non-orthogonal because e 1•e2 ≠ 0, or equivalently, because g ¯' is non-diagonal. C.2 Forward coordinate lines Here is a Maple plot of some x-space forward coordinate lines (parameters a = 2 and b = 1) ( C . 2 . 1 ) where ρ is the x'-space vertical axis. The coordinate lines in x-space are plotted using these equations, y = b ρi2-(x/a)2 // ellipses ρi= 1,2...10 10 ellipses y = x tanθ i // rays θi = 2π (i/20) , i = 1,2...20 20 rays (C.2.2) which are obtained from the forward transformation equations ρ2 = x2/a2 + y2/b2 tanθ = y / x . (C.1.1) C.3 Inverse coordinate lines Here is a Maple plot of some x'-space inverse coordinate lines (parameters a = 2 and b = 1) Appendix C: Elliptic Polar Coordinates 241 The coordinate lines in x'-space are plo tted using these equations (C.3.1) ρ = xi/(acosθ) x i = -10 to +10 20 blue curves ρ = yi/(bsinθ) y i = -10 to +10 20 red curves (C.3.2) which are obtained from the inverse transformation equations x = aρ cosθ y = bρ sinθ (C.1.3) The secθ and cscθ curve families appear to "change shape", but that is just what happens when functions are scaled up vertically but not horizontally. If one plot s one sine hump at different vertical scalings, the humps have different shapes. C.4 Drawing a contravariant vector V in x-space: the meaning of V 'n . A contravariant vector field V(x) can be expanded in these two ways, as shown in (6.6.9), V = V1(x) 1^ + V2(x) 2^ = Vx(x) x^ + Vy(x) y^ // un = n^ for Cartesian V = V'1(x') e1 + V'2(x') e2 = V'θ(x') eθ + V'ρ(x') eρ V'(x') = R(x) V(x) (C.4.1) where the V' n are the components of V transformed into x'-space where V becomes V'. The prime is not necessary on V' ρ but we maintain it as a reminder that it is an x'-space component. R( x) is the matrix of (2.1.6) and en are the tangent base vectors of Chapter 3. The fields V' n(x') are "the components of vector field V' in x'-space", since V' = R V , or V'i = RijVj. Moreover, these V' n(x') are "expressed in terms of curvilinear coordinates" x'. If one is asked to "express a vector V in curvilinear coordinates", one is usually being asked to write V as the second expansion above. The vectors en and V exist in x-space , and in the second expansion it just happens that the coefficients V' n(x') are the components of V', the transformed vector in x'-space, when it is expanded on the axis-aligned vectors e'n in x'-space. Appendix C: Elliptic Polar Coordinates 242 Here is a graphical representation of this vector V in x-space: (C.4.2) As advertised, the tangent base vectors are not at right angles. The V parallelogram accurately illustrates the equation V = V'θ eθ + V'ρ eρ. Since x-space is Cartesian, there is no distinction between Cartesian length (graphical length) and covariant length for vectors in x-space. Graphically, one could find the values for V' θ and V' ρ as follows: (1) for the point (x,y), compute the vectors eθ and eρ and compute their lengths | eθ| = h'θ and | eρ| = h'ρ ; (2) draw the parallelogram shown aligned with these vectors for some given V and find the edge lengths. The edges of the parallelogram are V' θ h'θ and V' ρ h'ρ so then the values of V' θ and V' ρ can be found. The alternative method is to compute R and use V' i = RijVj. C.5 Drawing a contravariant vector V' in x '-space: two "Views" As with previous examples, the drawing (C.4.2) is dr awn to the right of an x'-space drawing as follows: ( C . 5 . 1 ) In Chapter 3 the axis-aligned basis vectors e'n were introduced as e'n , n = 1,2...N // ( e'n)i = δn,i e'1 = (1,0,0...) etc (3.2.1) and en was shown to be a contravariant vector, Appendix C: Elliptic Polar Coordinates 243 e'n = R( x) en . (3.3.2) Applying matrix R(x) to the equation V = V'θ eθ + V'ρ eρ one gets the expansio n shown in Fig (C.5.1), V' = V'θ e'θ + V'ρ e'ρ (6.6.15) ( C . 5 . 2 ) which appears first in the list of expansions of V' in (6.6.15). There is no ambiguity concerning this last equation. Ambiguity can arise, however, when one tries to represent this equation graphically in x'-space. There are two very different "views" one can take of a drawing in x'-space. In the first view, we take x'-space to be "flat" (Cartesian) so that g' = 1. In the second view, we take x'-sp ace to be "curved" with g' ≠ 1. These views are really two different x'-spaces since the metric tensors are different. Cartesian View of x'-space (C.5.3) In this view of x'-space the length (norm) of a vector is given by | A|2 = δijAiAj = ΣAi2, so one has, since ( e'n)i = δn,i, g¯' = 1 | e'n| = 1 e'n = e^'n = n^' = the usual axis-aligned unit vectors in x'-space V' = V'θ θ^ + V'ρ ρ^ θ^ = e^'1 ρ^ = e^'2 . (C.5.4) The left-side of (C.5.1) is, in this view, a "normal Cartesian graph" and the vectors add up properly. For example Pythagoras tells us that | V'|2 = V'θ2 + V'ρ2 and θ^ • θ^ = 1 ρ^ • ρ^ = 1 θ^ • ρ^ = 0 . ( C . 5 . 5 ) Appendix C: Elliptic Polar Coordinates 244 This Cartesian View, which is x'-space with g ¯' = 1, is appropriate in applications in which it is not required or desired that norms and dot products be tensorial scal ars, as discussed at the end of Section 5.2. For example, when g ¯' is set to 1, one has | V'| ≠ |V| in (C.5.1). Another use of this view involves in tegration as will be seen below. Curvilinear View of x'-space (C.5.6) In this view of x'-space, one assumes that g ¯' takes a value which enforces the scalarity of norms and dot products between x-space and x'-space, which is to say, one takes g ¯' = ST g¯ S from (5.7.6) where g ¯ is the x-space metric tensor for x-space. Normally g ¯=1 (Cartesian x-space), so g ¯' = STS. In this Curvilinear View, then, the length | A'| of a contravariant vector A' is determined by | A'|2 = g¯'ijA'iA'j where g¯' = STS ≠ 1 | e'n| = |en| = h'n ≡ g¯ 'nn n = 1,2 for θ ,ρ (C.5.7) V' = V'θ e'θ + V'ρ e'ρ = (V' θ h'θ) e^'θ + (V'ρ h'ρ) e^'ρ e^'n ≡ e'n/ |en| = e'n/ h'n Vx2 + Vy2 = |V|2 = |V'|2 ≠ (V'θ h'θ)2 + (V'ρ h'ρ)2 // unless x' i are orthogonal coordinates This last inequality says that in the Curvilinear View the Pythagorean Theorem is invalid. In fact |V'|2 = g¯'ijV'iV'j = g¯'θθ V'θ2 + g¯'ρρ V'ρ2 + 2 g¯'θρV'θ V'ρ = (V' θ h'θ)2 + (V'ρ h'ρ)2 + 2 g¯'θρV'θ V'ρ . (C.5.8) In writing | e'n| = |en| and | V'|2 = |V|2 above, we use the rule shown in (5.10.4) which says | A'|2 = |A|2 for any contravariant vector A (|A|2 is a scalar ). Moreover, Appendix C: Elliptic Polar Coordinates 245 e^'θ • e^'θ = e'θ • e'θ / (h'θ2) = eθ • eθ / (h'θ2) = g¯'θθ / (h'θ2) = 1 e^'ρ • e^'ρ = e'ρ • e'ρ / (h'ρ2) = eρ • eρ / (h'ρ2) = g¯'ρρ / (h'ρ2) = 1 e^'θ • e^'ρ = e'θ • e'ρ / (h'θh'ρ) = eθ • eρ / (h'θh'ρ) = g¯'θρ / (h'θh'ρ) ≠ 0 <= !! (C.5.9) so the e^'n are unit vectors having unit covariant length, but e^'θ • e^'ρ ≠ 0 despite the fact that these vectors are drawn at right angles in the x'-space graph above, en = h'n e^'n. One might imagine trying to slant the lines in (C.5.6) to cause all intersection poi nts to have angles which match the metric tensor, which is to say, at each intersec tion point one would need an angle ψ where e^'θ • e^'ρ = cosψ . But in general e^'n • e^'m = g¯'nm / (h'nh'm) has a different value at every point, so such a graph would be quite complex. The upshot is that for a non-orthogonal system, the axes in x'-space are still drawn at right angles and the purpose of the graph is mainly to "locate" all the points x' which correspond to points x in x-space according to x' = F(x). The graph does successfully represent the idea that V' = V'ρ e'ρ + V'θ e'θ, but one must give up on Euclidean geometry for this vector su m triangle. It might be imagined that the x'-space graph is the projection onto the pl ane of paper of some vectors drawn on a curved surface emerging from the plane of paper, and that is then why Pythagoras is wrong. In the case of an orthogonal coordinate system (diagonal g ¯'), the 90 degree angles between the axes in x'- space are accurate representations of the fact that e^'n• e^'m = 0 when n ≠ m. And since scalars are preserved, one has in the Curvilinear View, | V'|2 = Σn (h'nV'n)2 = Σn V'n2 = | V|2 = Σn Vn2 // orthogonal only where V'n ≡ h'nV'n and V' = Σn V'n e^'n and V = Σn V'n e^n ( C . 5 . 1 0 ) One can then still apply regular Euclidean geometry to the vector addition N-piped in x'-space in the sense that | V'|2 = Σn (h'nV'n)2. Although the x'-space perpendicular-unit-vectors and Pythagorean paradoxes go away if the x' coordinates are orthogonal (as they would be for polar coordinates where a = b = 1), one still has the two x'-space views to keep in mind: the Cartesian View with g ¯' = 1 and the Curvilinear View with g ¯' = STS. C.6 Drawing the specific contravariant vector dx in x-space and x '-space Since d x is the primordial contravariant vector, everything stated in the last two Sections applies with V → dx and V' θ → dx'θ = dθ, V'ρ → dx'ρ = dρ, where we finally drop the primes on d θ and dρ . The expansions of d x and d x' are, dx = dθ eθ + dρ eρ // in x-space dx' = dθ e'θ + dρ e'ρ // in x'-space (C.6.1) For V = dx Fig (C.5.1) becomes, Appendix C: Elliptic Polar Coordinates 246 ( C . 6 . 2 ) It must be understood that now the vector arrows like d x are highly magnified a nd in reality are very small compared to, say, the curvature of the ellipse. From above, e'θ • e'θ = g¯'θθ e^'θ • e^'θ = 1 e'ρ • e'ρ = g¯'ρρ e^'ρ • e^'ρ = 1 e'ρ • e'θ = g¯'ρθ e^'θ • e^'ρ = g¯'θρ / (h'θh'ρ) (C.6.3) and once again the "right angle" in the x'-space picture is deceptive. The x'-space side of (C.6.2) is subj ect to the two "views" described above: Cartesian View of x'-space: (elliptical polar coordinates) g¯' = 1 | e'n| = 1 dx' = dθ θ^ + dρ ρ^ θ^ = e^'1 ρ^ = e^'2 θ^ • θ^ = 1 ρ^ • ρ^ = 1 θ^ • ρ^ = 0 . (C.6.4) Curvilinear View of x'-space: (elliptical polar coordinates) g¯' = STS | e'n| = |en| = h'n ≡ g¯ 'nn dx ' = dθ e'θ + dρ e'ρ = (dθ h'θ) e^'θ + (dρ h'ρ) e^'ρ (dx)2 + (dy)2 = |dx|2 = |dx'|2 = ( d θ h'θ)2 + (dρh'ρ)2 + 2 g¯'θρdθdρ = (dθ h'θ)2 + (dρ h'ρ)2 only if g ¯'θρ = 0 (orthogonal) . (C.6.5) Appendix C: Elliptic Polar Coordinates 247 Just to have a specific orthogonal coordinates example, we reduce (C.1.3) to polar coordinates by setting a = b = 1. In this case, (C.1.6) reduces to g ¯' = ⎝⎛ ⎠⎞ ρ2 0 0 1 as in (5.13.11) so h θ = ρ, hρ = 1 and g ¯'θρ = 0. We can then restate the two Views in this case Cartesian View of x'-space: (polar coordinates) g¯' = 1 | e'n| = 1 dx' = dθ θ^ + dρ ρ^ θ^ = e^'1 ρ^ = e^'2 θ^ • θ^ = 1 ρ^ • ρ^ = 1 θ^ • ρ^ = 0 . (ds)2 = |dx|2 = dx • dx = (dx)2 + (dy)2 // from x-space (which has g ¯ = 1 ) (ds') 2 = |dx'|2 = dx' • dx' = (dθ)2 + (dρ)2 ≠ |dx|2 = (ds)2 ( C . 6 . 6 ) Curvilinear View of x'-space: (polar coordinates) g¯' = STS = ⎝⎛ ⎠⎞ ρ2 0 0 1 | e'θ| = |eθ| = h'θ = ρ ≡ g¯ 'θθ | e'ρ| = |eρ| = h'ρ = 1 ≡ g¯ 'ρρ dx ' = dθ e'θ + dρ e'ρ = (dθ ρ) e^'θ + (dρ) e^'ρ (ds)2 = |dx|2 = d x • dx = (dx)2 + (dy)2 // from x-space (which has g ¯ = 1 ) (ds')2 = |dx'|2 = d x' • dx' = (dθ ρ)2 + (dρ)2 = | dx|2 = (ds)2 (C.6.7) C.7 Study of how dx transforms in the mapping between x-space and x '-space Consider this drawing which shows a representative set of vectors d x in x-space (the bars), along with the forward mappings (d x' = F(dx) or d x' = Rd x ) of the corresponding vectors d x' in x'-space. The vectors on the right all point up, those on the left point generally to the northeast. Appendix C: Elliptic Polar Coordinates 248 ( C . 7 . 1 ) Now select the red d x bar on the right and take it to be the dx of Fig (C.6.2). First determine the tangent base vectors eθ and eρ at the location of the red bar. Then setting d x = dθ eθ + dρ eρ, consider the value of the two numbers d θ and dρ for this red bar. Graphically, knowing which way eθ and eρ point at the bottom of the red d x, one expects d θ > 0 and d ρ > 0. The red d x' bar on the left has these Cartesian values dθ and dρ, and has a Cartesian-view length of |d x|2 = (dθ)2+(dρ)2. One can see from the picture that these Cartesian lengths vary for the 10 bars shown, though the leng ths are all the same in x-space. The Curvilinear-view lengths of the x'-space bars are all the same, and are equal to the Cartesian length of those bars in x-space since d x'•dx' = dx•dx. Consider now some bar mapping in the other direction: ( C . 7 . 2 ) Now the d x bars on the right all have different lengths. Thos e on the left have the same Cartesian length, which is what the drawing shows, but each one 's Curvilinear-view length matches that of its corresponding bar on the right. The ratio of the length of a bar on the right to the Cartesian length of the corresponding bar on the left is the scale factor h θ which recall is a function of location in space: Appendix C: Elliptic Polar Coordinates 249 bar on right = d x(1) = e1 dx'1 = e^1 h'1 dx'1 = eθ dθ = e^θ hθ dθ graph length = h θ dθ bar on left (Cartesian view) = d x'(1) = e'1 dx'1 = e^'1 dx'1 = e^θ dθ graph length = dθ ⇒ right bar length / left bar length = h θ = ρ a2sin2θ + b2cos2θ (increases with ρ) (C.7.3) If a=b, then h θ = ρ and the bar length on the right is then ρdθ as is obvious in polar coordinates. C.8 A Derivation of the Jacobian Integration Rule Consider now an integral ∫dθdρ f(θ,ρ). The tiny rectangles of area d θdρ, like the specific gray and orange ones highlighted on the left above, are regarded for the purposes of integration as being in the Cartesian view of x'-space. One then writes [ dA' is called d V' in Chapter 8 ] dA' ≡ dρdθ = the area of a differential patc h in Cartesian-view x'-space (C.8.1) This is the graphical area one sees in the picture. There is no need to define or consider any Curvilinear-view area in x'-space because the Ca rtesian-view area is being used. In the limiting process which defines the integration, each d θdρ patch on the left has the same area dρdθ. The interior of each patch on the left maps into some parallelogram patch on the right. One is not surprised to see that the patch areas on the right are different, though they map into patches on the left of the same Cartesian-view area. As shown in (8.4.d.3 ) the ratio of the two patch areas (volumes) is the absolute value of the Jacobian |J( x')|, (area of skewed patch on the right at location x) = |J( x')| dA' = |J( x')| dρdθ (C.8.2) This is not what we mean by "the Jacobian Integration Rule" in the Section title. That is coming below and it is going to involve the quantity dxdy. The mapping shown above between patches is an N=2 example of the general N-dimensional discussion in Section 8.2 which describes an orthogonal differential N-piped in (Cartesian-view) x'-space mapping into a non-orthogonal differential N-piped in x-space. Now back to the integration issue. There are two ways an integration can be done in Cartesian x-space: integral of f(x) = lim Σ i dA1(xi) f(xi) dA 1(xi) = patches shown on the right above integral of f(x) = lim Σi dA2(xi) f(xi) dA 2(xi) = dxdy (C.8.3) In the first integral , every patch dA 1(xi) on the right has a different shape and a different area as the integral is computed in the usual limiting-sum mann er. The gray and orange patches on the right are two of these many patches. Despite their non-uniform shap e and area, this rag-tag band of patches certainly "covers" the area being integrated over, and does so perfectly in the calculus limit. The area of one of these rag-tag patches is |J( x')|dA' = |J( x')|dθdρ and the areas are different because the Jacobian is a function of x = x(x') . Appendix C: Elliptic Polar Coordinates 250 In the second integral , every patch dA 2(xi) has the same area dxdy, so really dA 2(xi) does not depend on xi in this form of the integration. One such dxdy patch is shown in green above. The coverage of the dA 2 patches is of course also "perfect coverage" in the calculus limit. Since both integrals cover the same area perfectly, th ey both give the same result in the limiting process that defines the integral. This point is someti mes misunderstood. One is not just "replacing" a parallelogram patch such as the or ange one on the right with some dxdy patch that approximates it in area, like the green patch. The statement is about an integration. Thus one has lim Σi dA1(xi) f(xi) = lim Σi dA2(xi) f(xi) (C.8.4) or ∫[|J(x')| dθdρ] f(x(x')) = ∫[dxdy] f( x) ( C . 8 . 5 ) where on the left f( x) = f( x(x')) where x = F-1(x') ≡ x(x'). In the sense of distribution theory (Stakgold Chapters 1 and 5), one can then make this symbolic statement |J(θ,ρ)| dρdθ = d x d y ( C . 8 . 6 ) where the meaning of this symbolic equality is the integral statement above, ∫D dxdy f( x) = ∫D' dθdρ |J(x')| f(x(x')) , (C.8.7) valid for any integrable f( x) and any integration region D (region D' corresponds to D in x'-space.) Either of these last two equations constitute the "Jac obian Integration Rule" of the Section title. The integral on the left is well defined in 2D calcu lus, so the expression on the right shows how to "evaluate the integral on the left in curvilinear coordinates". At this point one may introduce a new but obvious symbol dA ≡ d x d y ( C . 8 . 8 ) so the above equality of integrals can be written ∫dA f( x) = ∫ dA' |J( x')| f(x(x')) |J( x')| dA' = dA (C.8.9) In N dimensions, dA and dA' are differential "volumes ", and the general Jacobian Integration rule takes the form, Appendix C: Elliptic Polar Coordinates 251 ∫dV f( x) = ∫ dV' |J( x')| f(x(x')) |J( x')| dV' = dV dV' ≡ dx'1dx'2....dx'N = the volume of an orthogonal differential N-piped in the Cartesian-view x'-space dV = dx 1dx2....dxN = the volume of an orthogonal differential N-piped in x-space. (C.8.10) Notice that these are not the two N-pipeds which "map into each other" as noted above. The N-piped dV has nothing to do that that mapping which involved a non-orthogonal N-piped in x-space. To finish off our sample N=2 case, recall from earlie r that for our polar elliptical coordinate system |J'(x')| = | det(S)| = ab ρ ( C . 8 . 1 1 ) and therefore ∫dxdy f(x,y) = ∫ dθdρ |J(x')| f(aρ cosθ, bρ sinθ) = ab ∫dθdρ ρ f(aρ cosθ, bρ sinθ) . (C.8.12) In the limit of regular polar coordinates, one then has a = b = 1 and ρ = r so ∫dxdy f(x,y) = ∫rdrdθ f(rcosθ, rsinθ ) ( C . 8 . 1 3 ) which is the familiar result. Appendix D: Tensor Densities 252 Appendix D: Tensor Densities and the ε tensor In this Section, DN means Developmental Notation. D.1 Definition of a tensor density Picture A is used in this Appendix along w ith the Standard Notation of Chapter 7. (D.1.1) First, equations (5.12.20) summarize facts about the Jacobian J which were converted to Standard Notation in (7.5.18) through (7.5.22). Here we add a symbol σ ≡ sign(J) and restate those facts in Standard Notation as J ≡ det(S i j) = σ sg' / sg = σ(sg'/sg)1/2 = σ(g'/g)1/2 ⇒ (g'/g)1/2 = σJ = |J| > 0 J2 = (g'/g) s = sign[det(g ij)] = sign(g) = sign(g') g = det(g ij) sg = |g| > 0 Si j ≡ (∂xi/∂x'j) σ = sign[det(Si j)] = sign(J) g' = det(g' ij) sg' = |g'| > 0 . (D.1.2) For proper Lorentz transformations of special relativity, det(S) = 1 so σ = +1. For curvilinear coordinates, one normally selects an ordering of the x i so that σ = +1, such as r, θ,φ in spherical coordinates. Nevertheless, we allow for the possibility of J < 0. Next, recall our generic sample tensor transforma tion (7.10.1), written two ways using (7.5.13) Sa b=Rba : T ' abc de = Ra a' Rb b' Rc c' Sd' d Se' e Ta'b'c' d'e' T ' abc de = Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' . (7.10.1) (D.1.3) T is a mixed rank-5 tensor, meaning it transforms as shown above with respect to the underlying transformation F. T is a regular standard-issue tensorial tensor. Now suppose instead that the object T were to transform like this, with J being the Jacobian noted above, T ' abc de = J-W Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' (D.1.4) where the extra factor J -W has been introduced. If T transforms this way, it is called a tensor density of weight W. Thus, an ordinary tensor is a tensor density of weight 0. The convention for the sign of W used here is th at of Weinberg p 99 Eq. (4.4.4), which equation has the following factor on the right side of a sample tensor density transform equation, Appendix D: Tensor Densities 253 |∂x'/∂x|W ≡ [det(∂x'/∂x)]+W = [ det(∂x'i/∂xk) ]+W = J-W . (D.1.5) Some authors use -W as the "weight" instead of +W, but we shall stick with Weinberg's convention. An immediate example of a tensor density is provided by (g'/g)1/2 = |J| rewritten as g' = J2 g = J-(-2) g ⇒ weight(g) = -2 g is a scalar density of weight - 2 . (D.1.6) so g ≡ det(gij) is a scalar density of weight -2. Notice that from g one can construct other scalar densities of other weights, for example : g'-1 = J-(2) g-1 ⇒ weight(g-1) = +2 g'-1 is a scalar density of weight +2 . (D.1.7) D.2 A few facts about tensor densities 1. It is pretty obvious that a sum of two index-similar tensor dens ities of weight W has weight W. ( D . 2 . 1 ) 2. Contracting indices within a tensor density does not alter its weight W. (D.2.2) If indices a and d are contracted in the (D.1.4) example above, one gets T ' abc ae = J-W Ra a' Rb b' Rc c' Rad' Ree' Ta'b'c' d'e' = J -W (Ra a'Rad') Rb b' Rc c' Ree' Ta'b'c' d'e' = J -W δa'd' Rb b' Rc c' Ree' Ta'b'c' d'e' = J -W Rb b' Rc c' Ree' Ta'b'c' a'e' . The factor J -W just sits there, impervious to contraction activities. 3. Going the other direction, when a larger tensor de nsity is formed from two sm aller ones, called a direct product or outer product , the weights get added . (D.2.3) Example 1: Start with two tensor densities A and B of weights W1 and W2, form the outer product: A'a = J-W1 Ra a'Aa' B' c d = J-W2 Rc c'Rdd' Bc' d' ⇒ (A'a B'c d) = J-(W1+W2) Ra a' Rc c'Rdd' (Aa' Bc' d') . Example 2: Start with |g'| = J2|g| from (D.1.2) and raise both sides to the power -W1/2 : |g'| -W1/2 = J-W1 |g|-W1/2 . // no R factors since scalar (D.2.3a) Appendix D: Tensor Densities 254 Thus, the factor ( |g|-W1/2) is a scalar density of weight W1. Let this be the first factor of an "outer product" where the second factor is the following tensor density of weight W2, B'c d = J-W2 Rc c'Rdd' Bc' d' // same as in Example 1 ⇒ (|g'|-W1/2B'c d ) = J-(W1+W2) Ra a' Rc c'Rdd' (|g|-W1/2 Bc' d') . (D.2.3b) If one selects W1 = –W2, the added factor neutralizes the weight of the tensor density to which it is prefixed, generating thereby a regular tensor (weight 0). So if tensor density B has weight W, ( | g ' | W/2 B'c d) = Ra a' Rc c'Rdd' (|g|W/2 Bc' d') (D.2.3c) and then (|g| W/2 Bi j) transforms under F as a regular tensor. (One should always keep in mind the fact that there is an underlying transformation x' = F(x) upon which the House of Tensor is built ). This is a standard method of converting a tens or density to a regular tensor. 4. Although sometimes authors take a differing stance for certain tensors like ε below, we shall assume that indices are raised and lowered on a tensor density in exactly the same way they are raised and lowered on an ordinary tensor of the same index structure (see Section 7.4). This means the gab raises an index and gab lowers a n i n d e x . ( D . 2 . 4 ) 5. Raising or lowering an index does not change the weight of a tensor density . (D.2.5) Again, using our generic example (D.1.4) above, T 'abc de = J-W Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' (1) T 'abc de = g'ex T 'abc dx (2) // raise last index e on T' in x'-space T a'b'c' d'e' = ge'e" Ta'b'c' d'e" . (3) // lower last index on T in x-space Therefore, T 'abc de = g'ex [ J-W Ra a' Rb b' Rc c' Rdd' Rxe' Ta'b'c' d'e'] // this is (2) + (1) above = g 'ex [ J-W Ra a' Rb b' Rc c' Rdd' Rxe' (ge'e" Ta'b'c' d'e")] // use (3) above = J-W Ra a' Rb b' Rc c' Rdd' (g'ex Rxe' ge'e") Ta'b'c' d'e" // regroup = J-W Ra a' Rb b' Rc c' Rdd' (Re e") Ta'b'c' d'e" // from last line in (7.5.9)† and again J -W passively watches all the action fly by. The weight of our generic tensor density with its last index raised is still W. † The last step uses (7.5.9) R ab = g'aa'Ra' b'gb'b but we raise a, lower b, and reverse two tilts to get Ra b = g'aa'Ra'b'gb'b . Appendix D: Tensor Densities 255 6. The covariant dot product of vector densities A and B of weights W and w is a scalar density of weight W + w and therefore A'•B' = J-(W+w) A•B . (D.2.6) Proof : First form the rank-2 tensor density AiBj which by (D.2.3) has weight W+w. Lower the second index and the mixed rank-2 tensor AiBj by item (D.2.5) still has weight W+w. Then contract to get A•B = AiBi and by (D.2.2), the weight is still W+w. Corollary : The magnitude of a vector density A of weight W is a scalar density of weight W, and therefore | A|' = J-W |A| . ( D . 2 . 7 ) Proof : |A|2 = A • A has weight 2W by (D.2.6), meaning | A'|2 = J-2W |A|2. Therefore | A'| = J-W |A| . 7. As J→1, tensor densities become true tensors. (D.2.8) One could imagine some limiting/morphing process on an underlying transformation F such that the linearized transformation matrix R approaches a rotation matrix at all points in space ( RR T= 1 and detR = 1, DN) and then J = detS → 1. In this case J-W → 1-W = 1 and therefore any tensor density, regardless of its weight W, becomes an ordinary tensor. Perhap s we should restrict this comment to underlying transformations F having J = detS > 0 since passing through detS = 0 is problematical. As was noted below (5.12.16), for an invertible x' = F(x), one does not have detS( x) = 0. at any x. Example : The cross product considered in Section D.7 belo w of N-1 contravariant vectors becomes in this limit an ordinary covariant vector. If g=1 in x-space, then g' = RRT = 1 in x'-space (DN) and then that resulting vector can be considered either contravarian t or covariant since both spaces are then Cartesian. This is the case with A = B x C under rotations in 3D sp ace. On can think of the εabc as moving in this limit from a tensor density of weight -1 to an ordinary tensor ( ε is treated in Section D.4 below). 8. If vector V is a vector density of weight W, then the four expansions shown in (7.13.10) become V = ΣnVn un with un • V = Vn V = ΣnVn un with un • V = Vn V = JW Σn V'n en with en • V = JW V'n V = JW Σn V'n en with en • V = JW V'n . (D.2.9) Proof : If V has weight W then, according to item 6 above, the four dot products shown on the right are scalar densities of weight W (the basis vectors all have weight 0). We then evaluate the four dot products : un • V = (un)i Vi = δn iVi = Vn // see (7.18.3) un • V = (un)i Vi = δniVn = Vn // see (7.18.3) ( en • V ) = JW (e'n • V' ) JW (e'n)iV'i = JW δn iV'i = JW V'n // see (7.18.1) ( en • V ) = JW (e'n • V' ) JW (e'n)iV'i = JW δniV'i = JW V'n // see (7.18.1) (D.2.10) Appendix D: Tensor Densities 256 In the last two lines, we use item 6 that A•B = J+(W+w) A'•B' with w=0. Thus the two un expansions are unaltered, but the en expansions pick up an extra factor of JW. For example, assume that V = Σmαm em with αm unknown. Then V • en = (Σmαm em)• en = Σmαm(em• en) = Σmαmδmn = αn Therefore αn = V • en = JW V'n so the expansion must be V = JW Σm V'n em . 9. This same idea applies to the more general tensor expansions of Appendix E. For example, if Aijk is a tensor density of weight W, one will have in place of (E.2.9), A = JW Σijk A'ijk (ei⊗ej⊗ek) . ( D . 2 . 1 1 ) Proof : According to item 6, the direct product space dot product shown below is a scalar density of weight W, so A • ( ea⊗eb⊗ec) = JW A' • (e'a⊗e'b⊗e'c) = JW A'ijk (e'a)i (e'b)j (e'b)k = JW A'abc If we write A = Σ ijk αijk (ei⊗ej⊗ek) with αijk unknown, we find as in the vector case above that the coefficients are αijk = JW A'abc. D.3 Theorem about Totally Antisymmetric Te nsors: there is really only one: εabc... The permutation tensor εabc...x has a number of indices equal to the dimension N of the space in which one is working. For N = 3 one has εabc, and εab or εabcd do not exist. The normal calibration of the tensor is ε123..N = 1. Each index swap causes a minus sign such as εabc = -εbac = - εcba , and if any pair of indices are the same, the result is zero, such as ε131 = 0, Due to this swapping property on any index, this tensor is called "totally antisymmetric". Theorem : Apart from a scalar factor, there exists only one totally antisymmetric (TA) tensor. (D.3.1) Proof: Suppose there were two TA tensors called εabc... and rabc.... If two or more of the indices are equal, both tensors are 0, so for such index sets, one can say rabc.. = f εabc.. where f is any finite function whatsoever. Consider now the case where all th e indices are distinct and therefore exhaust the set 123...N, and consider abc... to be a permutation of 123...N obtained by doing S pairwise swaps, abc... = P(123...) p = (-1)S . ( D . 3 . 2 ) If one were to associate a sign change with each swap, the total sign change would be p, the parity. Since ε and r are both TA tensors, each tensor can be "unw ound" back to a standard index order by doing these S swaps, and the swaps will cause a total sign of p relative to that standard order, so r abc.. = p r123... // for example, r2134.. = (-1)1 r1234.. (D.3.3) εabc.. = p ε123... Appendix D: Tensor Densities 257 Define scalar function f ≡ r123... / e123... , whatever it might be. Then r abc.. = p(f ε123...) εabc.. = p ε123... ( D . 3 . 4 ) and dividing these two equations one finds, r abc.. = f εabc.. ( D . 3 . 5 ) which has now been shown valid for all index sets abc.. . Therefore, any "other" totally antisymmetric tensor is just a scalar function times the ε tensor. Alternate proof assuming (D.11.8) below . Assume rabc.. = f εabc... Apply εabc.. to both sides and sum on abc.. to get εabc.. rabc.. = εabc.. (f εabc..) = (εabc.. εabc..)f = N! |g| f by (D.11.8). Therefore the assumed form rabc.. = f εabc.. is valid with f = (N!|g|)-1εabc.. rabc.. . Similar Example: For N= 3, if rank-4 tensor Mabcd is totally antisymmetric on indices abc, then Mabcd can be written in the form Mabcd = εabcqd where qd is a vector. In fact qd = (3!|g|)-1 εabcMabcd. D.4 The contravariant ε ten sor The "Levi-Civita symbol" or tensor is one that equals the permutation tensor when all indices are "up", as we shall see (a convention). One does not raise and lo wer indices of the mechanical permutation "tensor", but one does do this on the Levi-Civita te nsor, which we shall just call "the ε tensor". As we shall also see, the ε tensor is really a tensor de nsity of weight -1. When writing εabc.. one must have clearly in mind whether one means the permutation tensor or the Levi-Civita tensor. Unless indices are all up, they are in general not the same. Knowing nothing to start, assume that the totally antisymmetric εabc.. tensor transforms under F as a tensor density of some weight W which we hope to determine. Then ε' abc.. = J-W Ra a' Rb b' ... εa'b'c'.. . (D.4.1) Assume that εabc.. is the usual permutation tensor normalized to ε123...N = +1. This is the convention used by Weinberg p 99. This means each index swap changes the sign, and if two or more indices are the same, ε = 0. This is an important starting assumption, and from it most everything follows. Given this assumption, the RHS of (D.4.1) is to tally antisymmetric (TA). The argument is given once here and then used later se veral times. Consider an a ↔b swap. Then ε'bac.. = J-W Rb a' Ra b' ... εa'b'c'.. = J-W Rb b' Ra a' ... εb'a'c'.. // b'↔a' dummy indices = J-W Ra a' Rb b' ... (–εa'b'c'..) = – ε'abc.. . (D.4.2) Appendix D: Tensor Densities 258 The same result is true for any swap, thus RHS (D.4 .1) is TA. Since according to Section D.3 there is only one TA tensor available, apart from a scalar function factor, it follows that ε'abc.. = Kεabc... ( D . 4 . 3 ) where K is some scalar function, perhaps just a constant. Equation (D.4.1) above then reads K εabc... = J-W Ra a' Rb b' ... εa'b'c'.. . (D.4.4) Setting in the standard order, one finds that K ε 123... = J-W R1 a' R2 b' ... εa'b'c'.. ( D . 4 . 5 ) or K = J -W det(Ri j) = J-W (J)-1 = J-(W+1) // J = det(S) = 1/det(R) (D.4.6) so now ε' abc.. = Kεabc... = J-(W+1) εabc... (D.4.7) A second assumption is now made: that εabc.. (contravariant!) has the same value structure in any frame of reference, which is to say it is the same in x'-space as it is in x-space, ε' abc.. = εabc... . ( D . 4 . 8 ) where ε abc... is the usual permut ation tensor with ε123... = +1. This assumption is consistent with taking W = -1 in (D.4.7). Fact : The Levi-Civita tensor εabc... has weight W = -1. From (D.2.5) this is true regardless of the index positions on the ε t e n s o r . ( D . 4 . 9 ) Again, this follows the convention of Weinberg p 99. Some authors instead arrange for the above equation to be true for the covariant ε tensors, and use then ε 123...N = ε'123...N = +1, but we shall follow Weinberg. To summarize, assuming that εabc... is the usual permutation tensor normalized in the usual way, and assuming that ε'abc.. = εabc... so this tensor is the same in all frames or spaces, THEN one concludes that εabc... must transform as a rank-N tensor density of weight W = -1. That is to say, ε' abc.. = J Ra a' Rb b' ... εa'b'c'.. // this is (D.4.1) above with W = -1 (D.4.10) Viewed in this light, the tensor ε abc.. is known as the Levi-Civita tensor. Tullio Levi-Civita (1873-1941) . Italian, University of Padua 1892, with Ricci published the theory of tensor algebra in 1900 (see Refs.), which work assisted Einstein circa 1915 in formulating the theory of general relativity. The ε tensor bears his name. So metimes the affine connection (Appendix F) is called the Levi-Civita connection. Appendix D: Tensor Densities 259 D.5 Some facts about the ε tensor 1. Consider, with the assumption of (D.2.4) above applied to the ε tensor, ε abc... = gaa' gbb'..... εa'b'c'... // just as with any other tensor (D.5.1) This is again in the convention of Weinberg p 99 (4 .4.10). Although we show a ll indices lowered at once, one can lower them one at a time in the usual manner using g ij. Added-sign-s convention. Some authors make a special exception for the ε tensor and introduce an extra sign s into the above equation ( recall that s = -1 for special relativity) ε abc... = s gaa' gbb'..... εa'b'c'... s = sign[det(g ij)] . (D.5.2) This convention then invalidates the idea (7.4.11) that g ab lowers a tensor index on ε and gab raises a tensor index on ε. If that idea were valid, the above would read ε abc.. = s εabc and would force s =1. Furthermore, the meaning of a mixed ε tensor like εab becomes undefined in this convention because we don't know how to obtain εab from say ε ab. Since this convention is nevertheless used by some authors, we shall present some results below in this added-sign-s convention. Up-Equals-Down convention. Recall that we use the Weinberg convention (D.5.1), which says ε's indices work like those of any other tensor, and (D.4.8) which says ε'abc.. = εabc... . Some authors instead replace the "fiat rule" ε'abc.. = εabc... by a different fiat rule : ε^abc... = εabc... where we use a hat to distinguish their low index ε from our low index ε. Below we will show in the Weinberg convention that εabc... = g εabc... so we can then identify ε^abc... = (1/g) εabc... . The upper index version is the same, so ε^abc.. = εabc.. . In this convention one finds that ε^abc.. is a tensor density of we ight - 1 and then ε^abc... has weight 2 - 1 = +1. We mention this convention only because it exists, but we shall not express any of our results in this "up e quals down" convention. Continuing with (D.5.1), install the reference sequence to obtain ε 123.. = g1a' g2b'..... εa'b'c'... = det(gij) = g . // see (D.2.2) (D.5.3) Similarly, ε' 123.. = det(g' ij). To summarize, ε 123.. = det(g ij) = g // ε all-down index reference values ε'123.. = det(g' ij) = g ' . ( D . 5 . 4 ) In the "added sign s convention", these last two equations would have sg = |g| and sg' = |g'| on the right which means then these two ε values would be always positive. Appendix D: Tensor Densities 260 2. Take the same starting point as above εabc... = gaa' gbb'..... εa'b'c'... (D.5.1) The RHS is a totally antisymmetric in indices abc... . This is by argument (D.4.2) repeated in this context, εbac... = gba' gab'..... εa'b'c'... = gbb' gaa'..... εb'a'c'... = g bb' gaa'..... (-εa'b'c'...) = - gaa'gbb' ..... εa'b'c'... = - εabc... (D.5.5) Then according to Section D.3, the right side of the above can be written as RHS = C εabc... ( D . 5 . 6 ) since we there is only one TA tensor apart from scalar C. Therefore εabc... = C εabc... . ( D . 5 . 7 ) Insert the reference sequence ε 123... = C ε123... = C ( D . 5 . 8 ) But ε 123... = det(g ij) from (D.5.3), so C = det(g ij) = g ( D . 5 . 9 ) and then (D.5.1) says, for the "Weinberg convention", ε abc... = det(g ij) εabc... = g εabc... // relating all down to all up ε'abc... = det(g' ij) ε'abc... = g' ε'abc... // = g' εabc... . (D.5.10) where the second line follows by the same argument. Th ese equations relate all indices down to all up in the same space. Notice that both ε abc... and ε'abc... are totally antisymmetric. In the "added sign s convention" the above equations are instead ε abc... = |det(g ij)| εabc... = |g| εabc... // relating all down to all up ε'abc... = |det(g' ij)| ε'abc... = |g'| ε'abc... // = |g'| εabc... (D.5.11) 3. Divide the two Weinberg convention equations in (D.5.10) to find that ε'abc... = [det(g' ij)/ det(gij)] εabc... = (g'/g) εabc... ( D . 5 . 1 2 ) From (S.1.2) one has (g'/g) = J 2 so the conclusions regarding ε are these: ε' abc... = J2 εabc... = (g'/g) ε abc.. ε'abc... = εabc... = permutation tensor // general ε abc... = permutation tensor if g=1 (D.5.13) Appendix D: Tensor Densities 261 These conclusions are valid for the "added sign s convention" as well since det(g) and det(g') always have the same sign as shown in (D.1.2). Two comments: • Although we set ε' abc... = εabc... by fiat, we cannot similarly set ε'abc... = εabc... by fiat. This latter result comes out being ε'abc... = J2 εabc... as just shown. (D.5.14) • The fact that ε' abc... = J2 εabc... does not say that ε abc is a tensor density of weight -2 because there are no R factors showing (see tensor density definition (D.1.4) ) . (D.5.15) D.6 The covariant ε tensor : repeat Section D.4 as if its weight were not known According to (D.2.5), lowering indices does not change the weight of a tensor density. Eq (D.4.9) shows that ε abc.. is a tensor density of weight -1, so we know right away that εabc.. is also a tensor density of weight -1. Nevertheless, it is inte resting (and tests our cons istency) to see what happens when the same method used in Section D.4 for εabc... is applied to εabc... . We start by assuming εabc.. is a tensor density of some unknown weight W, ε'abc.. = J-W [ Raa' Rbb'.... εa'b'c'.. ] . (D.6.1) Eq (D.5.10) shows that ε a'b'c'.. is totally antisymmetric (TA) since we know εabc... is TA. The argument of (D.4.2) with up ↔down indices then shows from (D.6.1) that ε'abc... is TA as well. Since according to Section D.3 there is only one TA tensor available, apart from a scalar function factor, it follows that RHS(D.6.1) = K εabc.. so (D.6.1) then says, K ε abc.. = J-W [ Raa' Rbb'.... εa'b'c'.. ] . (D.6.2) Use (D.5.10) to set ε a'b'c'.. = det(gij) εa'b'c'.. inside the bracket, K ε abc.. = J-W [ Raa' Rbb'.... det(g ij) εa'b'c'..] , (D.6.3) and then install the reference sequence on both sides K ε 123.. = J-W [ R1a' R2b'.... det(g ij) εa'b'c'..] . (D.6.4) But (D.5.3) says ε 123.. = det(g ij), so cancel det(g ij) on both sides to get [ recall R ij = Sj i ] K = J -W [ R1a' R2b'.... εa'b'c'..] = J-W det(Rij) = J-W det(Sj i) = J-W J = J-(W-1) . (D.6.5) So here the result is K = J-(W-1) whereas in (D.4.6) the result was K = J-(W+1) . In the current case, since (D.6.1) and (D.6.2) have the same RHS, setting the LHS's equal says Appendix D: Tensor Densities 262 ε'abc.. = K εabc.. = J-(W-1) εabc.. ( D . 6 . 6 ) But (D.5.13) says that ε' abc.. = J2 εabc.. and therefore W = -1. Thr conclusion is that εabc... transforms with weight -1, the same as εabc..., so (D.6.1) becomes ε'abc.. = J [ Raa' Rbb'.... εa'b'c'.. ] . (D.6.7) D.7 Generalized cross products In (A.4.1) and (A.4.2) the following cross product of N-1 vectors is considered (now in Standard Notation, all indices are contracted except a) Q a ≡ εabc...x BbCcDd.....Xx or Q = B x C x D .... x X . (D.7.1) If the vectors B,C,D..X are all contravariant vectors, then applying the rule (D.2.3), one concludes that, since ε is a tensor density of weight -1 and since all the RHS vectors have weight 0, the object Q a is a covariant vector density of weight = -1 , and thus has this transformation rule Q' a = J RabQb . ( D . 7 . 2 ) Similarly, one may consider Qa ≡ εabc...x BbCcDd.....Xx . (D.7.3) If vectors B,C,D...X are covariant vectors, then Q a is a vector density of weight -1 and Q' a = J Ra bQb . (D.7.4) D.8 The tensorial nature of curl B It has just been shown that C = A x B [ Ca ≡ εabc AbBc ] is a vector density of weight -1, this being a special case of the generalized cros s product discussion above. As noted in the Example below (D.2.8), if gij = 1 and r is a (global) rotation, then g' ij = 1 as well. This means from (D.1.2) that g = g' = 1 and J = 1. Then the weight -1 transformation rule C' a = J RabCb becomes C' a = RabCb which is the same as a weight 0 rule, so then C can be regarded as a regular tensor in its transformati on properties (in this special case). [ Under inversion, if A→ -A and B→ -B, one gets C→ +C, so C is sometimes called a pseudotensor. ] One might conjecture that C = ∇ x B is also a vector density of weight -1, and that conjecture is correct as is now shown. Consider C n = εnab ∂aBb ( D . 8 . 1 ) Appendix D: Tensor Densities 263 where B is assumed to be an tensorial vector. It is he lpful to write this equati on in the following manner Cn = εnab [ ∂aBb – ∂bBa ] / 2 ( D . 8 . 2 ) where the second term is the same as the first term, since – εnab ∂bBa = εnba ∂bBa = εnab ∂aBb . (D.8.3) Recall now from (7.16.4) that the covari ant derivative of a vector is given by Bb;a = ∂aBb – Γc ab Bc ( D . 8 . 4 ) where the affine connection Γ c ab is symmetric under a ↔b. Therefore, Bb;a – Ba;b = [∂aBb – Γc ab Bc] - [∂bBa – Γc ba Bc] = ∂aBb – ∂bBa . (D.8.5) Therefore Cn can be expressed as C n = εnab [Bb;a – Ba;b ] / 2 ( D . 8 . 6 ) so by the same ε anti-symmetry noted above the final result is Cn = εnab Bb;a . ( D . 8 . 7 ) The major feature of B b;a -- as discussed in (7.16.5) -- is that it is a rank-2 tensor if B is a tensorial vector. The weight addition rule (D.2.3) can then be applied to εnab Bb;a. Since εnab has weight -1 and Bb;a has weight 0, the conclusion is that Cn is a vector density of weight -1. Thus, C = ∇ x B = curl B is a vector density of weight -1. D.9 Tensor E as a weight 0 version of ε : three conventions 1. Equations in the "Weinberg Convention" In this Section it is assumed as in (D.2.4) that g ij and gij raise and lower indices of the ε tensor just as they do for any other tensor (Weinberg c onvention). Above it was established that ε abc... = g εabc... ε123.. = +1 ε123... = g (D.5.10) ε'abc... = g'ε'abc... ε'123.. = +1 ε'123... = g' (D.5.10) ε' abc... = J2 εabc... = (g'/g) εabc... (D.5.13) ε is a rank-N tensor of weight W = -1 (D.4.9) and (D.6.7) (D.9.1) Appendix D: Tensor Densities 264 Again, just in passing, notice that ε' abc... = J2 εabc... does not say ε has weight -2 because the R factors are not present on the right side. Consider now the following new objects defined by (s g = |g|, s= sign(g) = sign(g') as in (D.1.2) ), Eabc... ≡ |g|-1/2 εabc... ⇒ E123... = |g| -1/2 g = |g| -1/2 s |g| = s |g|1/2 E'abc... ≡ |g'|-1/2 ε'abc.. ⇒ E'123... = |g'| -1/2 g' = |g'| -1/2 s |g'| = s |g'|1/2 . ( D . 9 . 2 ) From (D.1.2) g' = J 2g so that g transforms as a scalar density of weight -2. Since the sign of g and g' are the same, if follows that (sg) = |g| is also a scalar density of weight -2, and then the quantity |g|-1/2 transforms as a scalar density of weight +1, since |g'|-1/2 = J-1 |g|-1/2. Looking at the equation Eabc... ≡ |g|-1/2 εabc... above, and using the weight summation rule (D.2.3), one concludes that Eabc... transforms as a tensor of weight (+1) + (-1) = 0, and so Eabc... is an ordinary rank-N tensor. Comment : It was shown near (7.15.9) that a tensor dens ity equation with matching weights is "covariant", so one is not surprised to see the second line of (D.9.2) having the same form as the first line but everything is primed (s = s'). Raising indices on both sides of (D.9.2) gives th ese contravariant ordinary rank-N tensors, Eabc... ≡ |g|-1/2 εabc... ⇒ E123... = |g|-1/2 E'abc... ≡ |g'|-1/2 ε'abc... ⇒ E'123... = |g'|-1/2 . (D.9.3) To compare Eabc... and Eabc... we first evaluate Eabc... , Eabc... = |g|-1/2 εabc... // lower indices on both sides of 1st Eq of (D.9.3) since covariant = |g|-1/2 g εabc... // from (D.9.1) = s |g|-1/2 |g| εabc... // since g = s|g| by (D.1.2) = s |g|1/2 εabc... ( D . 9 . 4 ) We then pair this last equati on with the first of (D.9.3), Eabc... = s |g|1/2 εabc... (D.9.4) Eabc... = |g|-1/2 εabc... (D.9.3) (D.9.5) Dividing the two equations gives Eabc... / Eabc... = s |g| = g so that Eabc... = g Eabc... ( D . 9 . 6 ) We can then construct an E E product as follows Eabc... EABC... = [ |g|-1/2 εabc... ] [ |g|-1/2 εABC... ] // (D.9.3) and 1st line (D.9.4) = | g |-1 εabc... εABC... . (D.9.7) Appendix D: Tensor Densities 265 Next, we use standard ordering to get E123... = |g|-1/2 // from (D.9.3) E123... = s|g|+1/2 . // from (D.9.4) (D.9.8) Summarizing (D.9.8) and (D.9.6) and (D.9.7), E123... = |g|-1/2 E123... = s|g|+1/2 Eabc... = g Eabc... = s|g| Eabc... E'123... = |g'|-1/2 E'123... = s|g'|+1/2 Eabc... EABC... = |g|-1 εabc... εABC... E'abc... E'ABC... = |g'|-1 ε'abc... ε'ABC... (D.9.9) Since |g|-1 is a scalar density of weight +2 and each ε has weight -1 and each E has weight 0, one is happy to see the weights balance of the two si des of this pair of covariant equations. 2. Equations in the "added sign s convention" The previous section shows how things work out using the "Weinberg convention" noted at the start of Section D.5. Here is the previous section redone in the "added sign s convention" where ε abc... has an extra sign s as shown in (D.5.2). To ge t from Weinberg to added-sign-s, take εabc... → sεabc... . Doing this change in (D.5.10) for both εabc.. and ε'abc.. gives ε abc... = sgεabc... ε123.. = +1 ε123... = sg = |g| // g → sg ε'abc... = sg'ε'abc... ε'123.. = +1 ε'123... = sg' = |g'| // g' → sg' (D.9.10) ε' abc... = |J|2 εabc... = (g'/g) εabc... ε is rank-N tensor of weight W = -1 // same Consider the following new objects defined by ( sg = |g|, s= sign(g) = sign(g') as in (D.1.2) ) Eabc... ≡ |g| -1/2 εabc... ⇒ E123... = |g| -1/2 |g| = |g|1/2 E'abc... ≡ |g'| -1/2 ε'abc... ⇒ E'123... = |g'|-1/2 |g'| = |g'|1/2 . (D.9.11) By the same argument given above, Eabc... is an ordinary covariant tensor (ie, weight = 0). However, the indices cannot be raised by gij. In this convention then one must make independent definitions of the contravariant components as follows, Eabc... ≡ |g|-1/2 εabc... ⇒ E123... = |g|-1/2 E'abc... ≡ |g'|-1/2 ε'abc... ⇒ E'123... = |g'|-1/2 ( D . 9 . 1 2 ) To compare Eabc... and Eabc... , Appendix D: Tensor Densities 266 Eabc... = |g|-1/2 εabc... = |g|-1/2 |g| εabc... = |g|-1/2 |g| εabc... Eabc... = |g|-1/2 εabc... ( D . 9 . 1 3 ) so that E abc... = |g| Eabc... ( D . 9 . 1 4 ) Summarizing, E123... = |g|-1/2 E123... = |g|+1/2 Eabc... = |g| Eabc... E'123... = |g'|-1/2 E'123... = |g'|+1/2 E'abc... = |g'| E'abc... Eabc... EABC... = |g|-1 εabc... εABC... E'abc... E'ABC... = |g'|-1 ε'abc... ε'ABC... (D.9.15) In this "added sign s" convention, all these summarized results involve only |g| and there are no factors of s floating around. The cost of this benefit is a lack of true covariance (when s = -1), as demonstrated in Section D.11 below. 3. Equations in the "Ricci-Levi-Civita convention" Ricci and Levi-Civita use the "added s convention" but add a factor σ = sign(det(S)) into their definition of E ( see their paper p 135 or Hermann pp 31-21) so that Eabc... ≡ σ|g| -1/2 εabc... ⇒ E123... = σ|g| -1/2 |g| = σ|g|1/2 E'abc... ≡ σ|g'| -1/2 ε'abc... ⇒ E'123... = σ|g'|-1/2 |g'| = σ|g'|1/2 Eabc... ≡ σ|g|-1/2 εabc.. ⇒ E123... = σ|g|-1/2 E'abc... ≡ σ|g'|-1/2 ε'abc... ⇒ E'123... = σ|g'|-1/2 (D.9.16) Summarizing, E123... = σ|g|-1/2 E123... = σ|g|+1/2 Eabc... = |g| Eabc... E'123... = σ|g'|-1/2 E'123... = σ|g'|+1/2 E'abc... = |g'| E'abc... Eabc... EABC... = |g|-1 εabc... εABC... E'abc... E'ABC... = |g'|-1 ε'abc... ε'ABC... (D.9.17) Notice that in all three conventions, the last equation pair is the same . Since Ricci and Levi-Civita did not raise and lower i ndividual indices in their 1900 paper, they were not concerned about their convention bein g non-covariant in that sense. Appendix D: Tensor Densities 267 D.10 Representation of ε, εε and contracted εε as determinants This section is in de velopmental notation and ε is the permutation tensor . 1. Theorem about a certain permutation sum Consider the following object Q defined as a signed perm utation sum of the product of N matrix elements of a matrix M ij, Qabc..x ≡ ΣP p P2(Ma1Mb2 Mc3.....MxN) ( D . 1 0 . 1 ) In this equation, P 2 represents a permutation of the set of 2nd indices of the N matrix elements, and the sum is over all N! such permutations. There are many ways to arrive at a given permuta tion of 123...N by doing pairwise swaps, but for all these ways, the number of swaps S will be either even or odd. The parity p of a permutation is defined then as (-1) S and this p appears in the above sum. If one were to swap indices 2 ↔3 on the right above, each permutation would have S → S+1 since an extra swap is needed to undo 2↔ 3. Thus, all parities p → -p and in fact the whole object negates. But the swap 2↔3 is the same as b ↔c since M b3 Mc2 = Mc2 Mb3. Applying this argument to any pair of indices, one concludes that Q abc..x is totally antisymmetric and therefore by (D.3.1) can be written as K εabc...x : ΣP p P2(Ma1Mb2 Mc3.....MxN) = K εabc...x . (D.10.2) where in this Section ε abc...x is just the permutation tensor (not the Levi-Civita tensor). Setting abc..x to 123..N, one gets. ΣP p P2(M11M22 M33.....MNN) = K . ( D . 1 0 . 3 ) The left side of this last equation can be written as Σ P p P2(M11M22 M33.....MNN) = Σabc..x εabc...x M1aM2b M3c.....MNx (D.10.4) because p = εabc...x correctly assesses the parity of any give n permutation. But this object is simply det(M) so the conclusion is that K = det(M) and then ΣP p P2(Ma1Mb2 Mc3.....MxN) = det(M) εabc...x . (D.10.5) Consider now the following matrix where ab c..x is some permutation of 123...x, M (abc..) = Ma1 Ma2 Ma3 ... M aN M b1 Mb2 Mb3 ... M bN M c1 Mc2 Mc3 ... M cN ... M x1 Mx2 Mx3 ... MxN . (D.10.6) Appendix D: Tensor Densities 268 By rearranging the rows into their normal numerical order, one obtains matrix M, but incurs a sign from the various row swaps which sign is just εabc..x. Therefore, det(M(abc..) ) = εabc...x det(M) (D.10.7) and therefore we obtain the following "theorem" : Σ P p P2(Ma1Mb2 Mc3.....MxN) = det(M(abc..) ) = det(M) εabc...x . (D.10.8) The permutation sum is thus just the determinant of matrix M (abc..). The first term in the permutation sum, the term with an identity permutation, corresponds to the product of the diagonals of M(abc..). 2. Application of the theorem to M = δ: a representation of ε Apply the above theorem to matrix M = 1 ≡ δ, the identity matrix, so M ij = δi,j. Clearly det( δ) = 1 and one then has ΣP p P2(δa,1δb,2 δc,3.....δx,N) = det[δ(abc..) ] = εabc...x . (D.10.9) Thus is obtained a well-known representation of ε abc..x as a certain determinant of Kronecker deltas, ε abc...x = det[δ(abc..) ] where δ (abc..) = δa,1 δa,2 δa,3 ... δa,N = Ra δb,1 δb,2 δb,3 ... δb,N = Rb δc,1 δc,2 δc,3 ... δc,N = Rc ... δ x,1 δx,2 δx,3 ... δx,N = Rx (D.10.10) For future use, each row vector has been given a name like Ra where ( Ra)i = δa,i . The conclusion then is that (vertical bars here mean determinant) : (D.10.11) which is the same as εabc...x = ΣP p P2(δa,1δb,2 δc,3.....δx,N) . (D.10.12) Appendix D: Tensor Densities 269 3. Outer product of two ε tensors. Consider now ε abc...x = det ⎝⎜⎛ ⎠⎟⎞ Ra Rb ... Rx and εa'b'c'...x' = det ⎝⎜⎛ ⎠⎟⎞ Ra' Rb' ... Rx' . (D.10.13) Then εabc...x εa'b'c'...x' = det ⎝⎜⎛ ⎠⎟⎞ Ra Rb ... Rx det ⎝⎜⎛ ⎠⎟⎞ Ra' Rb' ... Rx' = det ⎝⎜⎛ ⎠⎟⎞ Ra Rb ... Rx det ( Ra' Rb' ... Rx') = d e t { ⎝⎜⎛ ⎠⎟⎞ Ra Rb ... Rx ( Ra' Rb' ... Rx') } (D.10.14) which is the determinant of this matrix Ra• Ra' Ra• Rb' Ra• Rc' ... Ra• Rx' Rb• Ra' Rb• Rb' Rb• Rc' ... Rb• Rx' Rc• Ra' Rc• Rb' Rc• Rc' ... Rc• Rx' ... Rx• Ra' Rx• Rb' Rx• Rc' ... Rx• Rx' . (D.10.15) A typical element of this matrix is given by Rc• Rb' = (Rc)i(Rb')i = δc,i δb',i = δc,b' (D.10.16) so that matrix can be written as δ a,a' δa,b' δa,c' .... δa,x' δb,a' δb,b' δb,c' .... δb,x' δc,a' δc,b' δc,c' .... δc,x' ≡ δ(abc..x; a'b'c'..x') .... δ x,a' δx,b' δx,c' .... δx,x' ( D . 1 0 . 1 7 ) where we have made up a name for this matrix as shown. The conclusion then is that Appendix D: Tensor Densities 270 ( D . 1 0 . 1 8 ) which is the same as ε abc...x εa'b'c'...x' = ΣP p P2(δa,a'δb,b'δc,c'.....δx,x') (D.10.19) because the right side of (D.10.19) is precisely th e determinant appearing in (D.10.18). As usual, the argument of P 2 is the product of the diagonal elements of the matrix of interest. 4. Contracting the first index of the outer product of two ε tensors. Consider what happens if one sums on the first index of the εε product: Σ a εabc...x εab'c'...x' . ( D . 1 0 . 2 0 ) For fixed given values of bc..x and b'c'...x' , there is only one way this sum can be non-zero. In that one way, bc..x and b'c'...x' must each be permutations of the set {12..N ex clude A} where A is the "hit value" of a in the sum on a. In the sum, only a = A contributes. Then using this hit value A and (D.10.19), Σ a εabc...x εab'c'...x' = εAbc...x εAb'c'...x' = Σ P p P2(δA,Aδb,b'δc,c'.....δx,x') = ΣP p P2(δb,b'δc,c'.....δx,x') (D.10.21) where in this last expression the sum can be regard ed as being over permutations where b'c'...x' is a permutation of b,c..x. Each of thes e lists of integers is in turn a permutation of {12..N exclude A}. Now, parity p = (-1)S where S is a number of swaps it takes to connect b'c'...x' with b,c..x, since a = a' = A. One might wonder if the overall sign of the RHS of the last equation is correct. A check of the first term in this sum which is just δb,b'δc,c'.....δx,x' shows that this overall sign is indeed correct. This first term must be positive because the product of two ε's is either +1 or 0. As an example, Σ a εabc εab'c' = ΣP p P2(δb,b'δc,c') = δb,b'δc,c' – δb,c'δc,b' . (D.10.22) The permutation sum shown on the right side of (D.10.21) is the determinant of δ (abc..x; a'b'c'..x') but with the first row and column crossed out. It can then be thought of as either the minor or cofactor of the element aa of this big δ matrix. Therefore, Σ a εabc...x εab'c'...x' = [cof δ(abc..x; a'b'c'..x')]aa (D.10.23) Appendix D: Tensor Densities 271 where the notation cofM refers to a matrix of cofactors with elements [cofM] ij. Don't confuse the a on the right side with the local dummy summation index a on the left side. The conclusion then is that (implied summation on a on the LHS) ( D . 1 0 . 2 4 ) 5. Contracting two or more indices of the outer product of two ε tensors. Consider what happens if one sums on the first two indices of the εε product: Σ a,b εabcd...x εabc'd'...x' . ( D . 1 0 . 2 5 ) For fixed given values of c,d..x and c'd'...x' , in or der for this double sum to be non-zero, the index sets cd..x and c'd'...x' must each be permutations of the se t {12..N exclude A,B} where A,B are a pair of hit values for the a and b sums. If a=A and b=B is a hit value, then so is a=B and a=A, so there are 2! contributing terms in the sum, and each term is +1. Therefore Σ a,b εabcd...x εabc'd'...x' = 2! εABc...x εABc'...x' = 2 ! ΣP p P2(δA,AδB,B'δc,c'δd,d'.....δx,x') = 2!ΣP p P2(δc,c' δd,d'.....δx,x') (D.10.26) where in this last expression the sum is over permutati ons where c'd'...x' is a permutation of c,d....x. Each of these lists of integers is in turn a permutation of {12..N exclude A,B}. Now parity p = (-1)S where S is a number of swaps it takes to connect c'd'...x' with c,d....x. Since the product of two ε's is either +1 or 0, the overall sign of the right side shown must be correct. As an example, Σ a,b εabc εabc' = 2!ΣP p P2(δc,c') = 2 δc,c' . (D.10.27) If c = c' = 2, then this says Σ a,b εab2 εab2 = ε132 ε132 + ε312 ε312 = 1 + 1 = 2 . (D.10.28) The permutation sum shown on the right side of (D.10.26) is the determinant of δ (abc..x; a'b'c'..x') but with the first 2 rows and columns crossed out. Therefore, Σ a,b εabcd...x εabc'd'...x' = 2! { [cof δ(abc..x; a'b'c'..x')]aa}bb . (D.10.29) The conclusion then is that (implied summation on a,b on the LHS) Appendix D: Tensor Densities 272 ( D . 1 0 . 3 0 ) Here a and b on the left are just dummy summation indices, whereas a and b on the right indicate that the right side is 2! {...} where {...} is the cofactor of a matrix which is the full δ (abc..x; a'b'c'..x') matrix but with the first two rows and columns crossed out . One can imagine doing th is crossing out in the picture of the full δ matrix shown in (D.10.18). The diagonal element of the first row and column is δ a,a' and of the second row and column δb,b' so we use the notation aa and bb to denote these rows and columns which are crossed out. This pattern continues as more indices are contracted . If three indices a,b,c are contracted, there will then be 3! hit values which are A,B,C and its permutati ons, and one just repeats the above discussion. The result will then be Σ a,b,c εabcd...x εabcd'...x' = 3! {{ [cof δ(abc..x; a'b'c'..x')]aa}bb}cc . (D.10.31) The conclusion then is that (implied summation on a,b,c on the LHS) ( D . 1 0 . 3 2 ) Eventually one arrives at a point where all but one of the indices are summed, so that ε abcd...x εabcd...x' = (N-1)! | δx,x'| = (N-1)! δ x,x' (D.10.33) an example being εabc2 εabc2 = 3! δ22 = 3! = ε1342 ε1342 + ε1432 ε1432 + 4 more terms = 1+1+4 = 6 . (D.10.34) The final point is that at which all indices are summed, with result εabcd...x εabcd...x = N ! ( D . 1 0 . 3 5 ) and example of which is ε abcεabc = ε1232 + ε2132 + 4 more terms = 1 + 1 + 4 = 6 . (D.10.36) Appendix D: Tensor Densities 273 6. Summary of Results • • • • εabcd...x εabcd...x' = (N-1)! δx,x' εabcd...x εabcd...x = N ! ( D . 1 0 . 3 7 ) The determinants appearing above are sometimes called generalized Kronecker deltas . For example, here is a sample determinant followed by our (easy to type) notation and then the official Kronecker delta notation: ≡ δ(cd...x; c',d'...x') ≡ δ c c'd d' ... x x' (D.10.38) Often the deltas are written in the form δc,c' = δc c' or δc c' Using both our notation and the generalized Kronecker delta notation, we rewrite (D.10.37) this way Appendix D: Tensor Densities 274 εabc..xεa'b'c'..x' = δ(abc...x; a'b'c'...x') = δ a a'b b'c c' .... x x' εabc..xεab'c'..x' = δ(bcd...x; b'c'd'...x') = δb b'c c' .... x x' εabcd..xεabc'd..x' = 2! δ(cd...x; c'd'...x') = 2! δc c'd d'.... x x' εabcde..xεabcd'e'..x' = 3! δ(de...x; d'e'...x') = 3! δd d'e e'.... x x' ••••• εabcd...x εabcd...x' = (N-1)! δ(x,x') = (N-1)! δx x' = (N-1)! δx,x' εabcd...x εabcd...x = N! (N = dimension of x-space) (D.10.39) Keep in mind that in all of Section D.10, ε has represented the permutation tensor. D.11 Covariant forms of the previous Section results The Section D.11 results were in effect all devel oped in Cartesian x-space where up and down indices on the ε's did not matter and we regarded ε as the permutation tensor. The rules for converting any of these results to covariant form are as follows: [ after conversion, ε becomes the Levi-Civita tensor where ε ab... is the only index position where it is th e same as the permutation tensor ] Weinberg convention: ( D . 1 1 . 1 ) • Make the replacement ε***** ε***** → |g|-1 ε***** ε***** = E***** E***** from (D.9.9). In this way we "continue off" the Cartesian equation (whe re g = 1) to obtain a covariant form. The objects E***** and E***** are true contravariant and covariant (weight = 0) rank-N tensor. This is vaguely reminiscent of "analytically continuing" a real function f(x) "off" the real axis to get a function of a complex variable f(z). When evaluated on the real ax is, f(z) and f(x) are the same. See the tensorization discussion in Section 15.2 for more detail. • Write the right side of the expression of interest replacing every δ a,b = δa b = ga b as shown in (7.4.19) . Then the right side will also be a true tensor. Example 1 : For g = 1, consider the εε product with no summed indices [first line of (D.10.37)] for N = 2: εabεa'b' = ⎪⎪ ⎪⎪ δaa' δab' δba' δbb' = δa,a' δb,b' – δa,b' δb,a' . // ε = permutation tensor (D.11.2) The covariant form is as follows, where now g is some arbitrary metric tensor for x-space, Appendix D: Tensor Densities 275 EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪ga a' ga b' gb a' gb b' = ga a' gb b' – ga b' gb a' . (D.11.3) The equation in x'-space would then be E'abE'a'b' = |g'|-1 ε'abε'a'b' = ⎪⎪ ⎪⎪g'a b g'a b' g'a' b g'a' b' = g'a b g'a b' – g'a' b g'a' b' (D.11.4) because true tensor equations are "cova riant" as discussed in Section 7.15 . One can raise and lower individual indices to get for example these valid tensor equations which are 3 members of the family of 4! = 24 tensor equations obtained by raising and lowering indices : EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪ga a' ga b' gb a' gb b' = ga a' gb b' – ga b' gb a' (D.11.5) Ea bEa'b' = |g|-1 εa bεa'b' = ⎪⎪ ⎪⎪ga a' ga b' gba' gbb' = ga a' gbb' – ga b' gba' (D.11.6) EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪gaa' gab' gba' gbb' = gaa' gbb' – gab' gba' (D.11.7) and of course in x'-space the equations are the same but everything is primed. For the Levi-Civita ε tensor, we then have this c ovariant version of (D.10.39) : Eabc..xEa'b'c'..x' = |g|-1εabc..xεa'b'c'..x' = g(abc...x; a'b'c'...x' ) Eabc..xEab'c'..x' = |g|-1εabc..xεab'c'..x' = g(bcd...x; b'c'd'...x' ) Eabcd..xEabc'd..x' = |g|-1εabcd..xεabc'd..x' = 2! g(cd...x; c'd'...x' ) Eabcde..xEabcd'e'..x' = |g|-1εabcde..xεabcd'e'..x' = 3! g(de...x; d'e'...x' ) ••••• Eabcd...x Eabcd...x' = |g|-1εabcd...x εabcd...x' = (N-1)!g(x, x') = (N-1)!gx x Ebcd...x Eabcd...x = |g|-1εabcd...x εabcd...x = N! (D.11.8) where g(...) is the determinant of a matrix of gα β elements. Eq. (D.11.5) provides an example of the first line of (D.11.8) for the N=2 dimensions. Each equati on in (D.11.8) is a true tensor equation allowing individual non-summed indices to be taken up and down on both sides. And of course the contraction tilts can be individually reversed as well. If all index positions are reversed, one gets a version of (D.11.8) where all matrix entries have the form g αβ instead of gα β, For example, the second last line would be, Eabcd...xEabcd...x' = |g|-1εabcd...xεabcd...x' = (N-1)!g( x,x') = (N-1)!g xx' (D.11.9) As in (7.4.19) one always has g αβ = gα β = δα β = δαβ = δα,β but of course g αβ ≠ δα,β . Appendix D: Tensor Densities 276 Since the above equations are true tensor equations , Section 7.15 says the equations are also valid in x'-space where every object in the equation is primed. Example 2 : For N = 3 the second line of (D.11.8) reads, EabcEab'c' = |g|-1εabcεab'c' = g(bc; b'c') = ⎪⎪ ⎪⎪gb b' gb c' gc b' gc c' = gb b'gc c' - gc b'gb c' . (D.11.10) Raising b' and c' gives EabcEab'c' = |g|-1εabcεab'c' = g(bc; b'c') = ⎪⎪ ⎪⎪gbb' gbc' gcb' gcc' = gbb'gcc' - gcb'gbc' . Shuffling indices, this can be written EsABEsA'B' = |g|-1εsABεsA'B' = g(AB; A'B') = ⎪⎪ ⎪⎪gAA' gAB' gBA' gBB' = gAA'gBB' - gBA'gAB' = EsABEsA'B' = |g|-1εsABεsA'B' . ( D . 1 1 . 1 1 ) Added-sign-s and Ricci-Levi-Civita conventions: Do the above two bullet items, then add an overall sign s to the right side, because ε abc.. = s g εabc... in these conventions instead of εabc.. = g εabc... so that εabc.. (Weinberg) = s εabc.. (added-sign). Example : The example above becomes ( ε a'b'→ s εa'b') EabEa'b' = |g|-1 εabεa'b' = s ⎪⎪ ⎪⎪ga a' ga b' gb a' gb b' = s ( ga a' gb b' – ga b' gb a') . (D.11.12) The second equation is undefined (when s=-1), because we don't know how to lower just one index on ε, hence on E, as discussed near (D.5.2). The third equation is EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪gaa' gab' gba' gbb' = gaa' gbb' – gab' gba' . (D.11.13) The first and third equations are tr ue tensor equations, except individual indices cannot be raised and lowered. If one were doing some significant work involving covariance and s=-1, it would certainly seem advisable to use the Weinberg convention since it is completely "covariant" for either sign of s. Appendix D: Tensor Densities 277 D.12 How determinants of rank-2 tensors transform In this document we have encountered onl y a few determinants of tensors (like g ij) and tensor-like objects (like Ri j and Si j). Nevertheless, we would like to know how the determinant of a rank-2 tensor transforms under x = F(x). To this end, we first rewrite the tr aditional determinant formula in a covariant form. Once this is done, the conclusions come quickly. Comment : The objects below like det(Mi j) and det(M ij) of course have no dependence on the "indices" i and j other than on the up/down pos ition of these indices. Elsewhere we write these objects as det(M* *) and det(M **) where the "wildcards" just show the nature of the matrix elements (down-tilt or all-down). Here, for technical reasons to be seen below, we maintain the notations det(Mi j) and det(M ij). We start with this mechanical statement of the determinant of a matrix M i j det(M i j) = εab..x M1 aM2 b....MN x ( D . 1 2 . 1 ) where ε is the permutation tensor. This fo rm is "mechanical" just in the sense that if you drew a picture of the matrix Mi j and mechanically evaluated det(Mi j), you obtain the above result. We can now restate this determinant, switching from the permutation tensor εabc...x to the index-all-up Levi-Civita tensor. As stated below (D.4.1), we take εabc..x to be equal to the permut ation tensor. We then have det(M i j) = εab..x M1 aM2 b....MN x ( D . 1 2 . 2 ) which certainly looks more "covariant" since all summe d indices appear to be contracted. But with fixed upper indices 1,2...N "hanging out", it is not quite clear what is going on here. Since a through x are all contracted, and since weight( ε) = - 1, one might be tempted to say det(M i j) is a scalar density of weight -1, but that is incorrect. To get a better view of things, it is useful to rewrite the above determinant as follows (proof follows) det(M i j) = (1/N!) εAB...X εab..x MA a MB b ...MX x . (D.12.3) Again, both ε's shown here are in effect permutation tensors. We shall now show that the right sides of the previo us two equations are exactly the same. First write the right hand side of (D.12.3) as RHS (D.12.3) = (1/N!) ε AB...X { εab..xMA a MB b ...MX x } . (D.12.4) The bracketed quantity can be expanded as, Q AB..X ≡ {εab..xMA a MB b ...MX x} = MA 1 MB 2 ...MX N + all signed permutations . (D.12.5) meaning all signed permutations of the upper indices. Notice that Appendix D: Tensor Densities 278 Q12..N ≡ {εab..xM1 a M2 b ...MN x} = M1 1 M2 2 ...MN N + all signed permutations . (D.12.6) This is just a mechanical evaluation of det(Mi j) and we conclude that Q 12..N = det(Mi j) . ( D . 1 2 . 7 ) The tensor Q AB..X is totally antisymmetric as this example demonstrates, Q BA..X = εabc..xMB aMA bMC c ...MX x = [ - εbac..x] MA bMB aMC c ...MX x = - [ ε abc..x] MA aMB bMC c ...MX x // a ↔b = - Q AB..X . ( D . 1 2 . 8 ) Then using theorem (D.3.1) we can write Q AB..X in this form. Q AB..X = K εAB..X . (D.12.9) To evaluate K, use the standard ordering 123..N to find from the above and (D.12.7), Q12..N = K ε12..N = K = det(Mi j) . (D.12.10) Then, since K = det(M i j) one finds that Q AB..X = det(Mi j) εAB..X . (D.12.11) Therefore (D.12.4) can be written RHS (D.12.3) = (1/N!)ε AB...X { εab..xMA a MB b ...MX x } = (1/N!) ε AB...X {QAB..X } = (1/N!) ε AB...X {det(Mi j) εAB..X} = det(Mi j) (1/N!) Σ AB..X (εAB...X)2 . (D.12.12) The sum Σ AB..X (εAB...X)2 is a "sum of ones" and there is a one for each permutation of AB..X. All other terms are zero. There are N! total permutations including the first, so ΣAB..X (εAB...X)2 = N ! ( D . 1 2 . 1 3 ) which agrees with the last equation of (D.10.37). Therefore, RHS (D.12.3) = det(M i j) (1/N!) Σ AB..X (εAB...X)2 = det(Mi j) . (D.12.14) Appendix D: Tensor Densities 279 This concludes our too-lengthy proof that the right sides of (D.12.2) and (D.12.3) are identical. Now we start with the proven result, det(M i j) = (1/N!) εab..x εAB...X MA a MB b ...MX x . (D.12.3) Eq (D.5.10) says that εABC... = g εABC... so the above can be written det(M i j) = (1/g) (1/N!) εab..x εAB...X MA a MB b ...MX x . (D.12.15) Now, finally, we have a form in which all tensor in dices are contracted with no loose ends. We can then use theorem (D.2.3) about the a dditivity of weights. Recall from (D.4.9) and (D.6.7) that both ε tensors shown in (D.12.15) have weight -1, and that the object 1/g has weight +2 from (D.1.7). Adding, we find that the object det(Mi j) has weight 0. We have therefore proven: (scalar = scalar density of weight 0) Theorem: The determinant det(Mi j) of a mixed rank-2 tensor Mi j transforms as a scalar under the transformation x = F(x) . ( D . 1 2 . 1 6 ) Comment : Since our Chapter 2 S matrix Si j is not a tensor, J = det(Si j) is not a scalar, and is fact not a tensor of any kind since it bridges x-space and x'-space. It is now straightforward to determine the transformati on nature of the other three rank-2 tensor types, and in fact to find simple relations between the four de terminants. In all these case s, we use of the up-down altering property of the g tensor of (7.4.11), the "up-tilt" or "down-tilt" version of matrix multiplication of (7.8.11), the matrix rule det(AB) = det(A)det(B), fact s (7.5.20) and (7.5.21), and the weight of g and 1/g as determined in (D.1.6) and (D.1.7) : det(M ij) = det(g iaMa j) = det(g ij) det(Mi j) = g det(Mi j). -2 -2 0 det(M ij) = det(M iagaj) = det(M ij)det(gij) = { g det(Mi j)}{g-1} = det(Mi j) 0 -2 0 +2 0 det(Mij) = det(giaMaj) = det(gij) det(Mij) = g-1 det(Mij) = g-1 det(Mi j) . (D.12.17) +2 +2 0 +2 0 The weights are shown under each line of equations. The conclusions are these: det(M ij) = det(Mi j) // scalar densities of weight 0 ( = scalar) det(M ij) = g det(Mi j) // scalar density of weight -2 g = det(g ij) det(Mij) = g-1 det(Mi j) // scalar density of weight +2 g-1 = det(gij) . (D.12.18) Appendix D: Tensor Densities 280 Here we have related each of the three partner determinants to the down-tilt determinant, and have shown the weight of each type of determinant. Example : We know that g ij is a rank-2 tensor from Section 5.7. Thus, we expect to find from (D.12.18) that, det(g ij) = det(gi j) ⇒ det( δij) = det(δi j) or 1 = 1 ok, weight 0 det(g ij) = g det(gi j) ⇒ g = g det( δi j) = g*1 = g ok, weight -2 det(gij) = g-1 det(gi j) ⇒ g-1 = g-1det(δi j) = g-1* 1 = g-1 ok, weight 2 . (D.12.19) Equations (D.12.18) are valid only for rank-2 tensors. They are not valid for R and S. Go back now to the first line of (D.12.18), det(M ij) = det(Mi j) = det([MT]ji) where MT is the "covariant transpose" of M as shown in (7.9.3). This shows that det(M **) = det([MT]**) or det(M ut) = det(MT ut) . // ut = up-tilt Since det(M) = det(M T) is valid for any matrix regardless of index position (MT is the "matrix transpose" of M), we find that det(MT ut) = det(MT ut) = det(M ut) . A similar argument beginning with det(Mi j) = det(M ij) = det([MT]j i) ⇒ det(M dt) = det(MT dt) shows that det(M T dt) = det(MT dt) = det(M dt) . // dt = down-tilt Since M T = MT for both-up or both-down index positions, we arrive at this intere sting generalization of the traditional determinant theorem det(M) = det(MT) : Fact: det(M) = det(MT) = det(MT) for all four possible index positions (D.12.20) where T is the matrix transpose and T is the covariant transpose (7.9.3). The scalar density weights of these determinants are given in (D.12.18) and do depend on the index positions. Appendix E: Tensor Expansions 281 Appendix E: Tensor Expansions: direct product, polyadic and operator notation E.1 Direct Product Notation This entire Section uses the general Picture A context where x-space need not be Cartesian, (E.1.1) The Standard Notation of Chapter 7 is used throughou t. Useful forms can be found in Section 7.18. The key tool required for the expression of tensor e xpansions is the notion of a direct product (tensor product, outer product) of n tensorial v ectors defined in this simple way, ( A⊗B⊗C ...)abc... ≡ AaBbCc..... ( A⊗B⊗C ...)a bc... ≡ AaBbCc..... etc . (E.1.2) The tensor A⊗B⊗C ... is nothing more than the outer product of vectors A,B,C as in (7.1.1) for contravariant vectors, but later extended to any mi xture of vector types. One can, however, construct outer products from more complicated tensors, see Lucht Tensor Products . Suppose one has A = vector F,G = rank-2 tensors M,N = rank-3 tensors . (E.1.3) Then here is a small sampling of outer pr oducts of tensors that can be formed, (F⊗A) abc = FabAc (F ⊗A⊗G)abcde = FabAcGde (F⊗ G)abcd = FabGcd (F⊗M⊗A )abcdef = FabMcdeAf (M⊗ N)abcdef = MabcNdef . (E.1.4) In any of these equations, any one or more of the in dices can be lowered on both sides to provide a valid equation since all objects in the list (E.1.3) are assumed to be true tensors. All the left-hand-side objects above are tensors because the right sides show that they transform as tensors. In what follows, only the (E.1.2) direct product of v ectors shall be considered. One can define the dot product of two direct-product-space vectors in this obvious manner, ( A⊗B⊗C ...) • (A'⊗B'⊗C' ... ) ≡ (A⊗B⊗C ...)abc... (A'⊗B'⊗C' ... )abc = A aBbCc..... A'aB'bC'c..... = A•A' B•B' C•C' ... (E.1.5) where of course the indices abc can be "tilted" in any way desired according to (7.11.3). Appendix E: Tensor Expansions 282 E.2 Tensor Expansions and Bases Preamble: A seeming paradox and how it is resolved Before attacking tensor expansions below, we wish to head off a possible confusion between a quantity transforming as a scalar under some transformation x' = F(x) versus transforming as a component of a tensor. The "paradox" is presente d in a few simple examples. Example 1 : Consider the vector r = (x,y,z) and a transformation r' = F(r) = R r which is a rotation. Assume x-space is Cartesian so g = 1. Consider the dot product s = r • x^ . As noted in (5.10.2), this dot product transforms as a scalar under F. In x'-space we still find that r' • x^' = s. After all, the projection of one vector onto another cannot change when the two vectors are rotated together. On the other hand, direct calculation shows that r • x^ = x, and x is a component of the vector r = (x,y,z). So how can one say that quantity r • x^ is a scalar when r • x^ = x which is a component of a vector? This type of question arises frequently in the study of tensor analysis: there seems to be a paradox that needs resolving. The physicist considers several different "thought experiments". In Experiment 1, two measurements are made. The fi rst is made in Frame S and one finds that s = x. The second measurement is made of rotated vector r' = R r in rotated Frame S' and there one finds that s = x'. The two measurements give the same number, so in Experiment 1 one finds that x' = x and the quantity so measured is a scalar. In this experiment x = r • x^ and x' = r' • x^'. In Experiment 2, everything is done in Frame S. One draws a vector r and measures x. One then rotates this vector within Frame S to get a new vector r' = R r where r' = (x',y',z') . One then measures x' and finds x' ≠ x. In this experiment, x = r • x^ and x' = r' • x^ where x^ has no prime. In Experiment 3 one observes unrotated r from rotated Frame S' so x = r • x^ and x' = r • x^' and this again results in x' ≠ x Thus, in these three experiments, x has the same de finition, but x' has three different definitions. That is why one can have x' = x in Experiment 1 and x' ≠ x in Experiments 2 and 3. The concept of r • x^ as a scalar applies to Experiment 1 where s = r • x^ = x = r' • x^' = x'. Example 2 . Consider the vector expansion (7.13.10) which says V = Σn V'n en with V'n = V • en . Here we have the same paradoxical situation: we know that s = V • en must be a scalar with respect to F, yet it is equal to the component of a contravariant vector V'n under F. If we write V = ΣnVnen, then in Frame S we find that V • en = Vn which is like r • x^ = x of the previous Example. In Frame S' we find instead that V' • e'n = V'i(e'n)i = V'iδn i = V'n which is like r' • x^' = x' in Example 1. In Experiment 1 outlined above, it is not a paradox to have s = en • V = Vn = V'n be a scalar. Example 3 . Below we expand a rank-3 tensor A = Σijk αijk (bi⊗bj⊗bk) and we find that αijk = A • (bi⊗bj⊗bk) where • is a the dot product (E.1.5). We know that this quantity αijk, analogous to s in the two Examples a bove, must be a scalar under x' = F(x). Yet when we take bn = en we find in (E.2.8) that αijk = A'ijk so our scalar αijk is equal to the component of a rank-3 tensor in x'-space. Once again, in Experiment 1 this is not a contradiction. It just happens that in x'-space the scalar αijk Appendix E: Tensor Expansions 283 appears as the value of a tensor component A'ijk in x'-space, just the way s in Example 1 appears as the value x' of vector r' in Frame S'. Tensor Expansions In what follows, we use the example of a rank-3 tensor and the reader can easily see how this applies to a rank-n tensor. Let bi be an arbitrary complete set of basis vectors in x-space. As shown in the notes following (6.2.8) there exists a unique set of dual ("reciprocal") basis vectors bi (also in x-space) such that bi• bj = δi j, where we now use the Standard Notation bn equations of (7.18.6). Consider then the following expansion of rank-3 tensor A with contravariant components Aabc, A = Σijk αijk (bi⊗bj⊗bk) where ( bi⊗bj⊗bk ...)abc = ( bi)a (bj)b (bk)c (E.1.2) Aabc = Σijk αijk (bi⊗bj⊗bk ...)abc = Σijk αijk (bi)a (bj)b (bk)c . (E.2.1) As we did in Example 3 (2.9.5) for en, we declare the bn vectors to be "contravariant by definition" by writing the rule ( b'n)i ≡ Ri j(bn)j. So then the vectors bn are true rank-1 tensors under x' = F(x). According to Section 7.1, the outer product ( bi)a (bj)b (bk)c is a true rank-3 contravariant tensor under F. Then (E.2.1) says that Aabc is a linear combination of these tensors, which we know is a tensor because the sum of two tensors of some type is a tensor of the same type. The expansion coefficients αijk can be obtained by dotting both sides of (E.2.1) with ( bi'⊗bj'⊗bk') and using (E.1.5), ( bi'⊗bj'⊗bk') • (bi⊗bj⊗bk) = bi'• bi bj'• bj bk'• bk = δi' iδj' jδk' k . (E.2.2) The result is then A • ( bi'⊗bj'⊗bk') = {Σijkαijk (bi⊗bj⊗bk)} • (bi'⊗bj'⊗bk') = Σijk αijk δi' iδj' jδk' k = αi'j'k' or unpriming indices, α ijk = A • (bi⊗bj⊗bk) . But since both A and ( bi⊗bj⊗bk) are elements of the triple direct-product space spanned by the vectors (bi⊗bj⊗bk), we use the dot product of (E.1.5) to claim that A • (bi⊗bj⊗bk) = Aabc (bi⊗bj⊗bk)abc = Aabc (bi)a (bj)b (bk)c . The coefficients αijk may then be written in all these ways : αijk = A • (bi⊗bj⊗bk) = Aabc (bi⊗bj⊗bk)abc = Aabc (bi)a (bj)b (bk)c (E.2.3) Appendix E: Tensor Expansions 284 where Aabc are the contravariant components of tensor A in x-space, and (bi)a are the covariant components of vector bi in x-space. As noted in Example 3 of the previous subsection, the object αijk transforms as a scalar , since it is the dot product of two dir ect-product-space vectors. This fact is especially obvious from the last expression in (E.2.3) wh ere all tensor indices are contracted so the result must be a scalar according to the neutralization rule (7.12.1). So αijk is a set of 33 =27 scalars. Two special cases attract our attention. The ( ui)a are axis-aligned basis vectors in x-space, as shown in (7.13.9) or (7.18.3). For these basis vectors, one has ( ui)a = δia and ( ui)a = δi a . If one considers expansion (E.2.1) with bn = un, then A = Σijk αijk (ui⊗uj⊗uk) where ( ui⊗uj⊗uk ...)abc = ( ui)a (uj)b (uk)c = δia δjb δkc (E.2.4) and the coefficients are found to be αijk = A • (ui⊗uj⊗uk) = Aabc δi a δj b δk c = Aijk , (E.2.5) so the scalar coefficients αijk are exactly the x-space contravariant components of the tensor A. If this seems paradoxical, see Example 1 above where we found that scalar s = r • x^ = x. Thus, A = Σijk Aijk (ui⊗uj⊗uk) . (E.2.6) On the other hand, if ei are the tangent base vectors in x-space (s ee Chapters 3), the dual vectors are the ei and from (7.13.1), ( ei)a = Sa i = Ria and ( ei)a = Sai = Ri a . If one considers the expansion A = Σijk αijk (ei⊗ej⊗ek) where ( ei⊗ej⊗ek ...)abc = ( ei)a (ej)b (ek)c = Ria Rjb Rkc (E.2.7) then the coefficients are found to be αijk = A • (ei⊗ej⊗ek) = Aabc (ei)a(ej)b(ek)c = Aabc Ri a Rj b Rk c = Ri a Rj b Rk c Aabc = A'ijk ( E . 2 . 8 ) and thus the scalar coefficients α ijk in this case are exactly the x'-space contravariant components of tensor A, as shown for M in (7.10.9). Thus, A = Σijk A'ijk (ei⊗ej⊗ek) . ( E . 2 . 9 ) Expansions like the above are the ge neralizations (to tensors of any ra nk) of these vector expansions stated in (7.13.12), Appendix E: Tensor Expansions 285 A = Σiαi bi αi = bi • A // arbitrary basis A = ΣiAi ui // axis aligned unit vectors A = ΣiA'i ei // tangent base vectors (E.2.10) where we continue to write rank-1 tensors (vectors) in bold font: A. To summarize, here is the general rank-n tensor expansion for an arbitrary basis, and then for the two specific bases just discussed: A = Σ ijk... αijk... (bi⊗bj⊗bk...) αijk... = Aabc... (bi)a (bj)b (bk)c... A = Σ ijk... Aijk... (ui⊗uj⊗uk...) Aijk... = contravariant components of A in x-space A = Σ ijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... = contravariant components of A in x'-space Tilting labels ( E . 2 . 1 1 ) In the above discussion, suppose we make the swap bi → bi . We quickly trace the steps: A = Σijk αijk (bi⊗bj⊗bk) where ( bi⊗bj⊗bk ...)abc = ( bi)a (bj)b (bk)c (E.2.1) A abc = Σijk αijk (bi⊗bj⊗bk ...)abc = Σijk αijk (bi)a (bj)b (bk)c = outer product sum (E.2.1 ) α ijk = A • (bi⊗bj⊗bk) = Aabc (bi⊗bj⊗bk)a bc = Aabc (bi)a (bj)b (bk)c = scalar (E.2.3) A = Σ ijk αijk (ui⊗uj⊗uk) // first special case (E.2.4) α ijk = A • (ui⊗uj⊗uk) = Aabc (ui)a (uj)b (uk)c c = Aabc δia δj b δk c = Aijk (E.2.5) A = Σ ijk αijk (ei⊗ej⊗ek) // second special case (E.2.7) α ijk = A • (ei⊗ej⊗ek) = Aabc (ei)a(ej)b(ek)c = Aabc Ria Rj b Rk c = A'ijk (E.2.8) Here αijk = Aabc (bi)a (bj)b (bk)c = Aabc (bi)a (bj)b (bk)c is a different set of scalars compared with the original αijk = Aabc (bi)a (bj)b (bk)c because in general bi ≠ bi. The sequence above shows that everything goes through as before, and so we may expand tensor A in either of these two ways, Appendix E: Tensor Expansions 286 A = Σijk αijk (bi⊗bj⊗bk) A = Σijk αijk (bi⊗bj⊗bk) . ( E . 2 . 1 2 ) The second expansion is really a mixed-basis expans ion as discussed in Section E.10 below. The main point of this exercise is to show that one may "reverse the tilt" of the i summation index and still have a viable tensor expansion. This is not an example of the "contraction tilt-re versal rule" (7.11.1) because the object α ijk is not a rank-3 tensor, it is a set of scalars. Moreover, index i is not a tensor index, it is a label on the basis vector bi. Nevertheless, in this case tilt-reversal is a llowed and in fact just serves to define a new set of coefficients αijk It turns out that the indices on αijk can be lowered by the matrix w' nm = bn • bm which appears in (7.18.6). To show this, consider α ijk = Aabc (bi)a (bj)b (bk)c = Aabc (bi)a (bj)b (bk)c // (E.2.3) then tensor tilt rule so w' siαijk = Aabc [w'sibi]a (bj)b (bk)c = Aabc [bs]a (bj)b (bk)c // bn = w'ni bi in (7.18.6) = α sjk // as shown in (E.2.3) a b o v e ( E . 2 . 1 3 ) which shows that w' ** lowers an index on αijk. Despite this ability of w' ** and w'** to lower and raise indices on the object family αijk, each element of this family (such as αijk) is a scalar. When bn = en, we find that g' then raises and lowers indices on αijk, but this does not make αijk a rank-3 tensor, because αijk transforms as a scalar under x' = F(x). We apologize for constantly hammering on this point, but it can be a source of confusion. Obviously one can tilt any or all of the indices in expansion (E.2.1). In particular, tilting all indices gives this alternate version of (E.2.11) A = Σ ijk... αijk... (bi⊗bj⊗bk...) αijk... = Aabc... (bi)a (bj)b (bk)c... A = Σ ijk... Aijk... (ui⊗uj⊗uk...) A ijk... = covariant components of A in x-space A = Σ ijk... A'ijk... (ei⊗ej⊗ek...) A' ijk... = covariant components of A in x'-space Orthonormal basis ( E . 2 . 1 4 ) If the basis vectors bi happen to be orthonormal, as defined by bi• bj = δi,j then bi = bi because the dual basis is unique. In this case w' ij = bi• bj = δij so the coefficient αijk is unchanged if any or all indices are lowered, see (E.2.13). An example of orthonormal basis vectors arises if bi = e^i ≡ ei/|ei| = ei/h'i and e^n • e^m = δn,m . In this case we find from (7.18.1) that g'nm = en • em = e^n • e^m h'mh'n = δn,m h'mh'n = h'n2δn,m ( E . 2 . 1 5 ) Appendix E: Tensor Expansions 287 so x'-space has the diagonal metric tensor g' nm = h'n2δn,m . A specific example arises with polar coordinates as shown in Fig (3.4.3) where e^1 = θ^ and e^2 = r^. Since the dual basis is unique, one has bi = bi = e^i = e^i †. In this case (E.2.1) and (E.2.3) state, A = Σijk αijk (e^i⊗e^j⊗e^k) αijk(any up/down) = Aabc (e^i)a (e^j)b (e^k)c . (E.2.16) On the other hand, one can expand A as in (E.2.9), using ei = h'ie^i , A = Σ ijk A'ijk (ei⊗ej⊗ek) = Σijk [A'ijk h'i h'j h'k ](e^i⊗e^j⊗e^k) . (E.2.17) Comparison of the last two equations shows that α ijk(any up/down) = Aabc (e^i)a (e^j)b (e^k)c = A'ijk h'i h'j h'k // no implied sums (E.2.18) Expansions on the unit versions of the tangent base vectors e^i are discussed more in Section E.8 below. † Is it really true that bi = bi ⇒ e^i = e^i where the latter two vectors are defined as unit vectors? This will serve as a little check on our notation. Since g' ab = h'a2δa,b we know that g'ab = h'a-2δa,b. Then g'nn = h'n2 and g'nn = h'n-2 . From (7.18.1) one has |en| = g'nn = h'n en = g'ni ei = g'nn en = h'n-2 en |en| = g'nn = (1/ h' n) en = h'n2 en so e^n ≡ en/|en| = en h'n-1 = [h'n2en] h'n-1 = en h'n = en / |en| ≡ e^n ⇒ e^n = e^n (E.2.19) Thus one is free to move the e^n label up or down at will. Tensor density expansions If A is a tensor density of weight W, the ge neral rule is to make this replacement: A'ijk... → J WA'ijk... ( E . 2 . 2 0 ) As justification for this rule, start with a regular tensor transformation for A, A' ijk... = Ri i' Rj j' Rk k'..... Ai'j'k'... ( E . 2 . 2 1 ) The rule then gives J W A'ijk... = Ri i' Rj j' Rk k'..... Ai'j'k'... ( E . 2 . 2 2 ) or A' ijk... = J-W Ri i' Rj j' Rk k'..... Ai'j'k'... ( E . 2 . 2 3 ) Appendix E: Tensor Expansions 288 which is the correct form for the transformation of a te nsor density of weight W as in the example (D.1.4). Using this rule, the expansion (E.2.9) of tensor density A would be written, A = J W Σijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... = contravariant components of A in x'-space ( E . 2 . 2 4 ) a result that is verified in (D.2.11). Taking components, A abc... = J W Σijk... A'ijk... (ei⊗ej⊗ek...)abc.. = J W Σijk... A'ijk... (ei)a (ej)b (ek)c ... = J W Σijk... A'ijk... Ria Rjb Rkc... // see (7.18.1) = J W Ria Rjb Rkc ...A'ijk ( E . 2 . 2 5 ) which is just the inverse of (E.2.23). Applied to a vector A of weight W, the expansion (E.2.24) becomes Aa = J WA'iei . ( E . 2 . 2 6 ) E.3 Polyadic Notation Some fields of study historically use "polyadic notation" as follows ( ABC ...) ≡ A⊗B⊗C ... (E.3.1) where the direct product notation was discussed above in Section E.1. It is sometimes a bit disturbing to modern readers to see bolded vectors stacked directly against each other, but the direct product makes the meaning clear. For arbitrary basis vectors, one would then have, for example, ( bibjbk...) ≡ bi⊗bj⊗bk ... . (E.3.2) Sometimes this basis vector notation is compressed even more, to wit, i j k ... ≡ (bibjbk...) ≡ bi⊗bj⊗bk ... ( E . 3 . 3 ) although this notation seems to be mostly used when the bi are the unit vectors ui. In all these notations, one must be aware that the symbols generally do not "commute". For example i j = ( bibj) = bi⊗bj ⇒ ( i j )nm = (bibj)nm = (bi⊗bj)nm = (bi)n (bj)m (E.3.4) ( j i )nm = (bjbi)nm = (bj⊗bi)nm = (bj)n (bi)m ≠ (i j )nm and therefore one cannot in general write i j = j i. The general expansion (E.2.1) now appears as Appendix E: Tensor Expansions 289 A = Σijk... αijk... (bi⊗bj⊗bk...) = Σijk... αijk... (bibjbk...) = Σijk... αijk... (i j k...) ( E . 3 . 5 ) where α ijk... = A • (bi⊗bj⊗bk...) = A • (bibjbk...) = A • (id jd kd ... ) = Aabc... (bi)a (bj)b (bk)c... (E.3.6) where we have just made up a notation id to stand for the dual vector bi. One can find further discussion of polyadic notation for example in Backus. E.4 Dyadic Products When two vectors A and B are combined in polyadic notation, the result is called a dyadic product ( AB) [ also known as a dyad or just a dyadic ] ( AB)ij ≡ AiBj // = ( A⊗B)ij = component of the matrix A⊗B = AB . (E.4.1) In this notation, the expansion (E.3.5) for a rank-2 tensor becomes A = Σ ij αij (bibj) αij = Aab(bi)a (bj)b = Aab (bibj)ab . (E.4.2) Notice from (7.1.1) that the dyadic product ( AB) is a rank-2 tensor if we assume that the underlying Ai and Bi are the x-space contravariant components of tensorial vectors A and B (which we normally assume). As a reminder, x-space need not be Cart esian. In Section E.7 it will be shown that the matrix (AB)ij can be associated with an operator (AB) in the un basis so ( AB)ij = <ui |(AB)| uj >, but this interpretation is not necessary for what follows. If vector A is replaced by the gradient operator ∇, one gets (∇B)ij ≡ ∇iBj = ∂iBj , // dyadic notation (E.4.3) but it is more common to define the operator ( ∇B) using a reverse dyadic notation (∇B)ij ≡ ∇jBi = ∂jBi = ∂Bi/∂xj // reverse dyadic notation (E.4.4) since the index order i,j on the far right matches the order on the left. Appendix E: Tensor Expansions 290 E.5 Matrix notation for dyadics (Cartesian Space) In Cartesian space g = 1 so for any vector a, we have ai = ai. If we think of b as a column vector with components b j, then the corresponding row vector bT contains those same elements b j, so we set (bT)j = bj. Therefore one can express the dyadic product in this more down-to-earth manner, ( ab)ij ≡ aibj = ai (bT)j = ( abT)ij ( E . 5 . 1 ) or ab = abT . (E.5.2) Here one knows that ab is a "dyadic" because there is no other meaning for two bolded column vectors abutting each other with no intervening operator, so no special notation like [ ab] is needed to indicate that ab is a dyadic. The object abT on the other hand has a well-defined meaning in matrix algebra, abT = ⎝⎛ ⎠⎞ a1 a2 (b1 b2) = ⎝⎛ ⎠⎞ a1b1 a1b2 a2b1 a2b2 = a matrix // same as matrix ab = a⊗b (E.5.3) and one sees that in fact ( ab)ij = (abT)ij = aibT j = aibj . (E.5.4) Meanwhile, the object aTb is just a number, aTb = (a1 a2) ⎝⎛ ⎠⎞ b1 b2 = a1b1 + a2b2 = a • b . // g = 1 . (E.5.5) This transpose notation can then be applied to the dyadic expansion of a 2x2 matrix A, A = Σij αij bibj = Σij αij bibjT = α11 b1 b1T + α12 b1 b2T ... . (E.5.6) In the special case that the bi are the unit vectors ui, and assuming N = 2 dimensions, one has A = Σnm Anm unum = Σnm Anm unumT = A11 u1 u1T + A12 u1 u2T + A21 u2 u1T + A22 u2 u2T = a matrix with A 12 in the upper right corner (E.5.7) where un is a column unit vector and unT is the corresponding row unit vector. For example, u1u2T= ⎝⎛ ⎠⎞ 1 0 ( 0 1) = ⎝⎛ ⎠⎞ 0 1 0 0 . (E.5.8) Obviously this matrix visualization is valid for any di mension N, not just N=2. For rank n > 2, however, this transpose-of-vector concept does not conveniently generalize. For n=3 the object uaubuc would be a Appendix E: Tensor Expansions 291 cube of zeros with a single 1 located at coordinates a,b,c, and so on for n > 3. One cannot write this as uaubucT for example. In associating of M = abT with a rank-2 tensor, we must limit our interest to Cartesian space where g = 1. That is because abT is the single matrix shown above in (E.5.5). If g ≠ 1, then there are four different matrices Mij, Mi j, Mij and Mij and the notation M = abT cannot support this fact. When g = 1 all four matrices are th e same matrix, and the x-space dot product is as shown in (E.5.5). g' ≠ 1 is still allowed. Ambiguity of ab: There is a certain ambiguity in (E.5.2) that ab = abT. Writing A = ab = abT, we imply that " ab" is the name of a matrix, an alternate to the name A. This matrix A = ab has matrix elements A ij = (ab)ij = aibj. On the other hand, in (E.3.1) we might say that B = ab = a⊗b which is a vector in a tensor product space. So this version of the object ab is not a matrix, it is a tensor product space vector. So A and B are different types of objects both indicated by the notation ab. However, the tensor components of the rank-2 tensor B are the same as the matrix elements of the matrix A. That is because (B) ij = [a⊗b]ij = aibj , being an outer product of two vectors. In Dirac nota tion described below in (E.7.4) we would write the objects A and B in this manner which stresses the distinction, A = | a><b| = the name of a matrix abT (or the corresponding operator in the Dirac Hilbert space) B = | a> ⊗ |b> = a vector in a tensor product space. (E.5.9) The matrix elements and tensor components (in the ui basis) would be A ij = <ui| A | uj> = < ui| a><b| uj> = aibj Bij = [< ui| ⊗ <uj| ] [ | a> ⊗ |b> ] = < ui|a><uj|b> = aibj . (E.5.10) E.6 Large and small dots used with dyadics (Cartesian Space) Sometimes a small-size dot • is used to indicate th e action of a dyadic (matrix) on a vector. If A is a dyadic (same symbol for matrix), and if c and d are vectors, then one defines: A • c ≡ Ac = a column vector ⇒ (A • c)i = (A c)i = Aijcj ⇒ A • c = ΣijAijcj ui c • A ≡ cTA = a row vector ⇒ (c • A)i = (cTA)i = cjAji ⇒ c • A = ΣijcjAji ui d • A • c = dTAc = a number = ΣijdiAijcj . (E.6.1) It then follows that, for the particular dyadic A = ab , ( ab) • c ≡ (ab) c = ( abT)c = a(bTc) = a (b • c) = (b • c) a = a column vector c • (ab) ≡ cT (ab) = cT(abT) = ( cTa) bT = ( c • a) bT = a row vector d • (ab) • c = dT (ab) c = dTabT c = (dTa)( bT c) = (d • a)(b • c) = a number . (E.6.2) Appendix E: Tensor Expansions 292 Here is more detail on the first line of the above group showing a skeletal matrix structure, ( ab) c = ( a bT)c = abTc = a(bTc) = a(b•c) { ⎝⎛ ⎠⎞ x x x x } ⎝⎛ ⎠⎞c1 c2 = {⎝⎛ ⎠⎞ a1 a2 (b1 b2)} ⎝⎛ ⎠⎞c1 c2 = ⎝⎛ ⎠⎞ a1 a2 (b1 b2) ⎝⎛ ⎠⎞c1 c2 = ⎝⎛ ⎠⎞ a1 a2 { (b1 b2) ⎝⎛ ⎠⎞c1 c2} = ⎝⎛ ⎠⎞ a1 a2 b•c (E.6.3) The same small dot is used to indicate the product of two dyadics, which is to say, matrix multiplication A•B ≡ A B . ( E . 6 . 4 ) Regarding this small size dot • : (1) from a matrix algebra point of view, it is completely superfluous except in the case c • A ≡ cTA ; (2) it is completely different from the dot • used in bTc = b•c. The next Section provides an explanation of the small dot as part of an operator interpretation for dyadics. E.7 Operators and Matrices for Ra nk-2 tensors: the bra-ket notation (Cartesian Space) We continue with the assumption that x-space is Cart esian, g = 1. The discussion below applies only to rank-2 tensors. One of our goals is to make contact with the dyadic discussion of Morse and Feshbach, one of the few places dyadic notation appears in a general mathematical physics book. Operator concept. As discussed in Section 5.10, x-space and x'-space of Picture A are both N-dimensional real Hilbert Spaces, with a scalar product indicated by the large dot •, and one can regard V as a vector in either space. The first line of (E.2.10) with A replaced by V states this expansion, V = Σiαi bi αi = bi • V // arbitrary basis (E.2.10) which we now rewrite, renaming the coefficients, V = Σi[V(b)]i bi [V(b)]i = bi • V (E.7.1) Moreover, one can regard a rank-2 tens or A as an "operator" in this Hilbert space, using notation (E.5.6), A = Σij [A(b)]ij bibjT . ( E . 7 . 2 ) In (E.2.10) and (E.5.6) the coefficients in these two expansions were called α i and αij, so here we are providing more descriptive names for these coefficients. Application of (bn)T on the left of (E.7.2) and bm on the right, and then a double use of (bn)Tbi = bn • bi = δn,i (see 7.18.6) gives, [A(b)]nm = (bn)T A bm . (E.7.3) Appendix E: Tensor Expansions 293 Here, one regards A as an operator in the x Hilbert space, whereas [A(b)]nm is a "matrix" which is associated with the operator A in the particular bn basis. If bn = un, then [A(u)]nm = Anm which is a particular matrix for this particular un basis. More generally, [A(b)]nm = (bT)n A bm = [(bT)n]i Aij [bm]j = [bm]i Aij [bm]j which is in general a completely different ma trix formed by linearly combining elements Aij. Bra-ket Notation. For the author of this document, the bra-ket notation commonly used in quantum mechanics (Paul Dirac 1939) provides a useful way to look at a rank-2 tensor A as an operator. It is true that in quantum mechanics one us ually deals with infinite-dimen sional Hilbert spaces and complex numbers, but the formalism applies just as well to r eal Hilbert spaces with finite dimensions (used for example to study "spin"). Here is how the bra-ket notation works: bi → |bi> // vector biT → <b i| // transpose vector bi → |bi> // vector ( bi)T → <bi| // transpose vector abT (E.5.3) → |a><b| // a matrix (outer product) aTb = a • b (E.5.5) → <a | b> // a number (inner product) a • b = b • a (7.4.14) → <a | b> = <b | a> for any a,b // real Hilbert Space V = Σi [V(b)]i bi (E.7.1) → |V> = Σi [V(b)]i |bi> // vector expansion ... [V(b)]i = bi • V (E.7.1) → [V(b)]i = <bi|V> // and coefficients bi • bj = δi j = δi,j (7.18.3) → <bi|bj> = δi,j // orthogonality = < b j|bi> = <bi|bj> = <bj|bi> Σi bi biT = 1 (7.18.6) † → Σi |bi><bi| = 1 = Σi |bi><bi| // completeness A = Σ ij [A(b)]ij bibjT (E.7.2) → A = Σ ij [A(b)]ij | bi> <bj| // tensor expansion ... [A (b)]ij = (bi)T A bj → [A(b)]ij = <bi | A | bj > // and coefficients bi• (Abj) = bj • (AT bi) (7.9.17) → <bi | A | bj > = <bj | AT | bi > // covariant transpose ( E . 7 . 4 ) † What appears in (7.18.6) for completeness is ( bn)i(bn)j = δij, but since g = 1 we can write this with indices down as ( bn)i(bn)j = δi,j which in matrix notation says Σi bi biT = 1 . Appendix E: Tensor Expansions 294 In this bra-ket notation, the N |b i> are a set of basis vectors which span an N-dimensional real Hilbert Space, while <b i| span the so-called adjoint (o r transpose in our case) Hilbert Space. One then refers to [A(b)]ij = <bi | A | bj > as "the matrix element of the operator A in the bi basis ", since the expansion (E.7.2) was A = Σij [A(b)]ij bibj = Σij [A(b)]ij bi⊗bj which is an expansion in the bi basis. In general, |b i> and |bi> are different vectors because bi and bi are different. For example en = g'nm em and our restriction to g = 1 certainly allows g' ≠ 1. [ Do not confuse with ( bi)a = (bi)a being the same; here a is a tensor index raised and lowered by g = 1, whereas i is a basis vector label and these are raised and lowered by w' as shown in (7.18.6).] In this notation, based on what was presented earlier, one can write, A nm = <un | A | um > = the x-space components of tensor A (basis un) raise/lower with g A'nm = <en | A | em > = the x'-space components of tensor A (basis en) raise/lower with g' [A(b)]nm = <bn | A | bm > = the matrix of A in the bn basis raise/lower with w ( E . 7 . 5 ) In the first of these three lines, one can raise and lower indices with gab and gab on both sides of the equation, but we are taking g = 1 so up and down here do not matter. On the second line this can be done with g' ab and g'ab. Eq (7.18.6) shows that bn = w'nm bm and conversely bn = w'nmbm where w' nm is the metric tensor g' nm one would get for some underlying transformation F b which causes bn to be its tangent base vectors en. So, on the third line above we can raise and lower indices on each side with wab and wab where w' nm ≡ bn • bm as in (7.18.6). Notice in the three equations of (E.7.5) that the operator A between the vertical bars is the exact same operator in each case. The matrices are different not b ecause the operator has changed, but because the basis vectors are different. In a more consistent notation one might write A nm = [A(u)]nm and A'nm = [A(e)]nm . Bases are related by a transformation. Consider again, [A(b)]nm = <bn | A | bm > = ( bn)T A bm = [bn]i Aij [bm]j = <bn|ui><ui|A|uj><uj|bm> . (E.7.6) We lower index m on both sides (using w' ab as noted above) and reverse the j tilt to get [A (b)]n m = <bn | A | bm > = ( bn)T A bm = [bn]i Ai j [bm]j = <bn|ui><ui|A|uj><uj|bm> . (E.7.7) One could then define the following tensor-like object, B n i ≡ [bn]i . ( E . 7 . 8 ) The first index on B is raised and lowered by w', while the second is raised and lowered by g, so this object is a bit like R and S in its non-tensor nature. Lowering n and raising i then gives Appendix E: Tensor Expansions 295 Bni = [bn]i = (BT)i n , (E.7.9) where we use the covariant transpose of a tilted matrix described in (7.9.3). One then has [A (b)]n m = Bn i Ai j(BT)j m . ( E . 7 . 1 0 ) Since all the matrices are tilted the same way and summed indices are contractions, this is one of the "legal" Standard Notation matrix multiplicati on forms like (7.8.6) and we then write, A (b) = BABT or more precisely [A(b) = BABT ]SN,dt (E.7.11) where SN,dt means Standard Notation, down-tilt, as described below (7.8.6). The matrix equation A(b) = BABT shows that the [A(b)]n m are related to the Ai j by a "congruence transformation" with a matrix B ni = [bn]i whose rows are the basis vectors bn. When bm = um , matrix B is the identity matrix, and when bm = em one has B ni = [en]i = Rni, so that B = R in this case. In Standard Notation the general R matrix is "covariant real orthogonal", [RRT = 1]SN,dt and [RT = R-1]SN,dt (see (7.9.3) and following text) so in fact one has for the bm = em basis, A (e) = B A BT = R A RT = R A R-1 = R A S . (E.7.12) Specifically in this case, [A (e)]n m = Rn iAi jSj m = Rn iRmjAi j = A'n m . (E.7.13) which is the expected result looking at the second line of (7.5.8). Comment : Recall that in Developmental Notation one has RRT = 1 and RT = R-1 only when R is a rotation, whereas in Standard Notation one has RRT = 1 and RT = R-1 for any R matrix, see (7.9.3). We used this fact above to show that a basis change from basis un to bn on operator A can be thought of as a congruence transformation by a matrix B whose rows are the vectors bn. This is a standard concept in linear algebra where one operates in Cartesian x-space. More on bra-ket notation and its relation to the small dyadic dot. Consider the following facts, where AT is the covariant transpose as in (7.9.3) and (7.9.17), <d | A | c > = dT A c = dT [A c ] = <d |Ac > <d | A | c > = dT A c = [ dT A] c = [AT d]T c = <ATd | c> (E.7.14) where |(Ac) > = a new Hilbert space vector which results when operator A is applied to |c> = A|c> <(ATd) | = a new transpose Hilbert space vector which results when A is applied to <d| = <d|A . Appendix E: Tensor Expansions 296 ( E . 7 . 1 5 ) Detail : In covariant notation [ dTA] = [AT d]T because [ dTA]i = ([AT d]T)i. To verify this, [ dTA]i = (dT)jAji = djAji = Ajidj ([AT d]T)i = [AT d]i = (AT)i jdj = Ajidj So one has this general idea that <d | A | c > = <d |Ac > = <ATd | c> A | c > = |(Ac)> , <d | A = <(A Td) | . (E.7.16) In this last line, the isolated A's are the same ope rator A sitting in the Hilbert space. This operator can "act" either to the right or to the left as shown. The object |(Ac)> ≡ |e> is some different vector in the Hilbert space (different from |c>), call it |e>, and the grouping (Ac) labels this vector. Similarly, <(ATd) | is some vector <f| in the transpose Hilbert space. The distinction between A as an abstract operator in the Hilbert space, and the A in (Ac) and (ATd) = (dTA)T as vectors in the Hilbert space is a subtle one. It is just this distinction that is implied by the small dot in the dyadic notation discussed in the previous section, and here is the correspondence between the dyadic notation and the bra-ket notation: A • c = A c d • A = (ATd)T = dTA d • A • c = dTAc A • B c A | c > = |Ac> <d | A = <(ATd)| <d | A | c > = <d | Ac > AB| c > . ( E . 7 . 1 7 ) In the rightmost column operator B is applied first to |c> to get vector |(Bc)>, and then operator A is applied to |(Bc)> to give yet another vector | (ABc)>. In bra-ket notation the product of two abstract operators is given just as AB, but in dyadic notation it is written A • B. Dyadics as operators. According to the above discussion, one can regard a dyadic ( AB), being a rank-2 tensor, as an operator and not as a matrix. The matrix Tnm = (AB)nm = AnBm is specific to the un basis in x-space, T nm = (AB)nm = <un |(AB)| um > = ( un)T A BT um = [ ( un)T]a Aa (BT)b [um]b = δn a Aa Bb δm b = AnBm . (E.7.18) In the generic bn basis one has [( AB)(b)]nm = <bn |(AB)| bm > . ( E . 7 . 1 9 ) It is to emphasize this operator view of a dyad ic that Morse and Feshb ach use fancy letters like U to represent dyadics. Then their small-dot notation U • B emphasizes the idea of an operator acting on a vector, equivalent to U| B>. Here are a few samples from their S ection 1.6 on dyadics. Under each clip we Appendix E: Tensor Expansions 297 have tried to relate the stated e quation(s) to our notation above. Like us, Morse and Feshbach are working in a Cartesian x-space, so up and down indices are the same for x-space objects. Note that un = un when g = 1. The M&F symbols U and an are our A and un : p 55 A • b = Σmn um Amn bn b • A = Σmn bm Amn un (E.6.1) or A |b> = Σmn |um><um|A|un><un|b> <b|A = Σmn<b|um><um|A|un><un| (E.7.20) p 55 A = Σnm unAnm um (E.5.7) ( E . 7 . 2 1 ) A-1 • A = A • A-1 = 1 // A-1 defined if matrix [A(u)]mn = Amn is invertible (E.7.22) p 57 0 • f = 0 1 • f = f ; 1 = Σn ununT (E.7.4) with bn = un or 0 | f> = |0> ; 1 | f> = | f> ; 1 = Σn | un><un| (E.7.4) (E.7.23) p 59 Us = Σnm [Us]nm unum = [Us]11u1u1 + [Us]12u1u2 + ... = [U s]11 i i + [Us]12i j + .. where i = u1, j = u2 , k = u3 and where [U s]ij = ⎝⎜⎜⎛ ⎠⎟⎟⎞ Ax Bz By Bz Ay Bx By Bx Az = a s ymmetric matrix (E.7.24) Notice the impressive name "idemfactor" for the identity operator 1 = Σi | ai><ai| = Σi aiaiT = Σiaiai where Σiaiai uses the dyadic notation (E.5.2) . Comment : A favorite bra-ket notation trick is to obtain useful results by inserting 1 = Σn | un><un | in opportune places, as in (E.7.20) above : Appendix E: Tensor Expansions 298 A |b> = 1 A 1 |b> = [ Σm | um><um | ] A [Σn | un><un | ] |b> = ΣnΣm | um><um |A| un><un |b> = ΣnΣm | um> Amn bn = Σnm Amn bn |um> ( E . 7 . 2 5 ) The vector A |b> = |Ab> has thus been expanded on the basis | u m>. E.8 Expansions of tensors on unit tangent base vectors: M and N We start with the general Picture A (and late r specialize to orthogonal coordinates), (E.8.1) In (E.2.9) it was established that one can expand a tensor A on the tangent base vectors en as A = Σijk... A' ijk... (ei⊗ej⊗ek...) A' ijk... = contravariant components of A in x'-space A' ijk... = Ri i'Rj j'Rk k'...... A i'j'k'... . // tensor transformation rule (E.8.2) We know from (7.13.6) that en = |en| e^n = h'n e^n so the above expansion fo r tensor A can be written A = Σ ijk... { h'ih'jh'k......A' ijk...} (e^i⊗e^j⊗e^k...) ≡ Σ ijk... [A(e^)]ijk... (e^i⊗e^j⊗e^k...) , (E.8.3) where the unit-vector expansion coefficients are given by [A (e^)]ijk... = { h'ih'jh'k......A' ijk...} // ≡ A' ijk... see (E.8.5) = h' ih'jh'k...... Ri i'Rj j'Rk k'...... A i'j'k'... = (h' i Ri i')( h'j Rj j')( h'k Rk k') ..... A i'j'k'... (E.8.4) Note that we have not at this point assumed that the { e^i} unit vectors are orthonormal, so g' can be non- diagonal. Appendix E: Tensor Expansions 299 Coefficient notation In a curvilinear coordinates application of these expa nsions, the expansion coefficients are usually written in the following manner, [A (e^)]ijk... = A' ijk... = Ax'ix'jx'k ...... ( E . 8 . 5 ) where the x' n are the names of the coordinates. For example, for a rank-4 tensor in spherical coordinates with coordinates x' 1 = r, x'2 = x'3 = θ and x'3 = φ one might write [A(e^)]2213 = A'2213 = Aθθrφ . (E.8.6) The script notation is the same shorthand we used in (7.13.12) where V = V'n en = V'n e^n . Note that there is a prime on V'n and a prime on A' ijk... above. The reason for this prime is that these quantities are associated with the curvilinear coordinates (variables) which in Picture A and B are called x 'i . For example we have V'n = h'n V'n . In the Moon & Spencer Picture of (14.11) with u-space (metric tensor g) on the left, the curvilinear coordinates are called ui. Since they then have no prime, we would use unprimed scripted variables. For example, Vn = hn Vn . This is the convention of Chapter 14, but in these Appendices we are using Pictures A and B. As shown below, one may interpret [A(e^)]ijk... = A'ijk... as the contravariant components of tensor A with respect to a certain transformation F M, and in this sense the up and down index position is significant. However, when we later assume the e^i are orthonormal, that forces g' ij to be diagonal and forces the condition e^i = e^i as shown below (E.2.14). Then the up or down index position on [A(e^)]ijk... makes no difference in expansion (E.8.3). In the spherical coordinates example where the e^i are orthonormal, we would normally write the above coefficient as A θθrφ. Matrices M and N It is convenient now to define some modified R matrices as M a b ≡ h'a Ra b ( E . 8 . 7 ) so that (E.8.4) becomes [A (e^)]ijk... = Mi i'Mj j'Mk k'...... A i'j'k'... . (E.8.8) As shown below, these matrices M are the R-matrices of a transformation called F H that takes x-space to a new x"-space in which the tangent base vectors are the unit vectors e^n, but we defer that interpretation to first get some facts laid out. Appendix E: Tensor Expansions 300 Defining Na b to be the inverse of Ma b, one has (verified just below), Na b ≡ h'b-1Sa b = h'b-1Rba ( E . 8 . 9 ) so the inversion of (E.8.8) is given by Aijk... = Ni i'Nj j'Nk k'...... [A(e^)]i'j'k'... . (E.8.10) To verify that this N is the correct inve rse of M, use (E.8.7) and (E.8.9) to show Ma kNk c = (h'a Ra k)( h'c-1Rck) = (h'a/ h'c) Ra k Rck = (h'a/ h'c)δa c = δa c (E.8.11) making use of orthogonality rule #3 of (7.6.4), R a k Rck = δa c . M and N can be written in terms of the en and en vectors as follows, M n i ≡ h'n Rn i = h'n(en)i // (7.13.1) Ni n = h'n-1Rni = h'n-1(en)i = ( e^n)i , // (7.13.1) (E.8.12) which says that N i n = { e^1 , e^2, .... } . (E.8.13) Thus the columns of Ni n are the unit tangent base vectors. Matrix H and x"-space In the discussion above one has x-space with basis vectors un and x'-space with basis vectors en (the tangent base vectors). It is useful then to defi ne x"-space as the space whose basis vectors are the e^n unit vectors, which are generally not orthogonal. The relation (E.8.7) Ma b ≡ h'a Ra b can be written as a down- tilt matrix equation, M = HR where H i j ≡ diag(h'1, h'2.....) . (E.8.14) It then follow that N = M -1 = R-1H-1 = SH-1 . ( E . 8 . 1 5 ) Finally, note that x = xiui = x'iei = x'i(h'ie^i) = x"ie^i ⇒ x"i = h'i x'i so x" = H x' . ( E . 8 . 1 6 ) Appendix E: Tensor Expansions 301 This shows that the transformation FH from x'-space to x"-space is linear with matrix H, x" = FH(x') = H x' . ( E . 8 . 1 7 ) Eq. (7.5.2) then says that the corresponding R-matrix is (R H)i k ≡ (∂x"i/∂x'k) = Hi k so RH = H and then SH = RH-1 = H-1. We can now show all three spaces in the same Picture as follows, (E.8.18) This picture shows that the transformation directly from x-space to x"-space is FM(x) = H F(x) . ( E . 8 . 1 9 ) Here F(x) is a (generally) non-linear transformation assumed to connect x'-space to x-space. This is then concatenated with linear transformation H to get non-linear transformation FM(x). We can now write A(e^) as A" and restate (E.8.3) and (E.8.8) above as A = Σijk... A" ijk... (e^i⊗e^j⊗e^k...) // rank-n tensor expanded on e^i⊗e^j⊗e^k... A " ijk... = Mi i'Mj j'Mk k'...... A i'j'k'... // rank-n tensor transformation Aijk... = Ni i'Nj j'Nk k'...... A" i'j'k'... // inverse of the above A " i = Mi j Aj // rank-1 tensor A " ij = Mi i'Mj j'A i'j' // rank-2 tensor where A" ijk... ≡ [A(e^)]ijk... ≡ A' ijk... (E.8.20) The word "tensor" suddenly has a new meaning in the above equations. The equations indicate objects being tensors with respect to this non-linear transformation FM(x) whose linearized-at-a-point matrix is R M = M = HR, where R is the linearized -at-a-point matrix version of F(x). H is the diagonal matrix of scale Appendix E: Tensor Expansions 302 factors h' i which are associated with g' ij in x'-space. The Aijk... are contravariant components of tensor A in x-space, while A"ijk... are the corresponding contravariant components of A in x"-space, all with respect to FM(x) and its matrix R M = M where, for example, d x" = M d x. One can compare the last line of (E .8.20) to the first line of the generic (7.5.8) showing how a rank-2 tensor transforms. The second line of (7.5.8) then tells us the manner in which the mixed tensor Aa b transforms under F M : A" a b = Ma a' Mbb' Aa' b' = Ma a'Aa' b'Mbb' . (E.8.21) If we use the Standard Notation cova riant transpose shown in (7.9.3) [R M = M], then M bb' = (MT)b' b and the above becomes A" a b = Ma a'Aa' b' (MT)b' b = [ MAMT ] a b from which we conclude that A" = MAM T // that is to say, [ A" = MAMT]SN,dt (E.8.22) Comment: This matrix result is a special case of the generic result (E.7.11) that A(b) = BABT when bn = e^n, but it is not obvious how this works out so here are some details. In order to compute Bn i ≡ [bn]i in (E.7.8) one has to know bn. One sees from (7.18.6) that bn = w'ni bi where w'nm = bn • bm = e^n • e^m = (h'nh'm)-1 en • em = (h'nh'm)-1g'nm ⇒ w'nm = (h'nh'm)g'nm . // since w nm and wnm are inverses (E.8.23) Therefore, bn = w'nm bm = (h'nh'm)g'nm e^m = h'ng'nm em = h'n en ⇒ Bn i ≡ [bn]i = h'n (en)i (E.8.24) But (E.8.12) says Mn i = h'n(en)i so we have shown that Bn i = Mn i so BABT = MAMT. The metric tensor g" in x"-space appearing in Fig (E.8.18) is this, g" ab = e"a• e"b = e^a • e^b = (h'ah'b)-1 ea • eb = (h'ah'b)-1 g'ab = w'ab , (E.8.25) so g" = w' in the Comment above. Appendix E: Tensor Expansions 303 At this point the x and x' spaces are completely general, and none of R, R H = H, RM = M is a rotation matrix. In the next Section, we shall specialize the above picture so that x-space is Cartesian with g = 1, and x'-space has orthogonal curvilinear coordinates x' which means g' is a diagonal matrix. In this scenario, F(x) is in general non-linear and so then is FM(x) = H F(x). Since the e^i now form a frame of orthonormal vectors, and since ui also form such a frame, one will not be surprised to find that M is now a rotation which relates these two frame sets. E.9 Application of Section E.8 to Orthogonal Curvilinear Coordinates We now switch to Picture B (g=1) and assume that the x' i are orthogonal coordinates, (E.9.1) and our three-frame picture (E.8.18) above then becomes (E.9.2) In this situation, x-space is Cartesian with g ab = gab = δa,b g' ab = h'a2 δa,b g'ab = h'a-2 δa,b. g" ab = (h'ah'b)-1 g'ab = (h'ah'b)-1h'a2 δa,b = δa,b // from (E.8.25) . (E.9.3) Notice that g" ab = g"( x)ab = e^a(x) • e^b(x) = δab so x"-space is locally Cartesian at any point x. For example, in polar coordinates one has r^ • θ^ = 0 at any location x. Since g = 1 and g" = 1, the up or down index position on a tensor object is not significant. Now, regardless of what R and S ar e, M is a "rotation", where we include in this term possible axis reflections. To show this fact, firs t note from (E.8.7) and (7.5.9) that Appendix E: Tensor Expansions 304 Mj k = (h'j Rj k) = (h'j g'jaRak) = (h'j [δja h'j-2]Rak) = (h'j-1Rjk) = (h'j-1Rjk) . (E.9.4) Using orthogonality rule #3 that R a b Rcb = δa c = δa,c it then follows that, Mi kMj k = (h'i Ri k) (h'j-1Rjk) = h'ih'j-1 ( Ri kRjk) = h'ih'j-1 δi,j = δi,j . (E.9.5) But Mi kMj k = δi,j is the Standard Notation statement that M is a "rotation", as was shown in (7.9.11). In developmental notation this appears as [MMT = 1]DN, as shown there. Since M is a rotation, N must be the inverse rotation since N = M-1. From (7.9.11) we then also know that M i k = Mik ( E . 9 . 6 ) which is the alternate form for Mi k being a rotation, Again, M is the R-matrix R M = M for F M. Relation between M and N. Looking at (E.9.5), Ma b Mc b = δa,c (E.9.5) and knowing that M a b (M-1)b c = δa c = δa,c ( E . 9 . 7 ) one concludes that (M-1)b c = Mc b . But (M-1)b c = Nb c so Nb c = Mc b // note: this does not say that N = MT in standard notation, see (7.9.3) (E.9.8) which can be verified from the above expressions for N and M. Since (E.9.6) must also be true for rotation N, we find that M ik = Mi k = Nk i = Nki // g = 1, g' = diagonal (E.9.9) Interpretation of N and M. Since en = S un ( this is (3.2.4) and (3.2.1) with e'n = un) and since en = h'ne^n , it follows that e^n = h'n-1en = h'n-1 S un or ( e^n)a = h'n-1 Sa b (un)b = h'n-1 Sa b δnb = h'n-1 Sa n = Na n (E.8.9) = Na bδb n = Na b(un)b or e^n = N un and ( e^n)a = Na n . // ⇒ un = M e^n (E.9.10) Since the un are the Cartesian unit vectors, it seems intuitiv ely obvious that the transformation that moves this frame of orthonormal unit vectors { un} into the orthonormal frame { e^n} must be a "rotation". Appendix E: Tensor Expansions 305 It was shown in (E.9.8) that Na n = Mn a , therefore Mn a = Na n = (e^n)a . ( E . 9 . 1 1 ) The rotation matrix N a n = (e^n)a has the orthonormal basis vectors e^n as its columns, while the rotation matrix Mn a = (e^n)a has the e^n as its rows. Recall from (E.2.15) and (E.2.19) that orthonormal e^n implies both e^n = e^n and g'nm = h'n2δn,m = diagonal. The relation un = M e^n can be written un = M( x) e^n(x) to emphasize that the rotation M( x) = RM(x) is really a different rotation at every point x, since the e^n(x) vary with x. This is very different from a global rotation which is the same at all points. For a global rotation R, F = R = linear and ∂iuj is a tensor. For R M being a rotation which varies from point to point, F M is non-linear just as F defining the curvilinear coordinates is non-linear, and ∂iuj fails to be a tensor under either F or F M. Matrix form of rank-2 tensor transformation: One implication of the above picture relates to tensor equations being covariant, as discussed in Section 7.15. If one has a tensor fi eld equation in x-space, Q ad c(x) = Hab(x)Tb c(x) Bd(x) , ( E . 9 . 1 2 a ) in which all the objects transform as tensors with respect to the underlying x" = FM(x) (and its linear approximation M( x) as in d x" = M(x) d x), then the equation is covariant and takes the same form in x"- space, Q"ad c(x") = H"ab(x")T"b c(x") B"d(x") . x" = FM(x) (E.9.12b) Using our script notation explained in (E.8.5) and (E.8.20), we rewrite the x"-space equation above as the third line below (primed script tensor names). The four th line is then appropriate to the Moon and Spencer Picture (14.1.1) where the curvilinear coordinates ui have no primes : Q ad c = HabTb c Bd x-space (general), Picture A or M&S (1.11) Q'ad c = H'abT'b c B'd x'-space (general), Picture A or B (1.11) Q'ad c = H'abT'b c B'd x"-space (orthogonal) , Picture A or B (1.11) Qad c = HabTb c Bd x"-space (orthogonal) , Picture M&S (14.1.1) (E.9.13) Comment : An x-space observer has axes un (Frame S) while an x"-space observer has axes e^n(x) (Frame S"), and these two sets of observation axes are related by un = M( x) e^n(x) where M( x) is a rotation. If the first equation describes something at location x in the realm of Newtonian mechanics, we expect the equation to have the same form in both Frame S and Frame S" which are related by this local rotation M(x). In other words, rotations are an invariance of Newtonian mechanics, and this means equations are covariant with respect to rotations. The above exampl e, which might apply to fluid dynamics, has this Appendix E: Tensor Expansions 306 covariance at each point x in the fluid, and it may happen that the rotation is a different rotation at different points x, but it is always a rotation. A simple interpretation of (E.9.13) is that the mutually orthogonal unit base vectors e^n form a rotated Cartesian frame of reference, so the x-space equations must have the same form in this rotated frame. This is why the first and last lines of (E.9.13) have the same form. Example: Polar Coordinates. In polar coordinates now with ordering r, θ = 1,2 one has S1 1 = (∂ x/∂r) = cosθ x = rcos θ S1 2 = (∂ x/∂θ) = -rsinθ y = r s i n θ S2 1 = (∂y/∂r) = sinθ S2 2 = (∂ y/∂θ) = rcosθ ( E . 9 . 1 4 ) Si j = ⎝⎛ ⎠⎞cosθ -rsinθ sinθ rcosθ Ri j = ⎝⎛ ⎠⎞cosθ sinθ -sinθ /r cosθ/r R = S-1 [g' = RRT]DN = ⎝⎛ ⎠⎞cosθ sinθ -sinθ /r cosθ/r ⎝⎛ ⎠⎞cosθ -sinθ /r sinθ cosθ/r = ⎝⎛ ⎠⎞ 1 0 0 1/r2 → g'ab = ⎝⎛ ⎠⎞ h'r-2 0 0 h'θ-2 so h r = 1 and h θ = r . The N and M matrices may be computed as follows: [ recall M = HR and N = SH-1 from (E.8.14,15) ] Ma b ≡ h'a Ra b = ⎝⎛ ⎠⎞ 1 0 0 r ⎝⎛ ⎠⎞cosθ sinθ -sinθ /r cosθ/r = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ = Rz(-θ) (E.9.15) Na b ≡ Sa b h'b-1 = ⎝⎛ ⎠⎞cosθ -rsinθ sinθ rcosθ ⎝⎛ ⎠⎞ 1 0 0 1/r = ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ = Rz(θ) (E.9.16) Therefore, the relation between a rank-2 tensor's e^n-expanded-form components [A(e^)]i j and the Cartesian form components Ai j is given (for polar coordinates) by (E.8.22) which says A" = A(e^) = A' = M A MT = M A MT ( E . 9 . 1 7 ) or ⎝⎛ ⎠⎞ Arr Arθ Aθr Aθθ = ⎝⎛ ⎠⎞ cosθ sinθ -sinθ cosθ ⎝⎛ ⎠⎞ A11 A12 A21 A22 ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ // verified in Lai p 316 Problem 5.71 where recall that up or down index position has no si gnificance in either x-space or x"-space and MT = MT as in (7.9.5). That is to say, one has A 12= A1 2 as well as A rθ = Ar θ, so one can think of the above as a down-tilt matrix equation. Appendix E: Tensor Expansions 307 Applications in Continuum Mechanics 1. In isotropic elastic stress analysis for states of plane stress and plane strain (see Lai p 251), the 3D Cartesian stress tensor T ij has a simple form in which the upper left four components can be represented as derivatives of a potential-like function called an Airy function φ, so that T 11 = ∂22φ, T22 = ∂12φ, and T12 = T21 = – ∂1∂2φ. In this case, equation (E.9.17) becomes ⎝⎛ ⎠⎞ Trr Trθ Tθr Tθθ = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ ⎝⎛ ⎠⎞∂22φ – ∂1∂2φ – ∂1∂2φ ∂12φ ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ // B = MAMT (E.9.18) where ∂ 1 = ∂/∂x1 = cosθ ∂r - (sinθ/r)∂θ ∂ 2 = ∂/∂x2 = sinθ ∂r + (cosθ/r)∂θ . Here is a Maple computation of (E.9.18): Appendix E: Tensor Expansions 308 // verified in Lai p 264 (5.27.3) (E.9.19) This then is a real-world example of using a rank-2 tensor in curvilinear coordinates expanded on the unit tangent base vectors. The mentioned plane of strain or stress has Cartesian coordinates x 1,x2 which are converted to polar coordinates r, θ. The third Cartesian coordinate x 3 is more or less ignored. 2. The relation between the stress tensor T ij and the infinitesimal strain tensor E ij for an isotropic material is stated in Cartesian coordinate x-space as ( a form of Hooke's Law generalizing F = -kx), T ij = λ tr(E)gij + 2μEij or the same thing: Tij = λ tr(E)gij + 2μEij where tr(E) = Σ i Eii = Eii with implied sum on i . (E.9.20) Here λ and μ are called Lamé's constants and g ij = gij = δi,j (see Lai p 208 (5.3.8)). Some components are (in the case that E 33 = 0), T11 = λ(E11+ E22) + 2μE11 // 1,2,3 = x,y,z T22 = λ(E11+ E22) + 2μE22 T12 = 2μE12 . ( E . 9 . 2 1 ) With respect to our transformation F M, (E.9.20) is a "true tensor equation" (tr(E) = Ek k = Ekk is scalar under rotations), so according to Section 7.15 it is "covariant" and in x"-space may be written Appendix E: Tensor Expansions 309 T"ij = λ tr(E")g"ij + 2μE"ij or T"ij == λ tr(E")δi,j+ 2μE"ij // g"ij = δi,j as in (E.9.3) or [T(e^)]ij = λ [T(e^)]kk δi,j + 2μ[E(e^)]ij . or T'ij = λ tr(E')δi,j + 2μE'j tr( E') = Σi E'ii . // E = script E (E.9.22) This notion of equation covariance was shown above in example (E.9 .13). Recall the notation example from (E.8.6) that, for spherical coordinates 1,2,3 = r, θ,φ, [A(e^)]2213 = A'2213 = Aθθrφ = Aθθrφ . (E.8.6) Correspondingly, we would set T'11 = Trr and so on to write the components of (E.9.21) this way in cylindrical coordinates, where 1,2,3 = r, θ,z, Trr = λ [ Err+ Eθθ] + 2μ Err // E zz = 0 Tθθ = λ [ Err+ Eθθ]) + 2μEθθ Trθ = 2μ Erθ . ( E . 9 . 2 3 ) This has the same form as (E.9.21) with 1 →r and 2→θ. These equations are consistent with Lai p 264 (5.27.7) with λ = -EYv/[(v+1)(2v-1)] and 2 μ = EY/(1+v). Statements of (E.9.23) are rare on the web, but here is one instance (Victor Saouma draft L ecture Notes on Continuum Mechanics, 1998), (E.9.24) 3. By way of contrast, the Cartesian-coordinates equation E ij = (∂iuj + ∂jui)/2, which relates strain tensor E ij to the vector displacement u of a continuum particle, is not a "true tensor equation", so Erθ ≠ (∂ruθ + ∂θur)/2. In fact, this relation is E = [( ∇u)T + (∇u)] / 2 and ( ∇u) for polar coordinates is computed in (G.6.6) and one ends up with E rθ = (∂ruθ + (1/r) ∂θur - uθ/r) / 2 . E.10 Tensor expansions in a mixed basis Recall the expansion (E.2.1) and (E.2.3) written for a general rank-n tensor A, A = Σijk... αijk... (bi⊗bj⊗bk...) αijk... = A • (bi⊗bj⊗bk...) (E.10.1) where αijk... are the coefficients of the expansion of A on the direct product basis shown. To make explicit the fact that the coefficients depend on the choice of basis, one might write (one b for each index, number of b's is the rank of the tensor) , Appendix E: Tensor Expansions 310 αijk... = [ A(b,b,b...)]ijk... . ( E . 1 0 . 2 ) The fact that the indices ijk... are "up" i ndicates that the b label stands for the bi basis and not bi. So here is an example showing the generic expansion of a rank-3 tensor, A = Σijk [ A(b,b,b)]ijk (bi⊗bj⊗bk) [ A(b,b,b)]ijk = A • (bi⊗bj⊗bk) . (E.10.3) Earlier we used the simpler notation [A(b)]ijk for the above coefficient, but now we want to show all the basis elements because now we want to cons ider a "mixed basis expansion" such as A = Σijk [ A(b,e,u)]ijk (bi⊗ej⊗uk) [ A(b,e,u)]ijk = A • (bi⊗ej⊗uk) . (E.10.4) This is a completely viable expansion since the b, e and u basis vectors are each a complete set within their part of the direct-product space. To verify the validity of this expansion, consider : [ A (b,e,u)]ijk = {A} • (bi⊗ej⊗uk) = { Σ i'j'k' [ A(b,e,u)]i'j'k' (bi'⊗ej'⊗uk')}• (bi⊗ej⊗uk) = { Σ i'j'k' [ A(b,e,u)]i'j'k' (bi'• bi) (ej'• ej) (uk'• uk) = { Σ i'j'k' [ A(b,e,u)]i'j'k' δi' iδj' jδk' k = [ A (b,e,u)]ijk . ( E . 1 0 . 5 ) In the case of a rank-2 tensor, one has the option of using the other notations discussed above, A = Σij [ A(b,u)]ij (bi⊗uj) = Σij [ A(b,u)]ij (biuj) direct product dyadic = Σij [ A(b,u)]ij (biujT) = Σij [ A(b,u)]ij |bi><uj| (E.10.6) matrix bra-ket where [ A (b,u)]ij = A • (bi⊗uj) = Aab (bi)a (uj)b = ( bi)T A (uj) = < bi| A | uj > . (E.10.7) One can of course use unit versions of the en basis vectors, e^n, and then one might write for example A = Σ ijk [ A(e^,e^,u)]ijk (e^i⊗e^j⊗uk) ( E . 1 0 . 8 ) where the hats are replicated into the superscript tensor label. Appendix F: Affine Connection 311 Appendix F: The Affine Connection Γc ab and Covariant Derivatives F.1 Definition and Interpretation of Γ : Γc ab = qc • (∂aqb) = Rc i(∂aRbi) In this Section we shall use a modified Picture C in which the quasi-Cartesian space on the right is called ξ-space instead of x(0)-space as in Picture C. The notation ξi for the coordinates of ξ-space seems traditional in general relativity work where the Γ object appears frequently. (F.1.1) Recall from the discussion near (1.10) that the metric tensor G is a diagonal ma trix whose elements are independently +1 or -1. Since the metric tensor tran sforms as a rank-2 tensor, we know from the last line of (7.5.8) (adapted from Picture A to Picture C1) that g ab = RaiRbjGij. Since G is diagonal, we write this as g ab = RaiRbiGii and gab = Ra iRb iGii ( F . 1 . 2 ) with a single implied sum on i. If G = 1, then the x i coordinates of x-space are "the curvilinear coordinates" and the ξi are "the Cartesian coordinates". In Picture C1 the tangent base vectors exist in ξ-space and we will call them qn. From (7.18.1) (adapted from Picture A to Picture C1) we know both that ( qn)i = Rni and that the dot product qn • qm = gnm where g is the metric tensor in x-space. In general qn = qn(ξ). However, since x = F(ξ), we are free instead to regard qn = qn(x), and that is what we shall do for Picture C1. If one moves a small amount d x in x-space, qn(x) (a vector in ξ-space) will change by some small amount. We are used to this idea in the Picture A c ontext of Fig (3.4.3) where the x-space tangent base vectors eθ = r θ^ and er = r^ both change if one takes θ→θ +dθ and r→r + dr in x'-space. So, for a small change dx j in x-space, the change in qn(x) is given by, d( qn)i = ∂j(qn)i dxj // ∂j ≡ ∂/∂xj or (d qn)i = (∂jqn)i dxj . // ∂x(fi) = ∂(fi) dx = ∂fi dx = (∂xfi) (F.1.3) Since qn and (∂jqn) are both vectors in ξ-space, and since the qk are known to form a complete basis in ξ- space, it must be possible to expand ( ∂jqn) on the qk with some appropriate coefficients, call them Γk jn : Appendix F: Affine Connection 312 (∂jqn) = Γk jn qk ⇒ (d qn)i = (∂jqn)dxa = Γk jn (qk)i dxa . (F.1.4) The coefficients Γk jn are known as the affine connection . They measure how the tangent basis vectors change in ξ-space as a function of x in x-space. One regards Γk jn(x) as an object associated with x-space, though it is not a tensor object as (F.6.3) below shows. Dotting the left equation into qm , using qm • qk = δm k as in (7.18.1), and then doing m → k gives Γk jn = qk • (∂jqn) = ( qk)i(∂jqn)i = Rk i(∂jRni) , ( F . 1 . 5 ) where from (7.18.1) and (7.5.16) (adjusted from Picture A to Picture C1) we have used ( qk)i = Rk i = ∂xk ∂ξi and ( qn)i = Rni = ∂ξi ∂xn . (F.1.6) Inserting the partial derivatives from (F.1.6) into (F.1.5) gives Γk jn = Rk i(∂jRni) = ∂xk ∂ξi ( ∂j ∂ξi ∂xn ) = ∂xk ∂ξi ∂2ξi ∂xj∂xn ( F . 1 . 7 ) Notice that (∂ jRni) = (∂nRji) / / s i n c e ∂2ξi ∂xj∂xn = ∂2ξi ∂xn∂xj (F.1.8) Form (F.1.7) shows that: Fact : Γk jn is symmetric on the lower two indices, so Γk jn = Γk nj (F.1.9) If we take k →λ, i→α, j→μ and n→ν , then (F.1.9) becomes or Γ λ μν = ∂xλ ∂ξα ∂2ξα ∂xμ∂xν and this equation appears as Weinbe rg p 100 (4.5.1), with traditional Greek indices used for relativity. We now restate some of the results a bove with more commonly used indices : (∂aqn) = Γk an qk → (∂aqb) = Γc ab qc Γk jn = qk • (∂jqn) → Γc ab = qc • (∂aqb) Appendix F: Affine Connection 313 so that (∂aqb) = Γc ab qc ( F.1.4) Γc ab = qc • (∂aqb) (F.1.5) Γc ab = Γc ba (F.1.9) Γc ab = Rc i(∂aRbi) (F.1.5) Γc ab = – Rbi (∂aRc i) ( F . 1 . 1 0 ) The last line will be derived in the next Section. One can convert the Γ-related equations from Picture C1 to Picture A, For example, since the tangent base vectors in x-space of Picture A are called en, (∂ aqb) = Γc ab qc → (∂'aeb) = Γ 'c ab ec Γc ab = qc • (∂aqb) → Γ'c ab = ec • (∂'aeb) ( F . 1 . 1 1 ) where Γ' = Γ'( x') and en = en(x'). F.2 Identities of the form (∂ aRd n) = – Re n Rd m (∂aRem) The identities (to be proven below) are : R d m (∂aRem) = – Rem (∂aRd m) 1 (∂ aRd n) = – Re n Rd m (∂aRem) 2 ∂a ≡ ∂/∂xa (∂ aRdn) = – Ren Rdm (∂aRe m) 3 . (F.2.1) Corollary: The first identity above allows the alternate form for the affine connection in terms of R, as quoted above in (F.1.10). Here we take d →c, m→i and e→b in identity #1, Rd m (∂aRem) = – Rem (∂aRd m) // identity 1 Rc i (∂aRbi) = – Rbi (∂aRc i) // same with new indices which then verifies (F.1.10). Appendix F: Affine Connection 314 Our context is: . (F.2.2) Proofs of identities 1,2,3: These identities are a simple consequen ce of the fact that RS = 1 which in standard notation is written δc b = Rc αRbα (orthogonality rule #3 in (7.6.4) ). So, 0 = ∂a(δd e) = ∂a(Rd mRem) = Rd m (∂aRem) + Rem (∂aRd m) . QED 1 (F.2.3) Apply Σ e Re n to both ends of (F.2.3) to get 0 = R e n Rd m (∂aRem) + (Re n Rem) (∂aRd m) = Re n Rd m (∂aRem) + δnm (∂aRd m) = Re n Rd m (∂aRem) + (∂aRd n) ⇒ (∂aRd n) = – Re n Rd m (∂aRem) . QED 2 (F.2.4) Alternatively, apply Σd Rdn to both ends of (F.2.3) to get 0 = (R dn Rd m) (∂aRem) + Rdn Rem (∂aRd m) = δn m (∂aRem) + Rdn Rem (∂aRd m) = (∂aRen) + Rdn Rem (∂aRd m) ⇒ (∂aRen) = – Rdn Rem (∂aRd m) now swap d and e: ⇒ (∂aRdn) = – Ren Rdm (∂aRe m) . QED 3 (F.2.5) F.3 Identities of the form (∂ cgab) = – [gan Γ b cn + gbn Γa cn] The derivatives of the metric tensor are given by ( ∂c = ∂/∂xc) (∂cgab) = – [gan Γ b cn + gbn Γa cn] 1 ( F . 3 . 1 ) (∂ cgab) = + [g an Γn cb + gbn Γn ca] 2 ( F . 3 . 2 ) ( F . 3 . 3 ) Appendix F: Affine Connection 315 Proof of 1: ( ∂cgab) = – [gan Γ b cn + gbn Γa cn] (F.3.1) Using (F.1.2) that gab = Ra iRb iGii , the LHS of (F.3.1) becomes , LHS = ( ∂ cgab) = ∂c(Ra iRb i)Gii = Ra i(∂cRb i)Gii + Rb i(∂cRa i)Gii . (F.3.4) For the RHS, the Γ objects can be replaced by their alternate definitions from (F.1.10), Γc ab = – Rbi (∂aRc i) // from (F.1.10) Γb cn = – Rni (∂cRb i) // b →n then c→b then a→c Γa cn = – Rni (∂cRa i) . // b →a ( F . 3 . 5 ) The RHS of the claimed identity (F.3.1) may then be written RHS = – g an Γ b cn – gbn Γa cn = { Ra kRn kGkk }{Rni (∂cRb i)} + {Rb kRn kGkk }{Rni(∂cRa i)} = R a k(Rn k Rni) (∂cRb i) Gkk + Rb k(Rn k Rni) (∂cRa i) Gkk = R a kδki (∂cRb i) Gkk + Rb kδki (∂cRa i) Gkk // orthog rule #2 of (7.6.4) = R a i (∂cRb i) Gii + Rb i (∂cRa i) Gii = ( ∂ cgab) // using (F.3.4) QED 1 (F.3.6) Proof of 2: ( ∂cgab) = + [g an Γn cb + gbn Γn ca] (F.3.2) Using (F.1.2) that g ab = RaiRbiGii, the LHS of (F.3.2) becomes, LHS = ( ∂cgab) = ∂c(RaiRbi)Gii = Rai (∂cRbi)Gii + Rbi (∂cRai)Gii . (F.3.7) For the RHS, the Γ objects can be replaced by their primary definitions in (F.1.10), Γc ab = Rc i(∂aRbi) Γn cb = Rn i(∂cRbi) // c →n then a→c then i→k Γn ca = Rn i(∂cRai) . // b →a ( F . 3 . 8 ) The RHS of the claimed identity (F.3.2) may then be written RHS = g an Γn cb + gbn Γn ca = { R akRnkGkk}{Rn i(∂cRbi)} +{RbkRnkGkk}{Rn i(∂cRai)} Appendix F: Affine Connection 316 = R ak (Rnk Rn i)(∂cRbi)Gkk + Rbk(Rnk Rn i)(∂cRai)Gkk = R ak δk i(∂cRbi)Gkk + Rbk∂k i(∂cRai)Gkk // orthog rule #1 of (7.6.4) = R ai (∂cRbi)Gii + Rbi (∂cRai)Gii = ( ∂ cgab) // using (F.3.7) QED 2 (F.3.9) F.4 Identity: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] The identity states that Γd ab may be expressed entirely in terms of the metric tensor, Γ d ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] . (F.4.1) Recall in our definition above, Γ d ab = Rd k(∂aRbk), that Γ was given in terms of R matrices. The following corollary concerns summing of the upper Γ index with a lower one, Γa an = (1/2) gad ∂ngad = (1/2)(1/g) ∂ng = (1/ |g| ) ∂n(|g| ) . (F.4.2) We shall prove the identity (F.4.1) first, and the corollary (F.4.2) after that. Proof of Identity: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] (F.4.3) We first rewrite (F.1.2) with new index names, g ab = RaeRbe Gee // sum on e gdc = Rd iRc i Gii . // sum on i (F.4.4) The first line below is computed from the first line in the pair above, then the next two lines below are obtained by doing forward cyclic inde x permutations of the first line : ∂cgab = [Rbe (∂cRae) + Rae(∂cRbe) ]Gee ∂agbc = [Rce (∂aRbe) + Rbe(∂aRce) ]Gee ∂bgca = [Rae (∂bRce) + Rce(∂bRae) ]Gee . (F.4.5) The last four lines can be appropriately inserted into the RHS of (F.4.3) to obtain Appendix F: Affine Connection 317 (RHS)d ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] = ( 1 / 2 ) (Rd iRc i Gii) Gee * [Rce (∂aRbe) + Rbe(∂aRce) + Rae (∂bRce) + Rce(∂bRae) – Rbe (∂cRae) – Rae(∂cRbe) ] . (F.4.6) 1 2 3 4 5 6 Due to the symmetry ( ∂ iRje) = (∂jRie) noted in (F.1.8), terms 2 and 5 cancel as do terms 3 and 6, while terms 1 and 4 are equal. Therefore, (RHS)d ab = (1/2) Rd iRc i Gii Gee * 2 Rce (∂aRbe) = R d i (Rce Rc i) Gii Gee (∂aRbe) = R d i δe i Gii Gee (∂aRbe) // orthog rule #1 of (7.6.4) = R d e Gee Gee (∂aRbe) = R d e(∂aRbe) / / Gee = Gee = ±1 = Γ d ab // from (F.1.10) = LHS of (F.4.3) QED (F.4.7) Proof of Corollary: The corollary is this (g ≡ det(gab) ) Γa an = (1/2) gad( ∂agnd + ∂ngad – ∂dgan ) = (1/2) gad ∂ngad = (1/2)(1/g) ∂ng = (1/ |g| ) ∂n(|g| ) . 1st 2nd 3rd 4th 5th ( F . 4 . 8 ) The 2nd expression is of course just (F.4.1) with c = b = n and implied sum on n. The first and third terms of the 2nd expression cancel due to symmetry of g: first term = gad ∂agnd = gda ∂dgan = gad ∂dgan = - third term (F.4.9) a ↔d gda = gad so we then have just Γa an = (1/2) gad(∂ngad) which is the 3rd expression shown in (F.4.8). To get the 4th expression, we must show that gab(∂ngab) = (1/g)∂ng. Here are the steps to prove this fact, where g ≡ det(gab) : Appendix F: Affine Connection 318 (1) gab = (g-1)ab = cof(gab)T/det(gab) = cof(g ab)/g ⇒ cof(g ab) = g gab . (F.4.10) (2) Apply g ba and sum on b to get: g bacof(gab) = g gbagab = g δaa = g . (F.4.11) (3) g = det(g ab) = gabcof(gab) ⇒ ∂g/∂gab = cof(g ab) = g gab * (F.4.12) (4) ∂ ng = ∂ g/∂xn = (∂ g/∂gab)( ∂gab/∂xn) = g gab (∂ngab) ⇒ (1/g) ∂ng = gab(∂ngab) (F.4.13) * In Step 3, notice that ∂ cof(gab)/∂gab = 0 because cof(g ab) is not a function of variable g ab. The element gab is one of those "crossed out" in obtaining the cofactor of g ab. Despite this fact, cof(g ab) is still an object having indices a,b. In contrast, g = det(g ab) = det(g **) has no indices! It remains only to obtain the 5th expression in (F.4.8), and that is easy to show: g-1/2 ∂n(g1/2) = g-1/2 (1/2) g-1/2 ∂n(g) = (1/2) (1/g) ( ∂ng) . g > 0 [-g] -1/2∂n([-g]1/2) = [-g]-1/2 (1/2)[-g]-1/2∂n([-g]) = (1/2)[-g]-1 ∂n([-g]) g < 0 which we summarize as |g| -1/2 ∂n(|g|1/2) = (1/2) (1/g) ( ∂ng ) ( F . 4 . 1 4 ) We close this Section with the following observations: Theorem : If x-space of Picture C1 of (F.1.1) has a metric tensor g ij(x) whose elements are constants independent of x, then Γc ab ≡ 0 . ( F . 4 . 1 5 ) Corollary : If the x-space in Figure C1 of (F.1.1) is Ca rtesian (g =1) or quasi-Cartesian (g = G), then Γc ab ≡ 0 . ( F . 4 . 1 6 ) Proof : The identity (F.4.1) says Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab], so if gij are constants, one must have Γd ab = 0. We shall often quote this Corollary with the phrase " Γ = 0 for a Cartesian space". Appendix F: Affine Connection 319 F.5 Picture D1 Context Our main context of interest is Picture A, . (F.5.1) For the proof to be presented in Section F.6, it is useful to think of Picture A as the top part of this Picture D1, (F.5.2) The relationship between the R's and S's are these R = R' R-1 = R' S ⇒ R' = R R S = R R'-1 = R S' . ( F . 5 . 3 ) The tangent and reciprocal base vectors in ξ -space associated with transformations F and F ' are these, based on (F.1.6), (qn)i = Rni ( qn)i = Rn i x-space (q'n)i = R'ni ( q'n)i = R'n i x'-space . (F.5.4) Now there are two affine connections, one for x-space and the other for x'-space, Γc ab ≡ (∂xc/∂ξi) (∂2ξi/∂xa∂xb) = Rc i ∂a (∂ξi/∂xb) = Rc i(∂aRbi) ∂a = ∂/∂xa = [ qc]i (∂a[qb]i) = qc • (∂aqb) ( F . 5 . 5 ) Γ 'c ab ≡ (∂x'c/∂ξi) (∂2ξi/∂x'a∂x'b) = R'c i ∂'a (∂ξi/∂x'b) = R'c i(∂'aR'bi) ∂'a = ∂/∂x'a = [ q'c]i (∂'a[q'b]i) = q'c • (∂'aq'b) . (F.5.6) The expressions appearing here in (F.5.5) are ju st converted versions of (F.1.7) and (F.1.5). Appendix F: Affine Connection 320 F.6 Relations between Γ and Γ ' The claimed relations are the following, in the context of Picture A shown above, Γ 'c ab = Rc d Raα Rbβ Γd αβ + Rc α (∂'aRbα) // Weinberg p 100 (4.5.2) (F.6.1) Γ 'c ab = Rc d Raα Rbβ Γd αβ – Rbβ(∂'aRc β) (F.6.2) Γ 'c ab = Rc d Raα Rbβ Γd αβ – Raα Rbβ (∂αRc β) // Weinberg p 102 (4.5.8) (F.6.3) The first terms are all the same, only the second term s vary. If the second term were not present, the relation would state that Γd αβ transforms as a mixed rank-3 tensor in the usual manner, as in example (7.10.1). Since the second term is present, Γd αβ is not a tensor (unless R bα(x) = constant, in which case the transformation F is linear so x' = F(x) = R x as mentioned in Section 2.8 ) . Proof of the first relation : We now make use of Picture D1 shown above. Start with the Γ ' definition expression shown in (F.5.6), Γ 'c ab ≡ R'c n(∂'aR'bn) = (R R)c n ∂'a(RR)bn = Rc dRd n ∂'a(RbβRβn) // R' = R R (F.5.3) = R c dRd nRbβ(∂'aRβn) + Rc d(Rd nRβn)( ∂'aRbβ) // Rd nRβn = δd β (7.6.4) = Rc dRd nRbβ([Raα∂α]Rβn) + Rc d(δd β)( ∂'aRbβ) // ∂'a = Raα∂α in first term only = Rc dRaαRbβRd n(∂αRβn) + Rc β (∂'aRbβ) // next use (F.5.5) for Rd n(∂αRβn) = Rc dRaαRbβ Γd αβ + Rc α (∂'aRbα) // Γd αβ ≡ Rd n(∂αRβn) QED (F.6.4) Magically, all the R's have gone away. The second term in the above relation can be written a different manner as follows. Consider, 0 = ∂'a(δc b) = ∂ 'a(Rc αRbα) = Rc α (∂'aRbα) + Rbα (∂'aRc α) ⇒ Rc α (∂'aRbα) = – Rbα(∂'aRc α) = – Rbβ(∂'aRc β) = – R bβ Raα(∂αRc β) (F.6.5) and this gives the other two relations stated above. F.7 Statement and Proof of the Covariant Derivative Theorem Many examples of this theorem will be given later. In this proof it is assumed that the tensor density of interest is purely covariant (all i ndices "down"). In the next Section it will then be shown how to adjust the theorem if one or more of the tensor density indices is "up". Appendix F: Affine Connection 321 Covariant Derivative Theorem: The covariant derivative B abc..x; α (as defined below) of a covariant tensor B abc..x of rank n and weight W transforms as a covari ant tensor density of rank n+1 and weight W . ( F . 7 . 1 ) The implication is that all the indices including α of B abc..x; α can be treated as ordinary tensor indices with respect to raising, lowering, contrac tion, and so on. The first term below in B abc..x; α is the regular derivative ∂α Babc..x , often written as B abc..x, α (comma, not semicolon), and this first term is not a tensor. Only when all the "correction terms" are included does the object become a tensor. The covariant derivative in x-space and then in x'-s pace is defined as follows: (Weinberg p 104 4.6.12) B abc..x; α ≡ ∂α Babc..x – Γn aαBnbc..x – Γn bαBanc..x – .... – Γn xαBabc..n // x-space del a-term b-term x-term + [ W / ( 2 g ) ] ( ∂αg) Babc..x (F.7.2) B'abc..x; α ≡ ∂'α B'abc..x – Γ 'n aαB'nbc..x – Γ 'n bαB'anc..x – .... – Γ 'n xαB'abc..n // x'-space del a-term b-term x-term + [ W / ( 2 g ' ) ] ( ∂' αg') B'abc..x (F.7.2)' The two definitions are the same except everything is pr imed in x'-space (except constant weight W). This is as one would expect if the x-space equation were a "true tensor equation" as discussed in Section 7.15 and were therefore "covariant". Although the Γ objects are not tensors themselves, the combination of terms shown in the definition of B abc..x; α is a rank n+1 tensor (a s will be demonstrated). As was shown in (F.4.2), (1/2)(1/g) ∂αg = Γκ κα so the W terms could be written as W Γκ κα Babc..x and W Γ 'κ κα B'abc..x , and this form is commonly seen in the literature on this subject. A proof of the theorem must then show that B abc..x; α as defined above in fact transforms as a tensor density of rank n+1 and weight W, which is to say, one must show that B'abc..x; α = J-W Raa'Rbb'..... Rxx' Rαα' Ba'b'c'..x'; α' (F.7.3) or, changing abc..x →ABC..X and then a'b'c'.. → abc.. , B'ABC..X; α = J-WRαα'{RAaRBb..... RXx} * B abc..x'; α' ( F . 7 . 4 ) or, in gory detail, Appendix F: Affine Connection 322 ∂'αB'ABC..X – Γ 'n AαB'nBC..X – Γ 'n BαB'AnC..X – ........... – Γ 'n XαB'ABC..n // LHS del' a'-term b'-term x'-term + [ W / ( 2 g ' ) ] ( ∂'αg') B'ABC..X = J -W Rαα'{RAaRBb..... RXx} * / / R H S {∂α'Babc..x – Γn aα'Bnbc..x – Γn bα'Banc..x – ........... – Γn xα'Babc..n del a-term b-term x-term + ( [ W / ( 2 g ) ] ( ∂α'g) Babc..x } (F.7.5) Proof: There is probably a faster way to prove that (F.7.5) is valid, but we shall be content with a brute force proof where we in essence evaluate both sides of (F .7.5) and show they are the same. In doing so, we make use of identities obtained in previous Secti ons. The proof is carried out in five steps. 1. Expand the LHS del' term and show del'-del matches the RHS del term. ∂'αB'ABC..X = (Rαβ∂β)( J-W RAaRBb..... RXx Babc..x ) del' = R αβ(∂β J-W) RAaRBb..... RXx Babc..x d e l ' - J + R αβ J-W (∂βRAa)RBb..... RXx Babc..x d e l ' - a + R αβ J-W RAa(∂βRBb)..... RXx Babc..x del'-b ... + R αβ J-W RAaRBb. ... (∂βRXx) Babc..x del'-x + R αβ J-W RAaRBb.............R Xx (∂β Babc..x ) del'-del (F.7.6) We claim that this last del'-del term of the LHS matches the del term on the RHS: del'-del LHS = R αβ J-W {RAaRBb...........R Xx} (∂β Babc..x ) (F.7.7) del RHS = J-W Rαα'{RAaRBb..... RXx} {∂α' Babc..x } (F.7.8) Setting α' = β shows that these two terms indeed match. 2. Show that the a-related terms balance. The a'-term on the LHS of (F.7.5) is – Γ 'n AαB'nBC..X . // a'-term of LHS (F.7.9) The connection (F.6.3) between Γ ' and Γ reads (then shuffle indices as shown), Γ 'c ab = Rc d Raα Rbβ Γd αβ – Raα Rbβ (∂αRc β) // α→κ , β→σ (F.6.3) Γ 'c ab = Rc d Raκ Rbσ Γd κσ – Raκ Rbσ (∂κRc σ) // c→n, a→A, b→α Γ 'n Aα = Rn d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ) . (F.7.10) Appendix F: Affine Connection 323 Now insert into (F.7.9) the expression (F.7.10) for Γ 'n Aα and the tensor transformation rule for B' nBC..X , – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { J-W Rnn'RBbRCc....Rxx Bn'bc..x } // a'-term (F.7.11) and to this we must add the contri bution del'-a shown in (F.7.6). Meanwhile, the RHS a-term in (F.7.5) is this J-W Rαα'{RAaRBb..... RXx}{– Γn aα'Bnbc..x } // now do n →n' = – { J -W Rαα'{RAaRBb..... RXx}{Γn' aα'Bn'bc..x } . // a-term of RHS (F.7.12) We want to show that a'-term + del'-a = a-term or LHS a'-term (F.7.11) + LHS del'-a te rm (F.7.6) = RHS a-term (F.7.12) or – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { J-W Rnn'RBbRCc....Rxx Bn'bc..x } + R αβ J-W (∂βRAn')RBb..... RXx Bn'bc..x // a →n' , this is the del'-a term = – J -W Rαα'{RAaRBb..... RXx}{Γn' aα'Bn'bc..x } ? ( F . 7 . 1 3 ) We can see that the factors J-W RBbRCc....Rxx Bn'bc..x are common to both sides so they can be removed to give a simpler relation which we must show to be valid: – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { Rnn' } + R αβ (∂βRAn') = – R αα'{RAa }{Γn' aα'} . ( F . 7 . 1 4 ) In the first term orthogonality (7.6.4) says R n d Rnn' = δdn' which pins d to n' in that term only, so the above becomes – RAκ Rασ Γn' κσ + Rnn'RAκ Rασ (∂κRn σ) + Rαβ (∂βRAn') = – R ασRAκ Γn' κσ . (F.7.15) The first term on the left cancels the only term on the right. Then do β→σ in the third term to get, RAκ Rασ {Rnn'(∂κRn σ)} + R ασ (∂σRAn') = 0 . (F.7.16) Now cancel the common R ασ factor and use the symmetry ( ∂κRn σ) = (∂σRn κ) to get (∂σRAn') = – Rnn'RAκ (∂σRn κ) . ( F . 7 . 1 7 ) Now in this order do σ→a, A→d, n→e, n'→n, κ→m to get Appendix F: Affine Connection 324 (∂aRdn) = –Ren Rdm (∂aRe m) . ( F . 7 . 1 8 ) But this is seen to be the third identity of (F.2.1 )! By reversing the above sequence of steps, one shows that the three a-related terms in the above LHS = RHS equation balance: a'-term + del'-a = a-term. or – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { J-W Rnn'RBbRCc....Rxx Bn'bc..x } + R αβ J-W (∂βRAn')RBb..... RXx Bn'bc..x // a →n' , this is the del'-a term = – J -W Rαα'{RAaRBb..... RXx}{Γn' aα'Bn'bc..x } ? (F.7.13) In reversing the sequence, one of course adds back the deleted common factors as well as the various implied index sums. It seems clearer to state the proof th is way rather than start with the last equality with no justification and artificially thread backwards to th e desired equation. This method is used as well in the next section. 3. Show that the b-related terms balance. We have proven that (F.7.13) is valid, and it is true for any tensor B n'bc..x , since this is a common factor on both sides. Eq (F.7.13) is therefore still valid if we replace the tensor B n'bc..x by the different tensor Q n'bc..x ≡ Bbn'c..x . Tensor Q is tensor B with the first two indices swapped. Make this replaceme nt, and after doing so, make the free index swap A ↔B and summation index swap a ↔b. One ends up then with this known-valid equation: – { R n d RBκ Rασ Γd κσ – RBκ Rασ (∂κRn σ)} { J-W Rnn'RAaRCc....Rxx Ban'c..x } + R αβ J-W (∂βRBn')RAa..... RXx Ban'c..x = – J-W Rαα'{RBbRAa..... RXx}{Γn' bα' Ban'c..x } . (F.7.19) As the reader may suspect, th e three terms in this last equation will ha ve this interpretation with respect to equation (F.7.5) : b'-term + del'-b = b-term . To verify this claim, we treat the terms one at a time. b'-term: First state the equation (F.7.9) = (F.7.11) on the first line below, then get the second line doing A↔B and dummy b↔ a : – Γ 'n AαB'nBC..X = – { Rn d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { J-W Rnn'RBbRCc....Rxx Bn'bc..x } – Γ 'n BαB'nAC..X = – { Rn d RBκ Rασ Γd κσ – RBκ Rασ (∂κRn σ)} { J-W Rnn'RAaRCc....Rxx Bn'ac..x } Since this second equation is valid for any tensor B n'ac..x , it is true for Q n'ac..x ≡ Ban'c..x where the first two indices are swapped. This gives, – Γ 'n BαB'AnC..X = – { Rn d RBκ Rασ Γd κσ – RBκ Rασ (∂κRn σ)} { J-W Rnn'RAaRCc....Rxx Ban'c..x } Appendix F: Affine Connection 325 The left side is the b'-term on the LHS of (F.7.5) an d the right side is the first term in (F.7.19). del'-b: The del'-b term in (F.7.6) is the first line below, R αβ J-W RAa(∂βRBb )..... RXx Babc..x // b →n Rαβ J-W RAa(∂βRBn')..... RXx Ban'c..x The second line matches the second term in (F.7.19). b-term : The b-term on the RHS of (F.7.5) is – J-W Rαα'{RAaRBb..... RXx}[ Γn bα'Banc..x] // n →n' – J-W Rαα'{RAaRBb..... RXx}[ Γn' bα'Ban'c..x ] The second line matches the right side of (F.7.19). We have thus shown that the b terms balance in equation (F.7.5), just the way the a terms balance. 4. Similarly, the c,d.....x terms match . Just repeat step 3 above for each extra index. 5. It remains to show that the three term s so far neglected in (F.7.5) match as well . These terms are LHS: R αβ(∂β J-W) RAaRBb..... RXx Babc..x // del'-J in (F.7.6) + [W/(2g')] ( ∂'αg') B'ABC..X // the LHS W term (F.7.20) RHS: J -W Rαα'{RAaRBb..... RXx} [W/(2g)] ( ∂α'g) Babc..x // the RHS W term That is, one must show that R αβ(∂β J-W) RAaRBb..... RXx Babc..x + [W/(2g')] ( ∂'αg') B'ABC..X = J -W Rαα'{RAaRBb..... RXx} [W/(2g)] ( ∂α'g) Babc..x . (F.7.21) Inserting the tensor transformation (F.7.4) of B' ABC..X in the second term gives R αβ(∂β J-W) RAaRBb..... RXx Babc..x + [W/(2g')] ( ∂'αg') J-W{RAaRBb..... RXx} * Babc..x = J -W Rαα'{RAaRBb..... RXx} [W/(2g)] ( ∂α'g) Babc..x . (F.7.22) Set β = α' in the first term to get Rαα'(∂α' J-W) RAaRBb..... RXx Babc..x + [W/(2g')] ( ∂'αg') J-W{RAaRBb..... RXx} * Babc..x = J -W Rαα'{RAaRBb..... RXx} [W/(2g)] ( ∂α'g) Babc..x . (F.7.23) Appendix F: Affine Connection 326 Next, remove the common factor R AaRBb..... RXx Babc..x (and associated implied sums) to get R αα' (∂α' J-W) + [W/(2g')] ( ∂'αg') J-W = J-W Rαα' [W/(2g)] ( ∂α'g) . (F.7.24) Since ∂'α = Rαα'∂α'(covariant vector transformation) this becomes Rαα' (∂α' J-W) + [W/(2g')] (R αα'∂α'g') J-W = J-W Rαα' [W/(2g)] ( ∂α'g) or (∂' α J-W) + [W/(2g')] ( ∂'αg') J-W = J-W [W/(2g)] ( ∂'αg) . (F.7.25) Move the second term to the RHS and do the left side derivative to get (-W) J-W-1(∂'α J) = J-W (W/2) [ ( ∂'αg)/g – (∂'αg')/g' ] (F.7.26) so it remains then to show that J -1(∂'α J) = (1/2)[ (∂ 'αg')/g' - (∂'αg ) / g ] . ( F . 7 . 2 7 ) From (5.12.14) one has J 2 = g'/g and J = (g'/g)1/2 so we need to show that ∂' α (g'/g)1/2 = (1/2) (g'/g)1/2[ (∂'αg')/g' - (∂'αg)/g ] or (1/2) (g'/g)-1/2 ∂'α (g'/g) = (1/2) (g'/g)1/2[ (∂'αg')/g' - (∂'αg)/g ] or ∂'α (g'/g) = (g'/g) [ ( ∂'αg')/g' - (∂ 'αg ) / g ] ( F . 7 . 2 8 ) Evaluation of the left side of (F.7.28) gives ∂' α (g'/g) = [ g( ∂'αg') - g'(∂'αg)]/g2 = (1/g) ( ∂'αg') - (g'/g2)(∂'αg) . Evaluation of right side of (F.7.28) gives (g'/g) [ ( ∂'αg')/g' - (∂ 'αg)/g ] = (1/g)( ∂'αg') - (g'/g2 ) (∂'αg) , Since these agree, (F.7.27) is valid and therefore wo rking backwards we find that (F.7.20) is valid. This concludes our lengthy 5-step proof of the Cova riant Derivative Theorem stated in (F.7.1). F.8 Rules for raising any index on a covariant deriv ative of a covariant tensor density. The Rules are stated in (F.8.9) an d (F.8.11), but will only make sense after looking at the examples presented here. Appendix F: Affine Connection 327 Consider the general form given in (F.7.2) for a covariant derivative Babc..x; α ≡ ∂α Babc..x – Γn aαBnbc..x – Γn bαBanc..x – .... – Γn xαBabc..n // x-space del a-term b-term x-term + ( W / 2 g ) ( ∂ αg) Babc..x . (F.8.1) Notice that there are n indices on B abc..x and there are n corresponding terms on the RHS in addition to the del and W terms. Each index of B abc..x thus has its own "correction te rm". What happens if one of the indices (say b) on B abc..x is raised? To find out, apply gβb to both sides. The effect of doing this is trivial for all terms except the del term and the b-term, since b is a regular tensor index on all such terms, so we get Baβ c..x;α ≡ gβb ∂α Babc..x – Γn aαBnβ c..x – gβb Γn bαBanc..x – .... – Γn xαBaβ c..n del a-term b-term x-term + ( W / 2 g ) ( ∂αg) Baβ c..x . (F.8.2) The del term can be written gβb (∂α Babc..x ) = ∂α (gβb Babc..x ) – (∂α gβb) Babc..x = ∂α Baβ c..x – (∂α gβb) Babc..x . (F.8.3) The second term del term here can be combined with the b-term to give del-2nd + b-term = – ( ∂ α gβb) Babc..x – gβb Γn bαBanc..x = – ( ∂α gβn) Banc..x – gβb Γn bαBanc..x // b →n in first term only = [ – ( ∂α gβn) – gβb Γn bα] Banc..x . (F.8.4) The identity (F.3.1) with n → i reads, (∂cgab) = – [gai Γ b ci + gbi Γa ci] or [– (∂ cgab) – gai Γ b ci] = gbi Γa ci // do c →α, b→n, a→β [– (∂αgβn) – gβi Γ n αi] = gni Γβ αi // then i →b on the left [– (∂αgβn) – gβb Γ n αb] = gni Γβ αi . (F.8.5) Replacing [...] in (F.8.4) by the last result gives, del-2nd + b-term = g ni Γβ αi Banc..x = Γβ αi Bai c..x = Γβ αn Ban c..x (F.8.6) so it has been shown that B aβ c..x;α ≡ ∂α Baβ c..x – Γn aαBnβ c..x + Γβ αn Ban c..x – ... – Γn xαBaβ c..n + W Γκ κα Bab c..x new b term (F.8.7) Here then is a comparison where β→b in the second line: Appendix F: Affine Connection 328 Babc..x; α ≡ ∂α Babc..x – Γn aαBnbc..x – Γn bαBanc..x – .... – Γn xαBabc..n + W Γκ κα Babc..x Bab c..x;α ≡ ∂α Bab c..x – Γn aαBnb c..x + Γb αn Ban c..x – .... – Γn xαBab c..n + W Γκ κα Bab c..x del a-term b-term x-term W-term ( F . 8 . 8 ) This demonstrates the following rule (s tated for q = b or any other index): Rule for raising some non-last index q: (1) In all terms, raise the q-position B index (in the q-term that index is called n) (2) in the q correction term, make the replacement – Γ n qα → + Γq nα ( = Γq αn ) (F.8.9) We refer to a correction term like the a-term and x-te rm in (F.8.8) as a "covariant correction term", and a correction term like the b-term as a "contravarian t correction term". Thus, for an arbitrary up/down placement of the abc..x indices, there will be a c ovariant correction term for each down index, and a contravariant correction term for each up index. Here is another example where x is also taken up: B ab c..x ;α ≡ ∂α Bab c..x – Γn aαBnb c..x + Γb αn Ban c..x – .... + Γx nαBab c..n + W Γκ κα Bab c..x del a-term b-term x-term W-term ( F . 8 . 1 0 ) where the contravariant correction terms are marked in red. Rule for raising the last index α: Raising the ; α index must be done "manually" so the first term will have gαα'∂α = ∂α' and all remaining terms will have explicit gαα' factors. (F.8.11) F.9 Examples of covariant derivative expressions Example J = 0 (covariant derivative of a scalar B) // J is the rank of the B tensor B;α = ∂α B covariant vector // = B ,α B;α = ∂α B contravariant vector // = B,α (F.9.1) Example J=1: (covariant derivative of a vector B) Ba;α = ∂α Ba – Γn aαBn covariant rank-2 tensor // 2nd term is sym on a ↔α (F.9.2) B a ;α = ∂α Ba + Γa αn Bn mixed rank-2 tensor (F.9.3) To obtain B a;α, write (F.9.2) as B a;β = ∂βBa – Γn aβBn and apply gαβ to both sides. The result is then Ba;α = ∂αBa – gαβΓn aβBn .Finlly, raise a and alter the correction term to get Ba;α = ∂αBa + gαβΓa nβBn . Ba;α = ∂α Ba – gαβΓn aβBn mixed rank-2 tensor (F.9.4) Ba;α = ∂α Ba + gαβΓa βn Bn contravariant rank-2 tensor (F.9.5) Appendix F: Affine Connection 329 Example J=2: (covariant derivative of a rank-2 tensor) Bab;α ≡ ∂α Bab – Γn aαBnb – Γn bαBan covariant rank-3 tensor (F.9.6) Ba b;α ≡ ∂α Ba b + Γa αn Bn b – Γn bαBa n etc. (F.9.7) Bab ;α ≡ ∂α Bab – Γn aαBnb + Γb αn Ban (F.9.8) B ab ;α ≡ ∂α Bab + Γa αn Bnb + Γb αn Ban . (F.9.9) Again, application of g αβ would give expressions for th e other four possibilities with ; α being "up". These "other possibilities" are always present, but we sha ll no longer mention them in the following examples. Example J=3: (covariant derivative of a rank-3 tensor) Babc;α ≡ ∂α Babc – Γn aαBnbc – Γn bαBanc – Γn cαBabn (F.9.10) B a bc;α ≡ ∂α Ba bc + Γa αn Bn bc – Γn bαBa nc – Γn cαBa bn (F.9.11) ... Babc ;α ≡ ∂α Babc + Γa nαBnbc + Γb nαBanc + Γc nαBabn (F.9.12) Special J=2 application to the metric tensor: gab;α ≡ ∂α gab – Γn aαgnb – Γn bαgan = 0 // by identity (F.3.2) (F.9.13) g a b;α ≡ ∂α ga b + Γa αn gn b – Γn bαga n = 0 + Γa αb – Γa bα = 0 // ga b = δa b (F.9.14) g ab ;α ≡ ∂α gab – Γn aαgnb + Γb αn gan = 0 – Γb aα + Γb αa = 0 (F.9.15) g ab ;α ≡ ∂α gab + Γa αn gnb + Γb αn gan = 0 // by identity (F.3.1) (F.9.16) The middle lines use the fact (7.4.19) that g a b = δa b. Since g ab;α is a tensor, knowing that any one of the above vanishes implies that all four lines vanish! The net result is g ab;α = ga b;α = gab ;α = gab ;α = 0 . // Weinberg p 105 (4.6.16,17,18) (F.9.17) The covariant derivative of any form of the metric tensor vanishes. As We inberg points one, one knows that in a quasi-Cartesian x-space g ab;α = 0 since g ab = Gaaδa,b = constant and Γ = 0. Then in any x'- space g'ab;α = 0 as well since g' ab;α = Raa' Rbb' Rαα' ga'b';α' . Appendix F: Affine Connection 330 Example J=2: (double covariant derivatives) Consider again the J=1 examples from above B a;α = ∂α Ba – Γn aαBn (F.9.2) Ba ;α = ∂α Ba + Γa αn Bn . (F.9.3) This applies to any vector B a. As the J=0 example shows, B ;a and B;a are bona-fide vectors (covariant and contravariant components of the same vector ) and therefore B;a;α = ∂α B;a – Γn aαB;n ( F . 9 . 1 8 ) B;a ;α = ∂α B;a + Γa αn B;n ( F . 9 . 1 9 ) where we have simply inserted a semicolon in each term. In these equations B is a scalar. Consider again the J=2 examples from above, B ab;α = ∂α Bab – Γn aαBnb – Γn bαBan (F.9.6) Ba b;α = ∂α Ba b + Γa αn Bn b – Γn bαBa n . (F.9.7) This applies to any rank-2 tensors B ab or Ba b. But B a;b and Ba ;b are bona-fide rank-2 tensors, and therefore we can restate the above as, B a;b;α = ∂α Ba;b – Γn aαBn;b – Γn bαBa;n ( F . 9 . 2 0 ) Ba ;b;α = ∂α Ba ;b + Γa αn Bn ;b – Γn bαBa ;n ( F . 9 . 2 1 ) In a similar manner one can derive expressions for triple covariant derivatives and beyond. For example B a;b;c;α = ∂α Ba;b;c – Γn aαBn;b;c – Γn bαBa;n;c – Γn cαBa;b;n . (F.9.22) The next examples are for tensor densities so there will be a W term as shown in (F.8.8). Example J = 0 (scalar density of weight W) // recall Γκ κα = (2g)-1∂αg from (F.4.2) B;α = ∂αB + W Γκ κα B ( F . 9 . 2 3 ) Example J = 1 (vector density of weight W) Ba;α = ∂α Ba – Γn aαBn + W Γκ κα Ba covariant rank-2 tensor density Ba ;α = ∂α Ba + Γa αn Bn + W Γκ κα Ba ( F . 9 . 2 4 ) Appendix F: Affine Connection 331 Example J = 2 (rank-2 tensor density of weight W) Bab;α = ∂α Bab – Γn aαBnb – Γn bαBan + W Γκ κα Bab covariant rank-3 tensor density (F.9.25) Ba b;α = ∂α Ba b + Γa αn Bn b – Γn bαBa n + W Γκ κα Ba b ( F . 9 . 2 6 ) Bab ;α = ∂α Bab – Γn aαBnb + Γb αn Ban + W Γκ κα Bab ( F . 9 . 2 7 ) Bab ;α = ∂α Bab + Γa αn Bnb + Γb αn Ban + W Γκ κα Bab . (F.9.28) As noted in Example 2 below (D.2.3), adding a factor g W/2 to a tensor density of weight W neutralizes the weight, and the result is a regular tensor. Here then are a few examples in which this is done. Since the product is a tensor, there are no W correction terms. Example J = 0 (covariant derivative of a scalar density B of weight W) (gW/2B);α = ∂α(gW/2B) covariant vector (F.9.29) (gW/2B);α = ∂α(gW/2B) contravariant vector (F.9.30) Example J=1: (covariant derivative of a vector density B of weight W) (gW/2Ba);α = ∂α (gW/2Ba) – gW/2 Γn aα Bn covariant rank-2 tensor (F.9.31) (gW/2Ba);α = ∂α (gW/2Ba) + gW/2 Γa αn Bn mixed rank-2 tensor (F.9.32) Example J=2: (covariant derivative of a tensor density B of weight W) (g W/2Bab);α = ∂α (gW/2Bab) – Γn aα(gW/2Bnb) – Γn bα(gW/2Ban) covariant rank-3 tensor (F.9.33) (gW/2Ba b);α = ∂α (gW/2Ba b) + Γa αn (gW/2Bn b) – Γn bα(gW/2Ba n) etc. (F.9.34) (gW/2Bab);α = ∂α (gW/2Bab) – Γn aα(gW/2Bnb) + Γb αn (gW/2Ban) (F.9.35) (gW/2Bab);α = ∂α (gW/2Bab) + Γa αn (gW/2Bnb) + Γb αn (gW/2Ban) (F.9.36) F.10 The Leibniz rule for the covariant derivat ive of the product of two tensor densities If A and B are arbitrary tensor densities each with an arbitrary set of up and down indices and arbitrary weight, then the claim of the Leibniz or product rule is this: (A----B----) ;α ≡ A----;α B---- + A---- B----;α // Weinberg p 105 (4.6.14) (F.10.1) Recall from the Covariant Derivative Th eorem (F.7.1) that A---- and A---- ;α have the same weight, call it WA. Similarly, B---- and B---- ;α have the same weight W B. According to the outer product rule (D.2.3), both terms on the RHS above have weight W A+WB and therefore this sum is the weight of the LHS (A--- -B----);α as well. Proof of (F.10.1): Start with the covariant derivative of a generic tensor A in the form of (F.8.8), Appendix F: Affine Connection 332 A---- ;α = A---- ,α + (A index correction terms) + W A Γκ κα A---- B---- ;α = B---- ,α + (B index correction terms) + W B Γκ κα B---- (F.10.2) where --- is any number of up/down indices. The "index correction terms" are those Γ terms discussed below (F.8.9). One can then write out the two terms on the RHS of (F.10.1) above as: A---- ;α B---- = A---- ,α B---- + (A index correction terms) B---- + [W A Γκ κα A---- ] B---- A---- B----;α = A---- B----,α + A---- (B index correction terms) + A---- [W B Γκ κα B---- ] . ( F . 1 0 . 3 ) Meanwhile, the LHS of (F.10.1) can be written as (A----B----) ;α = (A----B----) ,α + ( all index correction terms) + (W A+ WB) Γκ κα (A----B----) . ( F . 1 0 . 4 ) Momentarily ignoring the index correction terms, it is clear that the other terms match between LHS and RHS. The W terms match by visual inspection, wh ile the regular derivative terms match due to the "regular" Leibniz rule for the derivative of a product (A----B----) ,α = A---- ,α B---- + A---- B----,α (F.10.5) which is to say ∂α(A----B----) = (∂ αA----)B---- + A---- (∂α B----) . (F.10.6) Consider now the LHS terms calle d (all index correction terms) a bove. This set of terms can be partitioned into two groups, (all index correction terms) = (terms involving A indices) + (terms involving B indices) . (F.10.7) Let us pause to look at a simple example where IC T means we just show the index correction terms, (A abBcd);α|ICT = – Γn aα(AnbBcd) – Γn bα(AanBcd) + Γc αn (AabBnd) + Γd αn (AabBcn) . (F.10.8) The correction terms can be reordered in this way = { – Γ n aα(Anb) – Γn bα(Aan) } Bcd + Aab { Γc αnBnd + Γd αnBcn } = { index correction terms for A ab } Bcd + Aab { index correction terms for Bcd } . (F.10.9) Just so, in the general case one has (A----B----) ;α|ICT = ( all index correction terms) = { index correction terms for A---- } B---- + A---- { index correction terms for B---- } (F.10.10) and this then shows that the index correction term s on the two sides of (F.10.1) do in fact match. QED Appendix F: Affine Connection 333 Once the above product rule is verified, it is then easy to generalize just as for regular derivatives: (A----B----C----) ;α ≡ A----;α B---- C---- + A---- B----;α C---- + A---- B---- C---- ;α (F.10.11) Examples with two vectors: (AaBb);α = Aa;αBb + AaBb;α (AaBb);α = Aa ;αBb + AaBb;α (AaBb);α = Aa ;αBb + AaBb ;α (AaBb);α = Aa;αBb + AaBb ;α ( F . 1 0 . 1 2 ) Example with a scalar function A and a vector B : (AB b);α = A;αBb + ABb;α = A,αBb + ABb;α // A;α = A,α, (F.9.1) (F.10.13) A more general example: (AabcBde);α ≡ Aabc ;α Bde + Aabc Bde;α ( F . 1 0 . 1 4 ) Theorem : Any object X--- for which (X---) ;α = 0 can be "extracted" from a covariant derivative group which contains X---. That is to say (F.10.15) (A---- X--- B-----) ;α = X--- (A---- B-----) ;α Proof : By the Leibniz Rule (F.10.1), (A---- X--- B-----) ;α = (A---- B-----) ;α X--- + (A---- B-----) (X---);α = X--- (A---- B-----) ; We shall now apply this th eorem in a few examples. Example 1: Any constant can be extracted from a group. (π A---- B-----) ;α = π (A---- B-----) ;α // extract a constant (F.10.16) (ε ijk A---- B-----) ;α = εijk (A---- B-----) ;α // εijk = element of perm. tensor = ±1 or 0 (F.10.17) Example 2: Since by (F.7.19) g ab;α = 0 one can always extract g ab from a group. This applies to gab as well. And of course it applies to g ab = δab since δab is a constant. (A---- B-- a--);α = (gab A---- B--b--);α = gab(A---- B--b--);α (F.10.18) Appendix F: Affine Connection 334 This says that lowering (or raising) an index "commutes" with covariant di fferentiation -- one can lower an index ignoring the fact that : α is sitting there. But we already know this must be true because we know that the object (A----B--b--);α is a true tensor, and g ab can lower any index on a true tensor. Fact : (|g|s);α = 0, s = real This is |g| raised to any real power s (s is not an index) (F.10.19) Proof : Recall from (D.2.3a) that |g'| -W/2 = J-W |g|-W/2 . // no R factors since scalar (D.2.3a) Setting s = -W/2 gives |g'| s = J2s |g|s so |g'|s transforms as a scalar density of weight W = -2s. Recall next the derivative rule for a scalar density of weight W, B;α = ∂αB + W Γκ κα B (F.9.23) which we apply with B = |g|s to find, (|g|s);α = ∂α|g|s -2s Γκ κα |g|s = ∂α|g|s -2s { (1/2)(1/|g|)∂ α|g| } |g|s // (F.4.2) for Γκ κα = s|g|s-1 ∂α|g| - s |g|s-1∂α|g| = 0 . QED Example 3: Since by (F.10.19) (|g|s);α = 0, |g|s may be extracted from a covariant derivative group, ( |g| s A---- B-----) ;α = |g|s (A---- B-----) ;α ( F . 1 0 . 2 0 ) Appendix G: Expansion of ( ∇v) 335 Appendix G: Expansion of ( ∇v) in curvilinear coordinates (v = vector) G.1 Continuum Mechanics motivation This Appendix assumes the usual curvili near coordinates context, Picture B (G.1.1) Although the polyadic notation is regarded as archaic by some writers (eg, Wolfram), it is well embedded into the literature of continuum mechanics, a field awash in tensors. In the literature one sometimes se es, in Cartesian coordinates, (∇ A)ij ≡ ∂jAi ≡ Ai,j ( G . 1 . 2 ) where the indices are the reverse of th e normal dyadic definition of (E.4.1), ( BA)ij ≡ BiAj . ( G . 1 . 3 ) For example, in continuum mechanic s one encounters the so-called convective or material derivative of an arbitrary vector field A(x,t) in the Eulerian or spatial "view" of the motion of a blob of continuous matter, DA i/Dt = ∂tAi + ∇Ai • v = ∂tAi + (∂jAi) vj = ∂tAi + (∇ A)ij vj = ∂tAi + [(∇A) v]i ⇒ D A/Dt = ∂tA + (∇ A) v // = ∂tA + (v•∇)A = (∂t + v•∇) A = D/Dt ( A). (G.1.4) DAi/Dt is a historical notation for the total derivative dA i(x,t)/dt. Here v(x,t) is the velocity field of the moving matter blob. An example is acceleration a, where A = v, a = Dv/Dt = ∂tv + (∇ v) v // = ∂tv + (v•∇)v = (∂t + v•∇) v = D/Dt ( v). (G.1.5) see Lai p 76 (3.4.3) and p 78 (3.4.8). The object (∇ v), called the velocity gradient, is of great interest in fluid mechanics. Correspondingly, the object ( ∇u) is of great interest in the theory of elastic solids, where u is the displacement field. The equation (G.1.4) written with proper index positions, DA i/Dt = ∂tAi + (∇ A)ij vj (G.1.4) Appendix G: Expansion of ( ∇v) 336 becomes a true tensor equation if we make th e assumption shown in the next Section that ( ∇A)ij ≡ Ai;j . In this case, we know from the covariance theore m exemplified in (E.9.13) that, for orthogonal coordinates, this equation w ill appear in terms of e ^n-expanded coefficients (like vj = h'jvj) as follows: D A'i/Dt = ∂tA'i + (∇ A)'ij v'j ( G . 1 . 6 ) where (∇A)'ij = [(∇A)(e^)]ij. To actually use equation (G.1.6) in some system of orthogonal curvilinear coordinates, one has to compute the coefficients [( ∇A)(e^)]ij. This is done below in Section G.5 generally and Section G.6 for a few specific coordinate systems (where A is called v) . G.2 Expansion of ∇v on ei⊗ej by Method 1: Use the fact that v b;a is a tensor. The covariant derivative v b;a is discussed in Sections F.7, F.8 and F.9. We shall define a true tensor object (∇v) as vb;a so that, according to (F.9.2), (∇v)ba ≡ vb;a = [vb,a – Γ c ab vc] // v b,a ≡ ∂avb x-space (∇v)'ba ≡ v'b;a = [v'b,a – Γ 'c ab v'c] . // v' b,a ≡ ∂'av'b x'-space (G.2.1) If x-space is Cartesian, then Γ = 0 as in (F.4.16), in which case (∇v)ba = vb,a = ∂avb ( G . 2 . 2 ) so (∇ v)ba = vb;a aligns with our dyadic ( ∇v) object (E.4.4) in Cartesian space. As shown in (E.2.14) one can expand the rank-2 tensor v b;a in either of these ways, ∇v = Σij vi;j ui⊗uj v i;j = [vi,j – Γc ij vc] = vi,j = ∂jvi ∇v = Σij v'i;j ei⊗ej v' i;j = [v'i,j – Γ ' c ij v'c] (G.2.3) where the ui are the Cartesian basis vectors in x-space, while ei are the reciprocal base vectors. According to (F.5.6) with Fig (F.5.2), the affine connection Γ ' in x'-space is given by Γ 'c ab = R'c n(∂'aR'bn). When x-space is Cartesian, R = 1 and then R' = R (F.5.3), so Γ 'c ab = Rc i(∂'aRbi) Γ 'c ij = Rc k(∂'jRik) . ( G . 2 . 4 ) Inserting this into the second expansion of (G.2.3) gives ∇ v = Σij v'i;j ei⊗ej v' i;j = [(∂'jv'i) – Rc k(∂'jRik) v'c] or ∇v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = [(∂'jv'i) – Rc k(∂'jRik) v'c] . (G.2.5) Appendix G: Expansion of ( ∇v) 337 Note that the expression shown contai ns only x'-space coordinates and objects. The (e) superscript tells us that this matrix element of operator ( ∇v) is taken in the en basis, see Section E.7: [(∇v)(e)]ij = <ei|∇v| ej> = ( ei)T (∇v) ej = ei • (∇v) ej = ei • (∇v) • ej (G.2.6) bra-ket matrix dot of vectors dyadic Recall now the third identity from (F.2.1), (∂ aRdn) = – Ren Rdm (∂aRe m) 3 . (F.2.1) This was derived for Picture C1 of (F.1.1) with x on the left and ξ on the right. Adjusting this to Picture B of (G.1.1) with x' on the left and x on the right gives the first line below, and then an index shuffle gives the second line, (∂'aRdn) = – Ren Rdm (∂'aRe m) // a →j, d→i, n→k (∂'jRik) = – Rek Rim (∂'jRe m) . (G.2.7) Then, R c k(∂'jRik) = – Rc k Rek Rim (∂'jRe m) = – δc e Rim (∂'jRe m) = –Rim(∂'jRc m) (G.2.8) so that (G.2.5) can be written in this way, ∇ v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = v'i;j = [(∂'jv'i) + Rim(∂'jRc m) v'c] . (G.2.9) We shall use this second form for [( ∇ v)(e)]ij below. Footnote : A variation of the above development would be to start this way, using the expansion of (E.2.12) on the left, and using (F.9.3) on the right, ∇ v = Σij vi ;j ui⊗uj vi ;j = [vi ,j + Γi jc vc] = vi ,j = ∂jvi ∇v = Σij v'i ;j ei⊗ej v'i ;j = [v'i ,j + Γ ' i jc v'c] (G.2.10) Eq (G.2.4) says Γ 'c ab = Rc i(∂'aRbi), but (F.1.10) gives the alternate form shown on the first line below, and then an index shuffle gives the second line, Γ 'c ab = – Rbi (∂'aRc i) // i→ m, then c→i, a→j, b→ c Γ ' i jc = – Rcm (∂'jRi m) . Then (G.2.10) says ∇v = Σij [(∇v)(e)]i j ei⊗ej [(∇v)(e)]i j = v'i ;j = [(∂'jv'i) – Rcm (∂'jRi m) v'c] (G.2.11) which is similar to (G.2.5) but applies to the down-tilt index configuration. Appendix G: Expansion of ( ∇v) 338 G.3 Expansion of ∇v on ei⊗ej by Method 2: Use brute force. Method 1 is perhaps elegant in that it makes use of tensor transformations and the affine connection Γ. But a simple brute force method is really just as simple and does not require knowledge of Γ and covariant differentiation. Instead of using the ei⊗ej notation for basis vectors, here we use the alternate notation ei(ej)T which works for rank-2 tensor expansions. Recall that ei(ej)T is an NxN matrix as illustrated in (E.5.3) for N = 2. In this brute force method, start with the Cartesian-space expansion, (∇ v) = Σcd(∇v)dc uducT = Σcd(∂cvd) uducT . // note that ud = ud in Cartesian space (G.3.1) This expansion is the second line of (E.2 .14) using the matrix notation (E.5.3), ud ⊗ uc = uducT . To express things in x'-coordinates, first write ∂ cvd = (Ri c∂'i)( Rj dv'j) = Ri c [(∂'iRj d) v'j + Rj d (∂'iv'j)] . (G.3.2) The next step is to express the matrix uducT as a linear combination of eeefT. We start with the orthogonality rule # 1 of (7.6.4) . δn i = Rbn Rb i // implied sum on b, as usual (7.6.4) We know that ( en)i = Rn i (7.18.1) ( un)i = δn i (7.18.3) so the above orthogonality rule says: ( un)i = Rbn (eb)i ⇒ un = Rbn eb o r ud = Red ee . (G.3.3) Taking the transpose of both sides (sum of column vectors to sum of row vectors), uTd = Red eTe ⇒ ucT = Rfc efT . (G.3.4) Combining these last two results gives uducT = Red Rfc ee efT = Red Rfc ee efT . (G.3.5) The expansion (G.3.1) of ( ∇v) may then be written, Appendix G: Expansion of ( ∇v) 339 (∇v) = (∇v)dc uducT = (∂cvd) uducT = Ri c [(∂'iRj d) v'j + Rj d (∂'iv'j)] { Red Rfc ee efT } // (G.3.2) and (G.3.5) = ( R fc Ri c)[Red (∂'iRj d) v'j + (Red Rj d)(∂'iv'j)] ee efT = δfi [Red (∂'iRj d) v'j + δej (∂'iv'j)] ee efT // orthogonality twice = [ R ed (∂'fRj d) v'j + (∂ 'fv'e)] ee efT = Σef [(∇v)(e)]ef ee efT ( G . 3 . 6 ) where [(∇v)(e)]ef = (∂'fv'e) + Red (∂'fRj d) v'j ( G . 3 . 7 ) or [(∇ v)(e)]ij = (∂'jv'i) + Rim (∂'jRj m) v'j ( G . 3 . 8 ) This agrees with (G.2.9) obtain ed by the previous method. G.4 Expansion on e i⊗ej and e ^i⊗e^j Reversing the index tilts in (G.2.5) [as justified in (E.2.12)] one gets ∇ v = Σij v'i;j ei⊗ej where v'i;j = g'iag'jb v'a;b = g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] . // (G.2.9) (G.4.1) Then since ei = h'ie^i one gets yet another expansion (m ore generally see Section E.8 ), ∇v = Σij (v'i;j h'i h'j) e^i⊗e^j = Σij [(∇v)(e^)]ij e^i⊗e^j where [( ∇v)(e^)]ij = h'i h'j v'i;j = h'i h'j g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] . (G.4.2) Here is a summary of results so far: ∇ v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = [(∂'jv'i) + Rim(∂'jRc m) v'c] (G.2.9) ∇ v = Σij [(∇v)(e)]i j ei⊗ej [(∇v)(e)]i j = [(∂'jv'i) – Rcm (∂'jRi m) v'c] . (G.2.11) ∇ v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] (G.4.1) ∇v = Σij [(∇v)(e^)]ij e^i⊗e^j [(∇v)(e^)]ij = h' h'j g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] (G.4.2) (G.4.3) Appendix G: Expansion of ( ∇v) 340 One could replace v' a = g'adv'd in any of the above results. For example, the last object becomes [(∇v)(e^)]ij = h'i h'j g'iag'jb [(∂'b[g'adv'd]) + Ram(∂'bRc m) (g'cdv'd)] . (G.4.4) Another choice is to use the v'n components of v obtained when expanding v on the e^n , v = Σn v'n en = Σn v'n (h'ne^n) = Σn (h'n v'n) e^n = Σn v'n e^n ⇒ v'n ≡ h'n v'n (G.4.5) so one then replaces v'n = h'n-1v'n. The same object above then becomes [(∇v)(e^)]ij = h'i h'j g'iag'jb [(∂'b[g'ad h'd-1v'd]) + Ram(∂'bRc m) (g'cd h'd-1v'd)] . (G.4.6) As discussed in Section 14.7, the components v'n are convenient since they all have the same dimensions. Moreover, when a specific curvilinear system is selected, one can dispen se with the unpleasant font used in v'n and just write v'n = vx'(n) . For example, in spherical coordinates r, θ,φ : v'1 = vr v'2 = vθ v'3 = vφ v = Σn v'n e^n = vrr^ + vθθ^ + vφ φ^ . (G.4.7) In Chapter 14 the scripted variables have no primes because Picture (14.1.1) is being used instead of Picture B. G.5 Orthogonal coordinate systems In this case g' ab = h'a2δa,b and g'ab = h'a-2δa,b and things simplify. The four expansions (G.4.3) become (there is no change in th e first two expansions) ∇v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = [(∂'jv'i) + Rim(∂'jRc m) v'c] ∇v = Σij [(∇v)(e)]i j ei⊗ej [(∇v)(e)]i j = [(∂'jv'i) – Rcm (∂'jRi m) v'c] . ∇v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = h'i-2 h'j-2 [(∂'jv'i) + Rim(∂'jRc m) v'c] ∇v = Σij [(∇v)(e^)]ij e^i⊗e^j [(∇v)(e^)]ij = h'i-1 h'j-1 [(∂'jv'i) + Rim(∂'jRc m) v'c] (G.5.1) The (G.4.6) version of [(∇ v)(e^)]ij becomes, Pij ≡ [(∇v)(e^)]ij = h'i-1 h'j-1 [(∂'j[h'iv'i]) + Rim(∂'jRc m) (h'cv'c)] = h'i-1 h'j-1 [(∂'j[h'iv'i]) + Rid(∂'jRb d) (h'bv'b)] // m →d,c→b = h'i-1 h'j-1 [ (∂'jh'i) v'i + h'i(∂'jv'i) + h'i2Ri d(∂'jRb d) (h'bv'b)] (G.5.2) T2 T3 T1 where R id = g'iaRa b gbd = h'i2Ri d is used to put all R's into their down-tilt form. Appendix G: Expansion of ( ∇v) 341 G.6 Maple evaluation of ( ∇v) in several coordinate systems This object Pij shown above can be computed in Maple by a simple program which is easily modified for other orthogonal curvilinear systems. The first task is to obtain the R matrix from the inverse transformation equations x = F-1(x'), where we assume spherical coordina tes as an example. In this code, N = 3 and p means prime, as in hp = h'. (G.6.1) Then one needs the scale factors h n' from the metric tensor g ¯' = STS in (5.7.9) [developmental notation], Appendix G: Expansion of ( ∇v) 342 (G.6.2) The terms T1,T2 and T3 shown in (G.5.2) are then entered, P ij ≡ [(∇v)(e^)]ij = polh'i-1 h'j-1 [ (∂'jh'i) v'i + h'i(∂'jv'i) + h'i2Ri d(∂'jRb d) (h'bv'b)] (G.5.2) T2 T3 T1 (G.6.3) The terms are then added and simplified and out pops the result, (G.6.4) Below are some sample results (including the above): Appendix G: Expansion of ( ∇v) 343 (∇v) = Σij Pij e^i e^jT Pij = [(∇v)(e^)]ij (G.6.5) • Pij in polar coordinates (where 1,2 = r, θ) : // agrees with Lai p 57 (2.33.23) (G.6.6) • P ij in cylindrical coordinates (where 1,2,3 = r, θ,z) : // agrees with Lai p 60 (2.34.5) (G.6.7) The polar coordinates results are seen to be the upper-left 2x2 piece of the cylindrical results. • Pij in spherical coordinates (where 1,2,3 = r, θ,φ) : // agrees with Lai p 64 (2.35.25) (G.6.8) By entering the usual inverse transformation equations x' = F-1(x), and with suitable small alterations, the above Maple code can compute ∇v in any orthogonal curvilinear coor dinate system in any number of dimensions N. Appendix H: Expansion of div(T) 344 Appendix H: Expansion of div(T) in curvilinear coordinates (T = rank-2 tensor) H.1 Introduction The object of attention in this Appendix, divT , is expressed this way in Cartesian coordinates, (divT)i = ∂jTij , ( H . 1 . 1 ) where T ij is a rank-2 tensor. One can regard the above equation as describing the normal divergence of the "vector" which forms the ith row of the matrix T ij. In general (that is to say, under general transformations F), the rows of T ij are not tensorial vectors, and that is why (divT)i is not a tensorial scalar. Nor, in fact, are the (divT)i as defined above the components of a tensorial vector! As shown in Section H.3 below, divT will be redefined as a true tensorial vector which equals ∂jTij in Cartesian coordinates. H.2 Continuum Mechanics motivation The vector divT arises for example when Newton's 2nd Law F = ma is applied to a particle of continuous matter, in which case this law is know n as Cauchy's Equation of Motion, divT + ρB = ρa // Lai p 169 (4.7.4) (H.2.1) In this equation T is known as the Cauchy stress tensor, ρ is the mass density of the particle, B is any action-at-a-distance force (body force) per unit mass (such as gravity), and of course a is the acceleration of the particle. Under rotations and translations the quantity (divT) i = ∂jTij is a tensorial vector, ρ is a tensorial scalar, and a and B are tensorial vectors. As one would expect, divT + ρB = ρa is covariant in the sense of Section 7.15 under these kinds of transformations. More generally, if (divT)a is taken to mean Tab ;b as shown in the next Section. then (H.2.1) is a true tensor equation under any transformation x' = F(x). Then as shown in (E.9.13), equation (H.2.1) stated in terms of e ^n-expanded coefficients takes the form ( divT)' + ρB' = ρa' ( divT)'i = [(divT)(e)]i (H.2.2) where en are the tangent base vectors for some system of orthogonal curvilinear coordinates. One again, in order to use this equation, one must compute the coefficients [(divT)(e)]i and this task is done below. The result is rather complicated, as shown in (H.5.7). For spherical coordinates, the three components of (divT)'i = [(divT)(e)]i are displayed in (H.6.3), where each comp onent is seen to have seven terms. Appendix H: Expansion of div(T) 345 H.3 Expansion of divT on e n by Method 1: Use fact that Tab ;α is a tensor. As shown in (F.9.9), the following object transforms as a rank-3 tensor, T ab ;α ≡ ∂αTab + Γa αn Tnb + Γb αn Tan = ∂α Tab // x-space [g=1, Γ = 0 (F.4.16) ] T 'ab ;α ≡ ∂'αT 'ab + Γ 'a αn T 'nb + Γ 'b αn T 'an // x'-space (H.3.1) Contracting b with α yields the following tensorial vector equations, (divT)a = Tab ;b = ∂bTab // x-space , Γ = 0 (divT)'a = T 'ab ;b ≡ ∂'bT 'ab + Γ 'a bn T 'nb + Γ 'b bn T 'an // x'-space (H.3.2) Note that the Cartesian space statement (divT) a = ∂bTab is obtained, as noted in the opening comments above. As in (E.2.10), the vector divT can be expanded as divT = Σa(divT)'a ea (divT)'a = [(divT)(e)]a // divT in the en basis (H.3.3) Recall from (F.1.10) that Γ c ab = Rc i (∂aRbi) // for Picture C1 in (F.1.1) Γc ab = – Rbi (∂aRc i) . (F.1.10) Adjusting from Picture C1 to Picture A, the last equation above becomes the first one below. We then shuffle indices to get the second line, and contract a with b to get the third: Γ 'c ab = – Rbi (∂'aRc i) // do b →n then a→b then c→a Γ 'a bn = – Rni (∂'bRa i) // next, contract indices a and b Γ 'b bn = – Rni (∂'bRb i) . ( H . 3 . 4 ) Installing the last two lines into th e second line of (H.3.2) then gives (divT)'a = ∂'bT 'ab + Γ 'a bn T 'nb + Γ 'b bn T 'an = ∂' bT 'ab – Rni (∂'bRa i) T 'nb – Rni (∂'bRb i)T 'an . (H.3.5) The conclusion is that Appendix H: Expansion of div(T) 346 divT = Σa[(divT)(e)]a ea ( H . 3 . 6 ) [(divT)(e)]a = ∂'b T 'ab – Rni(∂'bRa i) T 'nb – Rni (∂'bRb i)T 'an // sum on n and b The vector divT = Σa(∂bTab)ua has thus been expressed in terms of x'-space coordinates and objects, and as usual the en are the tangent base vectors in x-space. H.4 Expansion of divT on e n by Method 2: Use brute force. Start with the known expansion of divT in Cartesian x-space, as in (E.2.10), divT = Σi [divT]i ui = Σi (∂jTij) ui . (H.4.1) Compute (∂jTij) = (Ra j∂'a)( Rb'i Rc'j T 'b'c') = R a j Rb'i Rc'j (∂'a T 'b'c') + Ra j Rb'i (∂'a Rc'j) T 'b'c' + Ra j Rc'j (∂'a Rb'i) T 'b'c' = ( R a j Rc'j) Rb'i (∂'a T 'b'c') + Ra j Rb'i (∂'a Rc'j) T 'b'c' + (Ra j Rc'j) (∂'a Rb'i) T 'b'c' = δa c' Rb'i (∂'a T 'b'c') + Ra j Rb'i (∂'a Rc'j) T 'b'c' + δa c' (∂'a Rb'i) T 'b'c' = R b'i (∂'a T 'b'a) + Ra j Rb'i (∂'a Rc'j) T 'b'c' + (∂'a Rb'i) T 'b'a . (H.4.2) In (G.3.3) it was shown that un = ΣbRbn eb. This is valid with indices tilte d the other way, as the reader can show mimicking the derivation of (G.3.3) or by raising and lowering labels using (7.18.1) en = g'ni ei and (7.18.3) un = gni ui and (7.5.9) R ab = g'aa'Ra' b' gb'b . The result is ui = Σe Rn i en . ( H . 4 . 3 ) Inserting (H.4.2) and (H.4.3) in to (H.4.1) gives, twice using orthogonality rules (7.6.4), divT = Σi (∂jTij) ui = { ( R n i Rb'i) (∂'a T 'b'a) + Ra j (Rn i Rb'i) (∂'a Rc'j) T 'b'c' + Rn i (∂'a Rb'i) T 'b'a} en = { δn b' (∂'a T 'b'a) + Ra j δn b' (∂'a Rc'j) T 'b'c' + Rn i (∂'a Rb'i) T 'b'a} en = { ( ∂'a T 'na) + Ra j (∂'a Rc'j) T 'nc' + Rn i (∂'a Rb'i) T 'b'a} en = Σ n[(divT)(e)]n en ( H . 4 . 4 ) where Appendix H: Expansion of div(T) 347 [(divT)(e)]n = (∂'a T 'na) + Ra j (∂'a Rc'j) T 'nc' + Rn i (∂'a Rb'i) T 'b'a . (H.4.5) The third identity in (F.2.1) says (∂ aRdn) = – Ren Rdm (∂aRe m) 3 . // for Picture C1 of (F.1.1) (F.2.1) Adjusted to Picture C1' of (F.1.4) this becomes the first line below, th en the next two lines are obtained by index shuffles, (∂' aRdn) = – R en Rdm (∂'aRe m) // do d →c', n→j (∂'aRc'j) = – Rej Rc'm (∂'aRe m) // do j → i (∂'aRb'i) = – Rei Rb'm (∂'aRe m) . ( H . 4 . 6 ) Inserting these last two lines into (H.4.5) gives, [(divT)(e)]n = (∂'a T 'na) + Ra j (∂'a Rc'j) T 'nc' + Rn i (∂'a Rb'i) T 'b'a (H.4.5) = ( ∂' a T 'na) – Ra j Rej Rc'm (∂'aRe m) T 'nc' – Rn i Rei Rb'm (∂'aRe m) T 'b'a = ( ∂' a T 'na) – δa eRc'm (∂'aRe m) T 'nc' – δn e Rb'm (∂'aRe m) T 'b'a // orthog. = ( ∂' a T 'na) – Rc'm (∂'aRa m) T 'nc' – Rb'm (∂'aRn m) T 'b'a . (H.4.7) Reverse the order of the last two terms, [(divT) (e)]n = (∂'a T 'na) – Rb'm (∂'aRn m) T 'b'a – Rc'm (∂'aRa m) T 'nc' . (H.4.8) Replace summation indices m →i, a→b, = ( ∂'b T 'nb) – Rb'i (∂'bRn i) T 'b'b – Rc'i (∂'bRb i) T 'nc' . (H.4.9) Finally, change n →a and then c' → n and b'→n, [(divT)(e)]a = (∂ 'b T 'ab) – Rni (∂'bRa i) T 'nb – Rni (∂'bRb i) T 'an . // sum on n and b (H.4.10) This is seen to match the result (H.3.6) of the previo us Section. The brute force method is in fact not too bad and requires no explicit us e of the affine connection Γ. Technical Note: In the above expression one can write for example R ni(∂'bRa i) = g'nmRm i(∂'bRa i). Then using the fact that Rm iRa i = g'ma one finds Rm i(∂'bRa i) + Ra i(∂'bRm i) = (∂'bg'ma) which then allows the replacement R ni(∂'bRa i) = g'nm (∂'b g'ma) – g'nm Ra i(∂'bRm i). This kind of transformation leads to an alternate form for divT , still with all down-tilt R matrices, but the index structure is different. This other form appears when one does the "brute force" method starting with ( ∂jTij) instead of ( ∂jTij). Of course in Cartesian space these two objects must be the same. Appendix H: Expansion of div(T) 348 H.5 Adjustment for T expanded on ( e^i⊗e^j) and divT expanded on e^a In the above, it has been assumed that Tij is a rank-2 tensor so that, in the notation of (E.2.11), T = ΣijTij(ui⊗uj) = ΣijT 'ij(ei⊗ej) . ( H . 5 . 1 ) If one is interested in an expansion of T on the unit vectors e^i where ei = h'i e^i this becomes T = Σ ij[T 'ijh'ih'j] (e^i⊗e^j) = Σij[T(e^)]ij (e^i⊗e^j) (H.5.2) One then has [T (e^)]ij = h'ih'jT 'ij ⇒ T 'ij = h'i-1 h'j-1 [T(e^)]ij . (H.5.3) If one is interested in this form of the T matrix el ements, then one is likely also interested in this expansion for divT , divT = Σa[(divT)(e)]a ea = Σn { [(divT)(e)]a h'a } e^a ≡ [(divT)(e^)]a e^a (H.5.4) where then [(divT) (e^)]a = h'a[(divT)(e)]a = h'a [(∂'b T 'ab) – Rni (∂'bRa i) T 'nb – Rni (∂'bRb i) T 'an] // from (H.4.10) = h'a * { ∂ 'b (h'a-1 h'b-1 [T(e^)]ab) – Rni (∂'bRa i) h'n-1 h'b-1 [T(e^)]nb – R ni (∂'bRb i) h'a-1 h'n-1 [T(e^)]an } // from (H.5.3) = h' a * { ∂ 'b (h'a-1 h'b-1) [T(e^)]ab – Rni (∂'bRa i) h'n-1 h'b-1 [T(e^)]nb // ∂(xy) = dx y + x dy + h' a-1 h'b-1 (∂'b [T(e^)]ab) – Rni (∂'bRb i) h'a-1 h'n-1 [T(e^)]an } . ( H . 5 . 5 ) Now ∂' b (h'a-1 h'b-1) = ∂ 'b(h'a h'b)-1 = – (h'a h'b)-2 ∂'b(h'a h'b) = – h'a-2 h'b-2 ∂'b(h'a h'b) so the above sequence for [(divT)(e^)]a continues, = h' a * { – h' a-2 h'b-2 ∂'b(h'a h'b) [T(e^)]ab – Rni (∂'bRa i) h'n-1 h'b-1 [T(e^)]nb + h' a-1 h'b-1 (∂'b [T(e^)]ab) – R ni (∂'bRb i) h'a-1 h'n-1 [T(e^)]an } = { – h' a-1 h'b-2 ∂'b(h'a h'b) [T(e^)]ab – h'a h'n-1 h'b-1 g'nm Rm i (∂'bRa i) [T(e^)]nb + h' b-1 (∂'b [T(e^)]ab) – h' n-1 g'nm Rm i (∂'bRb i) [T(e^)]an } (H.5.6) Appendix H: Expansion of div(T) 349 where recall that Rn i = Rni since x-space is Cartesian. In the last form only the down-tilt R matrix appears which simplifies calculation with Maple. This and all other results a bove are valid for general curvilinear coordinates, orthogona l as well as non-orthogonal. At this point we specialize to orthogonal systems, so (H.5.6) becomes T1 T3 [(divT)(e^)]a = { – h' a-1 h'b-2 ∂'b(h'a h'b) [T(e^)]ab – h'a h'b-1 h'n Rn i (∂'bRa i) [T(e^)]nb + h' b-1 (∂'b [T(e^)]ab) – h' n Rn i (∂'bRb i) [T(e^)]an } T2 T4 (H.5.7) Even for an orthogonal curvilinear coordinate system , the form of this tensor divergence is amazingly complicated. Here it is expressed in terms of the dow n-tilt R matrix and the scale factors, and as usual repeated indices are summed. H.6 Maple: divT in cylindrical and spherical coordinates The above object [(divT)(e^)]a can be evaluated by Maple code very similar to that shown in Appendix G for (∇ v). The main difference is the set of entry lines for the terms, (H.6.1) Appendix H: Expansion of div(T) 350 Here are some results for [(divT)(e^)]a = "divT a" : • cylindrical coordinates (where 1,2,3 = r, θ,z) : // agrees with Lai p 60 (2.34.8,9,10) ( H . 6 . 2 ) • For polar coordinates (divT) r,θ and are given by the first two lines above with the ∂ z terms set to 0. These polar results then agree with Lai p 58 (2.33.32,33). • spherical coordinates (where 1,2,3 = r,θ ,φ) : (H.6.3) The expressions above agree with Lai p 65 (2.35.33,34,35). Appendix I: Expansion of ( @B) 351 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates I.1 Introduction This appendix assumes the usual curvilinear coordinates context, Picture B, . (I.1.1) In this Section the vector Laplacian is computed in two different ways, each associated with a particular "tensorization" of its Cartesian form. The second method, though less pleasant than the first, gives insight into why the vector Laplacian always includes the scalar Laplacian of the field components. It is in this inclusive form that results are us ually stated in the literature. In passing, it should be noted that the vector Laplacia n is not just an idle mathematical curiosity. It shows up for example in the wave equations for electric and magnetic fields in a vacuum, (∇ 2 + k2)E(x) = 0 (∇2 + k2)B(x) = 0 k = ω/c E,B(x,t) = E,B(x)e-iωt (I.1.2) In continuum mechanics, it appears for example in the Navier/Cauchy equation which describes the small vector displacement field u in an isotropic elastic solid, ρo∂t2u = ρ0B + (λ+μ)∇e + μ ∇2u e = div u = dilatation // Lai p 216 (5.6.9) (I.1.3) or ρ o∂t2u = ρ0B + (λ+μ) (grad div u) + μ ∇2u Here ρ0 is the unperturbed mass density, B the body force, and λ and μ are the two Lamé constants which describe an isotropic elastic medium. The vector Laplacian makes another appearance in the better-known Navier-Stokes equation which descri bes the vector velocity field v in an incompressible Newtonian fluid, ρ [ ∂tv + (∇ v)v] = ρ B - ∇p + μ∇2v // Lai p 361 (6.7.6) (I.1.4) where ρ is the mass density and p is pressure. As in previous Appendices, for orthogonal curvilinear coordinates and based on (E.9.13) we can express the above equations in primed scripted e^n-expanded coefficients, (∇2B)' + k2B' = 0 // vector equation (Helmholtz) (∇2B)'i + k2B'i = 0 // components (I.1.5) ρ o∂t2u' = ρ0B' + (λ+μ) (grad div u )' + μ (∇2u)' // vector equation (Navier/Cauchy) ρo∂t2u'i = ρ0B'i + (λ+μ) (grad div u )'i + μ (∇2u)'i // components (I.1.6) Appendix I: Expansion of ( @B) 352 ρ [ ∂tv' + (∇ v)'v'] = ρB' - (∇p)' + μ(∇2v)' // vector equation (Navier-Stokes) ρ [ ∂tv'i + (∇ v)'i jv'j] = ρB'i - (∇p)'i + μ(∇2v)'i // components (I.1.7) In our non-script (e^)notation the objects appearing in the above equations may be written u'i = [u(e^)]i = h'iu'i // similarly for other vectors (∇ p)'i = [(∇p)(e^)]i = h'i ∂'ip'(x') where p'( x') = p( x) = scalar field, see (10.1.13) ( grad div u )'i = h'n ∂'n{ (1/ g' ) ∂'i (g' u 'i/h'i)} // see G in (13.1.17) (∇2B)'i = [(∇2B)(e^)]i = [(@B)(e^)]i = h'i [(@B)(e)]i = h'i (@B)'i = (@B)'i = treated in this Appendix. See (I.2.10) for ( @B)'i . (I.1.8) As noted in (E.8.5), the primes on scripted variables correspond to the primes on the names of the curvilinear coordinates which are x' i for Picture B. In Picture M&S of (14.1.1) the primes on scripted variables go away, but here we are using Picture B. I.2 Method 1 : a review In (13.3.4) it is shown that , in Cartesian coordinates, ∇ 2(Bn) = [ grad(div B) – curl(curl B) ]n ∇2 = ∂j∂j or ∇•∇ (B n) = [∇(∇• B) – ∇ x (∇ x B) ]n . ( I . 2 . 1 ) When all components are considered in a single equation, one could write ∇2(B) = grad(div B) – curl(curl B) or ∇•∇ (B) = ∇(∇• B) – ∇ x (∇ x B) ( I . 2 . 2 ) and this is frequently done (see examples cited in the previous Section). Since ∇2(B) ≠ ∂j∂j B in curvilinear coordinates, it seems notionally safer in our current context to use a different symbol for the vector Laplacian operator, and following Moon and Spencer we use @ so the second last equation above then says: @B = grad(div B) – curl (curl B) . ( I . 2 . 3 ) Since [@B]n agrees with ∇2(Bn) in Cartesian coordinates, and since we know how to write div, grad and curl in curvilinear coordinates (Chapters 9,10,12 or Chapter 15), the right hand side of the above equation Appendix I: Expansion of ( @B) 353 provides our "first method" of writing @B in curvilinear coordinates. In (15.7.8) it is shown that the proper "tensorization" of the above equation is given by (@B)n = (Bj ;j);n – g-1/2εnab(g-1/2εbdeBe;d);a @B = (@B)n un (I.2.4) and therefore, in x'-space ( x' are the curvilinear coordinates of interest), (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2ε'bdeB'e;d);a @B = (@B)'n en (I.2.5) where (B' j ;j) = [div B]' = [div B] (a scalar). The object @B is a normal vector (weight 0), assuming B is a normal vector. Doing various simplifying steps, we then arrive at the following expression (15.7.18) which lends itself to calculation: (@B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i) } @B = (@B)'n en – (1/ g' ) ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'a[g'bfB'f]) } (I.2.6) where ε'ncd is the usual permutation tensor ( ε'ncd = εncd). In this notation, B'i is an official x'-space contravariant vector component. We are often interested in working with vectors which are expanded on unit vector versions of the tangent base vectors. In such a unit vector expansi on of a vector, a script font has been used for the components. One has, B'n = B'n/h'n en = h'n e^n B = B'nen = B'ne^n h' n = scale factor (I.2.7) which leads to this rewrite of (I.2.6), (@B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i/h'i)} @B = (@B)'n en (I.2.8) – (1/ g' ) ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'a[g'bfB'f/h'f]) } or, as we will use it with unit vectors, (@B)'n = h'n ∂'n{(1/ g' ) ∂'i(g' B'i/h'i)} @B = (@B)'n e^n (I.2.9) – h' n (1/ g' ) ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'a[g'bfB'f/h'f]) } . Specializing to orthogonal coordinates yields the form we shall use below for computation in Maple, T1 (@B)'n = (1/h'n) ∂'n{(1/ g' ) ∂'i(g' B'i/h'i)} @B = (@B)'n e^n (I.2.10) – ( h ' n/g' ) ε'ncd ε'dab ∂'c { (1/g' ) h'd2(∂'a[h'bB'b]) } . T2 The second term could be written as two terms by reducing the product ε'ncd ε'dab into δδ - δδ in the usual manner, but Maple is happy to just "do it" as stat ed. And for non-orthogonal coordinates, the reduction of ε'ncd ε'eab is much less friendly (see (D.10.18)) and then one would want to use the εε product as is. Appendix I: Expansion of ( @B) 354 I.3 Method 1 for spherical coordinates: Maple speaks In this Section, the following notation is used: B'1 = Br B'2 = Bθ B'3 = Bφ B = B'ne^n = Bre^r + Bθe^θ + Bφe^φ = Brr^ + Bθθ^ + Bφφ^ . (I.3.1) Maple begins in the same manner as shown earlier in (G.6.1), the idea being that this code could be modified for any coordinate system, Appendix I: Expansion of ( @B) 355 (I.3.2) The last object is g' , where hp[n] = h' n . The next chunk of code computes the scalar Laplacian of an unspecified function f, and this expression will be used below in parsing the vector Laplacian results. The Laplacian expression comes from (11.6) with g'nm = (1/h'n2) δn,m : (I.3.3) The subs command used here (and more intensely below) removes the arguments of the function f for purely cosmetic reasons (see our Maple User's Guid e for details, operands section). The arguments are added in the first place to prevent Maple from thi nking the unspecified function f is a constant. Next comes a low-budget implementation of the permutation tensor eps(a,b,c) = ε abc, Appendix I: Expansion of ( @B) 356 (I.3.4) The two terms of ( @B)'n in (I.2.10) are then duly entered, along with their sum "starB" , ( I . 3 . 5 ) In the following code, we use this notation, similar to (I.3.1) for vector B, q1_ = ( @B)'1 = (@B)r q2_ = ( @B)'2 = (@B)θ q3_ = ( @B)'3 = (@B)φ ( I . 3 . 6 ) and in this notation Maple computes the components of the vector Laplacian : Appendix I: Expansion of ( @B) 357 ( I . 3 . 7 ) See comment below (I.3.3) about the Maple subs commands (purely cosmetic). I.4 Method 1 for spherical coordinates: putting results in traditional form It turns out that in each component of the vector Laplacian stated above, 5 of the 9 terms can be represented as if they were the scalar Laplacian act ing on the component in question. Here is how it works, where we now use the well-known spherical c oordinates Laplacian (which Maple has computed using (11.12) with 1,2,3 = r,θ .φ and h 1 = 1, h2 = r, h3 = rsinθ as shown above in (I.3.1) ), ∇2f = (1/r2)∂r(r2∂rf) + (1/r2sinθ)∂θ(sinθ∂θf) + (1/r2sin2θ)∂φ2f = 1 + 2 3+4 5 1 2 3 4 5 (I.4.1) Appendix I: Expansion of ( @B) 358 1 2 5 3 4 ( I . 4 . 2 ) Therefore ( @B)r = ∇2(Br) - (2/r2)Br - (2/r2)cot(θ )Bθ - (2/r2)∂θBθ -(2/r2sinθ) ∂φBφ = ∇ 2(Br) – (2/r2) [ Br + cotθ Bθ + ∂θBθ + cscθ ∂φBφ ] (I.4.3) 3 4 5 1 2 ( I . 4 . 4 ) Therefore (@B)θ = ∇2(Bθ) - (1/r2) [ - 2 ∂ θBr + cot2θ Bθ + Bθ + 2 cotθ cscθ∂φBφ ] = ∇ 2(Bθ) - (1/r2) [csc2θ Bθ - 2 ∂θBr + 2 cotθcscθ∂φBφ ] (I.4.5) 5 3 4 1 2 ( I . 4 . 6 ) (@B)φ = ∇2(Bφ) - (1/r2) [csc2θBφ - 2cscθ∂φBr - 2cotθ cscθ∂φBθ ] (I.4.7) Each component is seen to have nine terms. In this manner, we end up with the components of the vector Laplacian expressed in the traditional manner, ( @B)r = ∇2(Br) – (2/r2) [ Br + cotθ Bθ + ∂θBθ + cscθ ∂φBφ ] (@B)θ = ∇2(Bθ) – (1/r2) [csc2θ Bθ – 2 ∂θBr + 2cot θcscθ∂φBφ ] (@B)φ = ∇2(Bφ) – (1/r2) [csc2θ Bφ – 2cscθ∂φBr – 2cotθcscθ∂φBθ ] Appendix I: Expansion of ( @B) 359 where ∇2f = (1/r2)∂r(r2∂rf) + (1/r2sinθ)∂θ(sinθ∂θf) + (1/r2sin2θ)∂φ2f = (2/r) ∂rf + ∂r2f + (cotθ /r2)∂θf + (1/r2) ∂θ2f + (1/r2sin2θ)∂φ2f (I.4.8) The curious reader might wonder why, in each case, five of the nine terms of the vector Laplacian components can be represented by the scalar La placian acting on the component. This question is answered in the following Section. I.5 Method 2, Part A In the Method 1, described in Section I.2 above, we used this tensorization of the vector Laplacian, ( @B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a . (I.2.5) (I.5.1) An alternative tensorization is stated in (15.8.2), (@B)'n = B'n;j ;j . ( I . 5 . 2 ) These two tensors must be the same since the tensoriza tion of a Cartesian form equation is unique and this fact is verified in Section 7.8. Our Method 2 is to use this B'n;j ;j tensorization to compute once again the components of the vector Laplacian. Recall covariant derivative example (F.9.9), B ab ;α ≡ ∂α Bab + Γa αn Bnb + Γb αn Ban (F.9.9) Since B a;b is a rank-2 tensor one can apply the above example to Ba;b in place of Bab to get a result of identical form. The second line below is the desired index contraction (I.5.2) with a →n and b=α → j : B a;b ;α ≡ ∂α Ba;b + Γa αk Bk;b + Γb αk Ba;k ( I . 5 . 3 ) Bn;j ;j ≡ ∂j Bn;j + Γn jk Bk;j + Γj jk Bn;k // Γ'j jk = (1/ g ) ∂k(g ) as in (F.4.2) . Next, recall covariant derivative example (F.9.5), with a whole sequence of index shuffles following Ba;α = ∂α Ba + gαβΓa βn Bn // do β→b and n→s (F.9.5) Ba;α = ∂α Ba + gαbΓa bs Bs // do n→ a and α→j Bn;j = ∂j Bn + gjbΓn bs Bs // do n→ k Bk;j = ∂j Bk + gjbΓk bs Bs // do k→ n and j→k Bn;k = ∂k Bn + gkbΓn bs Bs ( I . 5 . 4 ) Insert the last two of these expressions in to the second line of (I.5.3) to get, Bn;j ;j = ∂j[∂j Bn + gjbΓn bs Bs] + Γn jk[∂j Bk + gjbΓk bs Bs] + Γj jk[∂k Bn + gkbΓn bs Bs] . (I.5.5) Appendix I: Expansion of ( @B) 360 Combine the second last term with the first, B n;j ;j = [∂j ∂j Bn + Γj jk(∂kBn )] + ∂j(gjbΓn bs Bs) + Γn jk[∂j Bk + gjbΓk bs Bs] + Γj jk[gkbΓn bs Bs] . (I.5.6) Since no terms have been dropped, we are still "cova riant" and in x'-space everything gets primed, B'n;j ;j = [∂'j ∂'j B'n + Γ 'j jk(∂'kB'n )] + ∂'j(g'jbΓ 'n bs B's) + Γ 'n jk[∂'j B'k + g'jbΓ 'k bs B's] + Γ 'j jk[g'kbΓ 'n bs B's] (I.5.7) Using Γ'j jk = (1/ g' ) ∂'k(g' ) from (F.4.2), the first two terms of (I.5.7) can be written this way, [∂'j ∂'j B'n + Γ 'j jk(∂'kB'n )] = [∂'j ∂'j B'n + (1/ g' ) ∂'k(g' ) (∂'kB'n )] = lap(B'n) (I.5.8) and we recognize this as being lap (B'n) based on (15.5.6), lap f = ∂'n∂'nf ' + (1/ g' ) ∂ 'n(g' ) (∂'nf ') = [1/ g' ] ∂ 'n [g' (∂'nf ')] . (15.5.6) (I.5.9) Therefore (I.5.7) can be written as, B'n;j ;j = lap(B'n) + ∂'j(g'jbΓ 'n bs B's) + Γ 'n jk[∂'j B'k + g'jbΓ 'k bs B's] + Γ 'j jk[g'kbΓ 'n bs B's] = lap(B' n) + E x t r a T e r m s ( I . 5 . 1 0 ) So we see why the scalar Laplacian of B' n appears in ( @B)n. Basically lap(B'n) is the B'n,j ,j term within B'n;j ;j , the only term that survives if Γ = 0. I.6 Method 2, Part B Unfortunately, we don't want to see lap(B'n) in (I.5.10), we want to see lap( B'n)! Consider then this rewrite of (I.5.9), using (I.2.7) that B'n = B'n/h'n, lap (B'n) = lap ( B'n/h'n) = [∂'j ∂'j (B''n/h'n) + (1/ g' ) ∂'k(g' ) (∂'k (B''n/h'n) )] . (I.6.1) One computes the pieces as follows: ∂' j ∂'j (B'n/h'n) = ∂ 'j ∂'j (B'nh'-1 n) = ∂'j [(∂'jB'n) h'-1 n + B'n(∂'j h'-1 n) ] = ( ∂'j∂'jB'n)h'-1 n + (∂ 'jB'n) (∂'j h'-1 n) + (∂'j B'n)(∂'j h'-1 n) + B'n (∂'j ∂'j h'-1 n) = ( ∂'j∂'jB'n) h'-1 n + 2(∂'jB'n) (∂'j h'-1 n) + B'n (∂'j ∂'j h'-1 n) (I.6.2) ∂' k (B'n/h'n) = [(∂'kB'n) h'-1 n + B'n(∂'k h'-1 n) ] (I.6.3) so that (I.6.1) becomes, Appendix I: Expansion of ( @B) 361 lap (B'n) = (∂'j∂'jB'n) h'-1 n + 2(∂'jB'n) (∂'j h'-1 n) + B'n (∂'j ∂'j h'-1 n) + ( 1 / g' ) ∂ 'k(g' )[(∂'kB'n) h'-1 n + B'n(∂'k h'-1 n) ] = h'-1 n {(∂'j∂'jB'n) + (1/ g' ) ∂'k(g' )(∂'kB'n) } + 2 ( ∂'jB'n) (∂'j h'-1 n) + B'n (∂'j ∂'j h'-1 n) + (1/ g' ) ∂ 'k(g' )B'n(∂'k h'-1 n) = h'-1 n lap ( B'n) // (I.5.9) for {..} + 2( ∂'jB'n) (∂'j h'-1 n) + B'n (∂'j ∂'j h'-1 n) + (1/ g' ) ∂ 'k(g' )B'n(∂'k h'-1 n) = h'-1 n lap ( B'n) + Other Terms . (I.6.4) The conclusion so far from (I.5.2) and (I.5.10) and (I.6.4), ( @B)'n = B'n;j ;j = lap(B'n) + Extra Terms (I.5.10) = [ h'-1 n lap(B'n) + Other Terms] + Extra Terms . (I.6.4) (I.6.5) As before, our interest is with the expansion @B = (@B)'n e^n. Since (@B) is a vector like any other vector, we write ( @B)'n = (@B)'n/h'n as in (I.2.7) so that (I.6.5) becomes, (@B)'n = lap( B'n) + h'n [Other Terms + Extra Terms] . (I.6.6) This result applies to any x'-space curvilinear coordinate system, orthogo nal or otherwise. Thus we have demonstrated why it is that lap( B'n) always appears as part of the vector Laplacian component ( @B)'n. We copy the Extras and Other Terms from (I.5.10) and (I.6.4), Extra Terms = ∂' j(g'jbΓ 'n bs B's) + Γ 'n jk[∂'j B'k + g'jbΓ 'k bs B's] + Γ 'j jk[g'kbΓ 'n bs B's] (I.6.7) Other Terms = 2( ∂'jB'n) (∂'j h-1 n) + B'n (∂'j ∂'j h-1 n) + (1/ g' ) ∂'k(g' )B'n(∂'k h-1 n) . (I.6.8) Rather than attempt algebraic simplification of the above terms, we will just throw them into Maple as is. Replacing B's = B's/h's in Extra Terms and "lowering" the diffe rential operators appropriately, one gets Extra Terms = ∂' j(g'jb Γ 'n bs B's/h's) + Γ 'n jkg'js∂'s(B'k/h'k) + Γ 'n jkg'jb Γ 'k bs B's/h's + Γ 'j jkg'kb Γ 'n bs B's/h's ET1 ET2 ET3 ET4 (I.6.9) Other Terms = 2(g'js ∂'sB'n) (∂'jh'n-1) + B'n (∂'j ∂'jh'n-1) + (1/ g' ) ∂ 'k(g' )B'n(∂'kh'n-1) . OT1 OT2 OT3 (I.6.10) Appendix I: Expansion of ( @B) 362 I.7 Method 2 for spherical coordinates: Maple speaks again In Method 1, there was no need in the Maple calcula tion for the affine connection object of (F.4.1), Γ 'd ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] (I.7.1) which in orthogonal coordinates simplifies to Γ 'd ab = (1/2) h' d-2 [δd b ∂'a(h'b2) + δd a∂'b(h'a2) – δab ∂'d(h'a2)] . (I.7.2) This last form shows that Γ 'd ab = 0 unless two indices match, in which case it might not vanish. For coordinate systems like spherical and cylindrical coordinates, Γ 'd ab is "sparsely populated", and this explains why won't be swamped by all those Extra Terms shown above. For spherical coordinates, only 9 of the 27 elements of the object Γd ab are non-zero : Γ '1 22 = -r Γ '2 12 = Γ '2 21 = 1/r // notation: Γ '1 22 = Γr θθ Γ '1 33 = -r sin2θ Γ '3 13 = Γ '3 31 = 1/r Γ '2 33 = -cosθsinθ Γ '3 23 = Γ '3 32 = cotθ // 1,2,3 = r, θ,φ = radius, polar, azimuthal Γ r = ⎣⎢⎢⎡ ⎦⎥⎥⎤ 0 0 0 0 -r 0 0 0 -rsin2θ Γθ = ⎣⎢⎡ ⎦⎥⎤ 0 1/r 0 1/r 0 0 0 0 -sin θcosθ Γφ = ⎣⎢⎡ ⎦⎥⎤ 0 0 1/r 0 0 cot θ 1/r cotθ 0 (I.7.3) Note that each matrix is symmetric since Γa bc = Γa cb. To maintain generality, however, we let Maple compute Γ d ab = G(d,a,b) from (I.7.1), so Γ' d ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] (I.7.4) The Extra Terms are then entered, as shown above : (I.7.5) Appendix I: Expansion of ( @B) 363 Extra Terms = ∂'j(g'jb Γ 'n bs B's/h's) + Γ 'n jkg'js∂'s(B'k/h'k) + Γ 'n jkg'jb Γ 'k bs B's/h's + Γ 'j jkg'kb Γ 'n bs B's/h's ET1 ET2 ET3 ET4 ( I . 7 . 6 ) And then the Other Terms are entered as well, Other Terms = 2(g' js ∂'sB'n) (∂'jh'n-1) + B'n (∂'j ∂'jh'n-1) + (1/ g' ) ∂ 'k(g' )B'n(∂'kh'n-1) (I.7.7) OT1 OT2 OT3 ( I . 7 . 8 ) Appendix I: Expansion of ( @B) 364 Maple now computes the non- lap( B'n) terms shown in (I.6.6), called starBx(n) in the code, (@B)'n = lap( B'n) + h'n [Other Terms + Extra Terms] . (I.6.6) The final results are then generated : ( I . 7 . 9 ) Recall that results of Method 1 were, ( @B)r = ∇2(Br) – (2/r2) [ Br + cotθ Bθ + ∂θBθ + cscθ ∂φBφ ] (@B)θ = ∇2(Bθ) – (1/r2) [csc2θ Bθ – 2 ∂θBr + 2cot θcscθ∂φBφ ] (@B)φ = ∇2(Bφ) – (1/r2) [csc2θ Bφ – 2cscθ∂φBr – 2cotθcscθ∂φBθ ] (I.4.8) One then sees that the non- scalar-Laplacian terms just computed as (I.7.9) by Method 2 exactly match the non-∇2 terms in (I.4.8) as found by Method 1. I.8 Results for Cylindrical Coordinates from both methods The Maple program was easily modified for this system. Here are the results: B'1 = Br B'2 = Bφ B'3 = Bz B = B'ne^n = Bre^r + Bφe^φ + Bφe^z = Brr^ + Bθφ^ + Bφz^ . (I.8.1) The scalar Laplacian of an unspecified function f : ∇ 2f = (1/r)∂r(rf) + (1/r2)∂φ2f + ∂z2f = 1 + 2 3 4 Appendix I: Expansion of ( @B) 365 ( I . 8 . 2 ) 1 2 3 4 The components of the vector Laplacian are found by Method 1 to be (I.8.3) 1 2 4 3 Therefore, (@B)r = ∇2(Br) - Br/r2 - 2∂φBφ/r2 . ( I . 8 . 4 ) (I.8.5) 3 4 1 2 Therefore, ( @B)φ = ∇2(Bφ) - Bφ/r2 + 2∂φBr/r2 . ( I . 8 . 6 ) (I.8.7) 4 3 1 2 Therefore, ( @B)z = ∇2(Bz) ( I . 8 . 8 ) Appendix I: Expansion of ( @B) 366 as befits a component which is Cartesian. In th is manner, Method 1 produces the components of the vector Laplacian expressed in the traditional manner, (@B)r = ∇2(Br) – (1/r2)[Br + 2∂φBφ] (@B)φ = ∇2(Bφ) – (1/r2)[Bφ – 2∂φBr] (@B)z = ∇2(Bz) where ∇2f = (1/r)∂r(rf) + (1/r2)∂φ2f + ∂z2f = ∂2 rf +(1/r)∂rf + (1/r2)∂φ2f + ∂z2f (I.8.9) Method 2 produces the following results for the terms which are added to the scalar Laplacian, (I.8.10) and these are seen to agree with non- ∇2 terms in (I.8.9), (@B)r = ∇2(Br) – (1/r2)[Br + 2∂φBφ] (@B)φ = ∇2(Bφ) – (1/r2)[Bφ – 2∂φBr] (@B)z = ∇2(Bz) (I.8.9) In cylindrical coordinates the Γ object is even sparser than in spherical coordinates. One has Γ 'd ab = (1/2) h' d-2 [δd b ∂'a(h'b2) + δd a∂'b(h'a2) – δab ∂'d(h'a2)] . Γ '1 22 = -r // notation: Γ '1 22 = Γr φφ Γ '2 12 = 1/r Γ '2 21 = 1/r // 1,2,3 = r, φ,z r φ z Γr = ⎣⎢⎡ ⎦⎥⎤ 0 0 0 0 -r 0 0 0 0 Γφ = ⎣⎢⎡ ⎦⎥⎤ 0 1/r 0 1/r 0 0 0 0 0 r φ z Γz = ⎣⎢⎡ ⎦⎥⎤ 0 0 0 0 0 0 0 0 0 (I.8.11) Appendix I: Expansion of ( @B) 367 so only 3 of 27 components are non-vanishing. Hand Calculation of Γ' for cylindrical coordinates From a simple polar coordinates picture one knows that d r^ = dφ φ^ d φ^ = -dφ r^ . ( I . 8 . 1 2 ) Then from (3.4.2) with θ → φ, we have this situation fo r cylindrical coordinates, er = e^r = r^ ⇒ der = dr^ = dφ φ^ = dφ (eφ/r) = (1/r)d φ eφ eφ = r φ^ ⇒ deφ = d(r φ^) = r dφ^ + dr φ^ = r ( -dφ r^) + dr φ^ = -rdφ er + (dr/r) eφ ez = z^ ⇒ d ez = 0 . (I.8.13) Therefore only 3 of 9 partial derivatives are non-zero, ∂ rer = 0 ∂reφ = (1/r) eφ ∂zer = 0 ∂φer = (1/r) eφ ∂φeφ = -rer ∂zeφ = 0 ∂zer = 0 ∂zeφ = 0 ∂zez = 0 . (I.8.14) Since we are in the Picture A context of Fig (3.4.3), we use (F.1.11) which says Γ'c ab = ec • (∂'aeb) . (F.1.11) Then Γ' c ab = g'cd ed • (∂'aeb) = h'c-2 ec • (∂'aeb) . // no sum on c (I.8.15) For our situation with x' = (r,φ,z), we find that for Γ'c ab to not vanish, one must have ab = φr, rφ and φφ since all other ( ∂'jeb) = 0 in (I.8.14). In each of these cases only one Γ component survives due the orthogonality of the base vectors : Γ' φ φφ = hφ-2 eφ • (∂φeφ) = r-2 eφ • [ -rer] = 0 Γ' z φφ = hz-2 ez • (∂φeφ) = 1 * ez • [ -rer] = 0 Γ'r φφ = hr-2 er • (∂φeφ) = 1 * er • [ -rer] = -r r^ • r^ = - r Γ' φ rφ = hφ-2 eφ • (∂reφ) = r-2 eφ • [(1/r) eφ] = r-2 r φ^ • φ^ = (1/r) Γ' φ φr = hφ-2 eφ • (∂φer) = r-2 eφ • [ (1/r) eφ] = r-2 r φ^ • φ^ = (1/r) (I.8.16) These results agree with (I.8.11). Appendix J: Expansion of ( ∇T) 368 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor) J.1 Total time derivative as prototype equation This appendix assumes the usual curvilinear coordinates context, Picture B : (J.1.1) The total time derivative of a contravariant rank-2 tensor field Tij(x,t) can be written as dT ij(x,t)/dt = ∂Tij/∂t + (∂Tij/∂xk) (∂xk/dt) = ∂ Tij/∂t + (∂Tij/∂xk) vk or d tTij = ∂tTij + (∂kTij) vk . // d t ≡ d/dt, ∂t ≡ ∂/∂t, ∂k ≡ ∂/∂xk (J.1.2) We take this as a useful prototype equation for two r easons. First, it contains our object of interest, which is the gradient of a rank-2 tensor, ∂kTij. Second, this equation plays a role in the continuum mechanics of non-Newtonian fluids as discussed in Section J.6 below. One can define, in Cartesian coordinates, a ( ∇T) object : (∇T)ij k ≡ ∂kTij ( J . 1 . 3 ) so that, from (J.1.2), d tTij = ∂tTij + (∇ T)ij k vk . (J.1.4) Comment: Our convention has been to bold vectors and not to bold other tensors. In line with this idea, we shall write ∇T where the grad is bolded and the T is not bolded. In order to express the above equation in curvilinear coordinates, it must be "tensorized" in the sense of Section 15.2 so that the equation is covariant. Thus, ( ∇T) ij k must be regarded as components of a (mixed) rank-3 tensor which, in Ca rtesian coordinates, are equal to ∂kTij. Since vk are the components of a tensorial vector, ( ∇T)ij kvk transforms as a rank-2 tensor, and th en all terms in the above equation are rank-2 tensors and the equation is then covariant and therefore a ppears this way in x'-space, dtT'ij = ∂tT'ij + (∇ T)'ij k v'k . (J.1.5) In our usual formalism, x'-space is the space of some generic curvilinear coordinates x'n (not necessarily orthogonal) and then (J.1.5) tells us the form ta ken by the time derivative equation in curvilinear coordinates, and it remains only to compute the objects ( ∇T')ij k. Appendix J: Expansion of ( ∇T) 369 For convenience, we can lower the tensorial ij indices on the above tensor equations to get dtTij = ∂tTij + (∇T)ijk vk (∇T)ijk ≡ ∂kTij // Cartesian coordinates d tT'ij = ∂tT'ij + (∇ T)'ijk v'k // curvilinear coordinates (J.1.6) and then we can deal with the pure covariant tensor components ( ∇T)' ijk . As in previous appendices, we shall be interested in (J.1.6) stated in e^n-expanded coefficients according to the covariance idea of (E.9.13), dtT'ij = ∂tT'ij + (∇ T)'ijk v'k ( J . 1 . 7 ) where the components ( ∇T)'ijk = (∇T)'ijk = [(∇T)(e^)]ijk are computed in (J.7.8) below and evaluated for specific orthogonal coordinate systems in subsequent equations. J.2 Computation of components ( ∇T)'ijk The covariant derivative is discussed in Sections F. 7- F.9. We shall define a true tensor object ( ∇T)abc as Tab;α which is defined in (F.9.6), (∇T)abc ≡ Tab;c ≡ Tab,c – Γn acTnb – Γn bcTan = Tab,c = ∂cTab // x-space (∇T)'abc ≡ T'ab;c ≡ T'ab,c – Γ'n acT'nb – Γ'n bcT'an . // x'-space (J.2.1) Recall that T ab,c is a shorthand for ∂cTab and that Γc ab is the affine connection which tells how the basis vectors change as one moves around in space. In Cartesia n x-space (first line above, see also Picture C1 in Fig (F.1.1)) the basis vectors are the fixed un which don't change, so Γc ab ≡ 0 as in (F.4.16). In curvilinear x'-space, Γ'c ab ≠ 0. Since T ab;c transforms as a true rank-3 tensor, its defining equation is "covariant" (Section 7.15) so that in x'-space the eq uation has exactly the same form but everything is primed. The above two lines characterize the process of "t ensorization": we find a true "tensorial tensor" Tab;c which agrees with ( ∇T)abc = ∂cTab in Cartesian space. The tensorized version of T ab,c is unique (as shown in Section 15.2), and it is T ab;c . Since this tensor is given by T' ab;c in x'-space, we use the second line above to compute the components of the tensor ( ∇T) object in x'-space, which is to say, in curvilinear coordinates. It is a simple matter to have Maple compute Γ'c ab for any curvilinear coordinate system, and then the second line in (J.2.1) reports out the components ( ∇T)'abc, (∇T)'abc = ∂'cT'ab – Γ'n acT'nb – Γ'n bcT'an Γ 'd ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] // (F.4.1) (J.2.2) Appendix J: Expansion of ( ∇T) 370 where g' ab is the metric tensor in x'-space. Then we know from (J.1.6) how to write the time derivative equation in any curvilinear coordinates, dtT'ij = ∂tT'ij + (∇ T)'ijk v'k or d tT'ij = ∂tT'ij + (∇ T)'ij k v'k ( ∇T)'ij k = g'ii'g'jj'(∇T)'i'j'k . (J.2.3) In (I.3.2) we show Maple code to co mpute the covariant metric tensor g' ij from the curvilinear coordinates' defining equa tions (such as x = rsin θcosφ for sphericals). Then (I.7.5) shows the extra code for computing g'ij and Γ 'd ab . The reader can then add a few extra lines to have Maple compute the desired (∇T)'abc using the equations (J.2.2) above. Later in this Appendix we shall use Maple to compute certain related quantities which we can then verify against a known source. J.3 Tensor expansions of ∇T on the u n and en base vectors It is useful at this point to write out th e tensor expansions for the rank-3 tensor ( ∇T) to show exactly where the components ( ∇T)'abc appear. As shown in (E.2.11) and (E.2.14) one can expand the rank-3 tensor (∇ T) in various ways. The first line below shows e xpansions on x-space basis vectors, while the second line shows expansion on the reciprocal and tangent base vectors (which also exist in x-space) : ∇T = Σijk Tij;k ui⊗uj⊗uk = Σijk [(∇T)(u)]ijk ui⊗uj⊗uk = Σijk [(∇T)(u)]ijk ui⊗uj⊗uk ∇T = Σijk T'ij;k ei⊗ej⊗ek = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek . ( J . 3 . 1 ) As usual, up and down index positions on T are the same for the first line in Cartesian space, but are significant on the second line (for non-orthogona l coordinates). The superscripts on the ( ∇T)(u) and (∇T)(e) components indicate which basis vectors are being expanded upon. One normally just writes (∇T)(u) = (∇ T), and (∇T)(e) = (∇ T)' where the prime indicates the cu rvilinear x'-space. So, we now have three different notations for the expansion components: Tij;k = [(∇T)(u)]ijk = (∇T)ijk T' ij;k = [(∇T)(e)]ijk = (∇ T)'ijk . (J.3.2) Just to fill things out, here are the corresponding expa nsions for rank-2 tensor T, again from (E.2.11) and (E.2.14), T = Σ ij Tij ui⊗uj = Σij [T(u)]ij ui⊗uj = Σij [T(u)]ij ui⊗uj Tij = [T(u)]ij T = Σij T'ij ei⊗ej = Σij [T(e)]ij ei⊗ej = Σij [T(e)]ij ei⊗ej T'ij = [T(e)]ij (J.3.3) and then for a rank-1 tensor, as in (7.13.10), v = Σi vi ui = Σi [v(u)]i ui = Σi [v(u)]i ui = Σi viui vi = [v(u)]i v = Σi v'i ei = Σi [v(e)]i ei = Σi [v(e)]i ei = Σi v'iei v' i = [v(e)]i . (J.3.4) Appendix J: Expansion of ( ∇T) 371 J.4 Tensor expansions of ∇T on the e ^n base vectors In practical applications, it is sometimes useful to deal with components of te nsors which are expanded on the unit versions of the tangent base vectors, e^n ≡ en/ |en| = en/h'n . On the one hand, this introduces major complications (see below) since such compon ents are non-covariant (neither contravariant nor covariant nor mixed). On the other hand, since the unit vectors are all dimensionless, all tensor components have the same dimensions , which is very useful in any pr actical engineering work. We shall refer to such tensor components as "unit-base-vector components". This subject is addressed below (E.2.14) and in Sectio n E.8. We start with one of the (J.3.1) expansions, ∇T = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek = Σijk (∇T)'ijk ei⊗ej⊗ek (J.4.1) and process it in this manner , = Σ ijk [(∇T)(e)]ijk (h'ie^i) ⊗ (h'je^j) ⊗ (h'ke^k) = Σ ijk { h'i h'j h'k [(∇T)(e)]ijk } e^i⊗e^j⊗e^k = Σ ijk [(∇T)(e^)]ijk e^i⊗e^j⊗e^k where [(∇T)(e^)]ijk = h'i h'j h'k [(∇T)(e)]ijk . (J.4.2) If our x'-space coordi nate system happens NOT to be orthogonal, then the up/down position of the indices on [(∇T)(e^)]ijk has significance. This is so because, as s hown near (E.2.11), these indices are lowered by w'nm = e^n • e^m . For an orthogonal system, one has w' nm = e^n • e^m = δn,m and then the up/down positions of the indices on [(∇T)(e^)]ijk all indicate the same number. One should not confuse this with the fact that the up/down position of an i ndex on any true x'-space tensor like ( ∇T)'ijk is always significant, whether or not a coordinate sy stem is orthogonal, because in general g' ab ≠ δa,b. Now for convenience later, we make one more definition, (∇T)'ijk ≡ [(∇T)(e^)]ijk = h'i h'j h'k [(∇T)(e)]ijk = h'i h'j h'k (∇T)'ijk (J.4.3) where we follow our convention that the components of unit-base-vector tensors are written in script, see (E.8.5). (The symbol T is a script T , not a "tau" τ. ) Similarly, we can write the unit-base-vector expansion for a rank-2 tensor T, T = Σ ij [T(e^)]ij e^i⊗e^j where [T(e^)]ij = h'i h'j [T(e)]ij (J.4.4) with the definition T'ij ≡ [T(e^)]ij = h'i h'j [T(e)]ij = h'i h'j T'ij . (J.4.5) Finally, for a vector v, Appendix J: Expansion of ( ∇T) 372 v = Σi [v(e^)]i e^i where [v(e^)]i = h'i [v(e)]i ( J . 4 . 6 ) with the definition v'i ≡ [v(e^)]i = h'i [v(e)]i = h'i v'i . (J.4.7) In terms of the scripted unit-base-vector components, our three expansions are: ∇T = Σijk (∇T)'ijk e^i⊗e^j⊗e^k ( ∇T)'ijk = h'i h'j h'k (∇T)'ijk T = Σij T'ij e^i⊗e^j T'ij = h'i h'j T'ij v = Σi v'i e^i v'i = h'i v'i . (J.4.8) The scripted tensor components have indices wh ich are integers, i = 1,2...N. Once we actually select a particular curvilinear coordinate system, one can re place the indices with curvilinear coordinate names and then, since those names indicate that one is talking about x'-sp ace components, and since we just assume we are dealing with unit-base-vector compone nts, both the script and the prime position can be dropped. For example, in spherical coordinates with 1,2,3 = r, θ,φ one can write (∇T)'123 = (∇T)rθφ T'12 = Trθ T'11 = Trr v'3 = vφ v'1 = vr (J.4.9) As noted below (J.4.2), if a curvilinear coordinate syst em is orthogonal (and this is generally the case, and is certainly the case for spherical coordinates), th en the up/down position does not matter and one writes (∇T)'123 = (∇T)'123 = (∇ T)rθφ T'12 = Trθ T'11 = Trr v'3 = vφ v'1 = vr . (J.4.10) Notice that the scripted forms are al ways necessary when summations like Σijk are involved, unless one is willing to write out all the terms in the sum, which is a bit clumsy. The continuum mechanics book of Lai et al ., which we shall refer to below, avoids summati ons in curvilinear coordinates and thus has no need for our scripted components. J.5 Total time derivative equation written in unit-base-vector curvilinear components Recall from (J.2.3) our prototype time deriva tive equation of interest in x'-space, d tT'ij = ∂tT'ij + (∇ T)'ij k v'k . (J.5.1) Appendix J: Expansion of ( ∇T) 373 How does one write this equation in terms of unit-ba se-vector tensor components? We know that for orthogonal coordinates the answer is (J.1.7) based on covariance. Here we shall answer the question for general coordinates, and shall show that (J.1.7) "obtains" for orthogonal coordinates. From (J.4.5) and (J.4.7) we had (no implied sums here) T'ij = h'i h'j T'ij v'k = h'k v'k ( J . 5 . 2 ) so (J.5.1) can be processed as follows: d tT'ij = ∂tT'ij + (∇ T)'ij k v'k / / ( J . 5 . 1 ) h' i h'j dtT'ij = h'i h'j ∂tT'ij + h'i h'j (∇T)'ij k h'k-1 h'k v'k // mult thru by h' i h'j d t(h'ih'j T'ij) = ∂t(h'ih'j T'ij) + h'i h'j h'k-1 (∇T)'ij k (h'k v'k) // h' n = h'n(x'), no t dt T'ij = ∂t T'ij + [ h'i h'j h'k-1 (∇T)'ij k ] v'k // (J.5.2) d t T'ij = ∂t T'ij + Q'ij k v'k Q'ij k ≡ h'i h'j h'k-1 (∇T)'ij k . (J.5.3) Comment: Note that d th'n(x) = ∂th'n(x) = 0 and not d th'n(x) = ∂th'n(x) + (∇h'n) v. The reason is that the field h'n(x) is not an Eulerian fluid property like A 1 in (J.6.9) below, it is a property of space at point x. Recall that no assumption has been made that the curvilinear coordinates are orthogonal. Section J.2 showed how the ( ∇T)'ij k can be computed for any curvilinear coordinate system, and thus one can compute the Q' ijk shown above. Our question is now answered: (J.5.3) shows how one writes the total time derivative equation in arbitr ary curvilinear coordinates. If we now assume the coordinates are orthogonal, then (∇T)'ij k = gkk'(∇T)'ijk' = hk2δk,k' (∇T)'ijk' = hk2(∇T)'ijk (J.5.4) and then Q' ij k ≡ h'i h'j h'k-1 (∇T)'ij k = h'i h'j h'k-1 hk2(∇T)'ijk = h'i h'j h'k(∇T)'ijk = (∇T)ijk . // (J.4.3) (J.5.5) As noted below (J.4.2), for orthogonal coordinates the up/dow n index position on ( ∇T)ijk makes no difference, so for example, ( ∇T)ijk = (∇T)ij k . Installing Q'ij k = (∇T)ij k into (J.5.3) then gives, dt T'ij = ∂t T'ij + (∇T)'ij k v'k (∇T)'ij k ≡ h'i h'j h'k (∇T)'ijk . // orthogonal (J.5.6) Appendix J: Expansion of ( ∇T) 374 which is (J.1.7) which was deduced by covarian ce. Once again, indices go up and down for free on T'ij and (∇T)ijk, but not on T'ij and (∇T)'ijk . We shall assume from now on that the curvilinear coordinates are orthogonal. As a reminder, here is what the above equation says if i = j = 1, d t T'11 = ∂t T'11 + (∇T)'11 k v'k . (J.5.7) Using the notation scheme just described above in (J.4 .9) and (J.4.10), in spherical coordinates one would write the above equation as, d t Trr = ∂t Trr + (∇T)rrr vr + (∇ T)rrθ vθ + (∇ T)rrφ vφ . (J.5.8) J.6 Shorthand notations and a continuum mechanics application Consider our original Cartesian ti me derivative equation (J.1.2), dtTij = ∂tTij + (∇ T)ij k vk . (J.6.1) One could regard (∇T) ij k as the kth component of a vector (∇ T)ij labeled by fixed values i and j, so that [(∇T)ij]k = (∇T)ij k . ( J . 6 . 2 ) Then one can write (J.6.1) as d tTij = ∂tTij + (∇ T)ij • v . ( J . 6 . 3 ) The next step is to suppress the ij labels, since the equation above is true for any i and j, d tT = ∂tT + (∇T) • v . ( J . 6 . 4 ) This is only a shorthand notation , no precision justification is required. We can do the same thing to the equation written for orthogonal unit-ba se-vector curvilinear coordinates : dt T'ij = ∂t T'ij + (∇T)'ij k v'k (J.5.6) [(∇T)'ij]k = (∇T)'ij k // vector with index k dt T'ij = ∂t T'ij + (∇T)'ij • v'k // dot notation d t T' = ∂t T' + (∇T)' • v' // indices stripped (J.6.5) and then we have Appendix J: Expansion of ( ∇T) 375 dtT = ∂tT + (∇T) • v dt T' = ∂tT' + (∇T)' • v' . ( J . 6 . 6 ) In the book of Lai et al ., the • is omitted and the shorthand notations are written, dtT = ∂tT + (∇T) v dt T' = ∂tT' + (∇T)' v' ( J . 6 . 7 ) where one imagines that ( ∇T) and (∇T)' are 3-index operators which act on a 1-index object to generate a 2-index object. Again, it is just a shorthand. The real meaning of these equations is dtTij = ∂tTij + (∇T)ij k vk dt T'ij = ∂tT'ij + (∇T)'ij k v'k . (J.6.8) Example: An application of our total time derivative equa tion appears in the first line of Lai p 470 , DA1/Dt = ∂A1/∂t + (∇ A1) v ( J . 6 . 9 ) where D/Dt is the way a total time derivative is expre ssed in continuum mechanics (D/Dt = d/dt). It is the convective or material derivative for Eulerian-picture functions (like A 1(x,t)), meaning that the second term registers a time change in a fluid property at the fixed point x due to "new fluid" with velocity v passing through that point (a point being a tiny differential volume of fluid). In the notation of (J.6.7), this would be written dtA'1 = ∂tA'1 + (∇A1)' v' . ( J . 6 . 1 0 ) The detailed meaning is dt (A'1)ij = ∂t (A'1)ij + (∇A1)'ij k v'k ( J . 6 . 1 1 ) and for i = j = 1 this says (spherical coordinates) , dt[(A1)rr] = ∂t[(A1)rr] + (∇A1)rrr vr + (∇ A1)rrθ vθ + (∇ A1)rrφ vφ . (J.6.12) The tensor A 1 is the " first Rivlin-Ericksen tensor " associated with the flow of non-Newtonian fluids (rheology). The A i tensors, briefly mentioned in Section K.4, are derivatives of a certain deformation tensor called C t, and computation of the A i is done in the following iterative manner, Ai+1 = DtAi + Ai(∇v) + (∇v)TAi , // Lai p 468 (8.11.3) (J.6.13) where (∇v) is the gradient-of-vector object treated in our Appendix G, v being the fluid velocity field. In particular, A 2 = DtA1 + A1(∇v) + (∇v)TA1 which involves our object of interest D tA1 = dtA1. So, in order to compute A 2 in curvilinear coordinates, one needs d t(A'1)ij which involves the ( ∇A1)'ijk. In order to use the above equation in practice, one has to know for example that ( ∇A1)rrθ = [ ∂ θ(A1)rr - (A1)θr - (A1)rθ]/r, a fact that is certainly not immediately obvious (but see the rr θ entry in (J.7.9) below). Appendix J: Expansion of ( ∇T) 376 So, our next task is to compute the (∇T)'ijk which appear in our prototype equation above, dt T' = ∂t T' + (∇T)' v' // shorthand notation (J.6.7) dt T'ij = ∂t T'ij + (∇T)'ij k v'k (J.5.6) ( J . 6 . 1 4 ) J.7 Maple computation of the (∇T)'ijk components for spherical coordinates Recall from (J.4.3) that, for arbitrary curvilinear coordinates, ( ∇T)'ijk ≡ h'i h'j h'k (∇T)'ijk ( J . 7 . 1 ) and from (J.2.2), (∇T )'abc = ∂'cT'ab – Γ'n acT'nb – Γ'n bcT'an ( J . 7 . 2 ) Γ 'd ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] . g' = metric tensor for x'-space But we are now assuming only orthogonal coordinates, so g'ab = δa,bha2 g'ab = δa,bha-2 ( J . 7 . 3 ) ⇒ (∇T)'ijk = g'ii'g'jj'g'kk'(∇T)'i'j'k' = (h'i h'j h'k)-2 (∇T)'ijk (J.7.4) and then from (J.7.1), ( ∇T)'ijk = (∇T)'ijk = (h'i h'j h'k)-1 (∇T)'ijk . ( J . 7 . 5 ) Remember that up/down index position does not matter on ( ∇T)'ijk for orthogonal coordinates. We want the result expressed in terms of the T'ij and not the T' ij, so from (J.4.8), T'ij = (h'i h'j) T'ij = (h'i h'j) g'ii' g'jj'T'i'j' = (h'i h'j)-1 T'ij ⇒ T'ij = (h'i h'jT'ij) . ( J . 7 . 6 ) Then (J.7.2) reads, (∇T )'abc = ∂'cT'ab – Γ'n acT'nb – Γ'n bcT'an // do abc →ijk rename (∇T)'ijk = ∂'kT'ij – Γ'n ikT'nj – Γ'n jkT'in (∇T)'ijk = ∂'k(h'i h'jT'ij) – Γ'n ik(h'n h'jT'nj) – Γ'n jk(h'i h'nT'in) // (J.7.6) Appendix J: Expansion of ( ∇T) 377 = h'i h'j(∂'kT'ij) + ∂'k(h'i h'j) T'ij – Γ'n ik(h'n h'jT'nj) – Γ'n jk(h'i h'nT'in) (J.7.7) where we break the ∂'k term in two pieces for Maple technical reasons. So the Maple program will compute from (J.7.5) and (J.7.7), ( ∇T)'ijk = (h'i h'j h'k)-1 (∇T)'ijk ( J . 7 . 8 ) where ( ∇T )'ijk = h'i h'j(∂'kT'ij) + ∂'k(h'i h'j) T'ij – Γ'n ik(h'n h'jT'nj) – Γ'n jk(h'i h'nT'in) . T1 T2 T3 T4 For now we assume spherical coordinates. Maple first computes g ** = gcov using (I.3.2) and g** = gcontra using (I.7.5). Then the affine connection is com puted from these metric tensors, just as in (I.7.5), ( J . 7 . 9 ) Then the terms of (J.7.8) are entered ( T'ij = Te[i,j], h' i = hp[i], Γ 'd ab = G(d,a,b), etc.) ( J . 7 . 1 0 ) The terms are then added, multiplied by (h' i h'j h'k)-1 according to (J.7.8), and then displayed. In this code then, DT(i,j,k) is ( ∇T)'ijk as computed on the first line. In th e triple loop, the complicated resulting expressions are cleaned up in a series of cosmetic trigonometric filtering steps: DT(i,j,k) → f → g → h[k] and it is then the simplified h[k] expressions which are printed out. Appendix J: Expansion of ( ∇T) 378 ( J . 7 . 1 1 ) and here are the resulting values for ( ∇T)'ijk (for example, ( ∇T)'112 = (∇T)rrθ ) ( ∇T)'ijk for Spherical Coordinates (J.7.12) Appendix J: Expansion of ( ∇T) 379 The strange " symbols in the above table should be ignored, just a Maple print command artifact. These results agree with the spherical co ordinates table given in Lai p 505. The divT results of Appendix H can be verified from the above table using ( divT)'i = ∂'jT'ij = (∇T)'ijj = Σj(∇T)'ijj . (J.7.13) For example, (divT) r = (∇ T)rrr + (∇T)rθθ + (∇ T)rφφ = ∂rTrr + (1/r)[∂θTrθ - Tθθ + Trr] + (1/r)[ csc θ ∂φTrφ - Tφφ + Trr + cotθ Trθ] = ∂ rTrr + (1/r)∂θTrθ - Tθθ/r + (2/r)T rr + (1/rsin θ) ∂φTrφ - Tφφ/r + cotθ Trθ /r (J.7.14) and we quote from (H.6.3), . (H.6.3) Appendix J: Expansion of ( ∇T) 380 J.8 Maple computation of the (∇T)'ijk components for cylindrical coordinates Making four small edits to the spherical coordi nates Maple program converts it to a cylindrical coordinates program. Here are the resulting values for ( ∇T)'ijk (for example, ( ∇T)'123 = (∇T)rθz ) ( ∇T)'ijk for Cylindrical Coordinates (J.8.1) and these expressions agree with those in the table on page 504 of Lai. The same comment made about divT at the end of the previous Section applies here as well. The object Γ 'd ab has 3 of 27 components non zero as shown in (I.8.11), only 2 of which are distinct. Appendix J: Expansion of ( ∇T) 381 It should be emphasized that this same simple Maple code can be used to compute the ( ∇T)'ijk for any system of orthogonal curvilinear coordinates in any number of dimensions N. For non-orthogonal coordinates the results would be obtain ed from these more general equations, (∇T)'ijk = h'i h'j h'k (∇T)'ijk (J.7.1) (∇T)' ijk = g'ai g'bj g'ck (∇T)'abc // raise indices (∇T)'abc = ∂'cT'ab – Γ'n acT'nb – Γ'n bcT'an (J.7.2) Γ 'd ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] . g' = metric tensor for x'-space ( J . 8 . 2 ) J.9 The Lai Method of computing (∇T)'ijk for orthogonal coordinates This method appears in Lai pp 501-505. It differs fr om the method given above in Section J.8 mainly because it uses a different affine connection, but of co urse it gives the same results. We shall present Lai's computation of the components (∇T)'ijk in the context of Picture B, (J.9.1) At the very end we will translate the result to Picture C1 of (F.1.1) where x instead of x' are the curvilinear coordinates. For Picture B, one has from (F.1.11), (∂' jen) = Γ 'k jn ek ⇒ (d en)i = (∂ 'jen)i dx'j = Γ'k jn (ek)i dx'j (J.9.3) Γ' k jn = ek • (∂'jen) ( J . 9 . 4 ) where the en are our usual Chapter 3 tangent base vectors and we regard en = en(x'). It is possible to define a different affine connection Γ^'k jn in this manner, (∂'je^n) = Γ^'k jn e^k ⇒ (d e^n)i = Γ^'k jn (e^k)i dx'j (J.9.4) Γ ^'k jn = e^k • (∂'je^n) . ( J . 9 . 5 ) This new affine connection Γ ^'k jn describes how the unit vectors e^n vary with x'. Now consider the tensor expansion from (J .4.8), where all objects are functions of x', Appendix J: Expansion of ( ∇T) 382 T = Σij T'ij e^i⊗e^j = T'ij e^i⊗e^j . ( J . 9 . 6 ) The object T = T( x') is a vector in a double direct-product space spanned by e^i⊗e^j. Apply ∂'k to get ∂' kT = (∂'kT'ij) e^i⊗e^j + T'ij(∂'ke^i)⊗e^j + T'ij e^i⊗(∂'ke^j) . (J.9.7) Now do index shuffles on (J.9.4), (∂' je^n) = Γ^'k jn e^k // do k→ s then j → k and n → i (∂'ke^i) = Γ^'s ki e^s // do i →j (∂'ke^j) = Γ^'s kj e^s . ( J . 9 . 8 ) Insert the last two lines into (J.9.7) to get ∂'kT = (∂'kT'ij) e^i⊗e^j + T'ijΓ^'s ki e^s⊗e^j + T'ij e^i⊗ Γ^'s kj e^s In the second term swap s ↔ i and in the third term swap s ↔ j (dummy summation indices) to get, ∂'kT = (∂'kT'ij) e^i⊗e^j + T'sjΓ^'i ks e^i⊗e^j + T'is e^i⊗ Γ^'j ks e^j = [ ∂' kT'ij + T'sjΓ^'i ks + T'isΓ^'j ks ] e^i⊗e^j, (J.9.9) so ∂'kT is another vector in the same double direct-product space. To economize below, we write the above as ∂' kT = Qij k e^i⊗e^j where Qij k ≡ [∂'kT'ij + T'sjΓ^'i ks + T'isΓ^'j ks ] . (J.9.10) Next, dot both sides of (J.9.10) with the vector ua ⊗ ub where the un are our usual Cartesian space axis- aligned unit vectors ( un)i = (un)i = δi,n . On the left side one gets (∂' kT) • ua ⊗ ub = (∂'kT)ab = ∂'kTab ( J . 9 . 1 1 ) where Tab are the contravariant components of the T tensor in x-space. On the right side one gets e^i⊗e^j • ua ⊗ ub = (e^i•ua) (e^j•ub) = ( e^i)a (e^j)b . (J.9.12) Thus (J.9.10) becomes Appendix J: Expansion of ( ∇T) 383 ∂'kTab = Qij k (e^i)a (e^j)b . (J.9.13) We know from (7.6.8) that ∂c = Rk c ∂'k, so applying Rk c to both sides of (J.9.13) gives ∂cTab = Rk c (e^i)a (e^j)b Qij k . ( J . 9 . 1 4 ) Since x-space is Cartesian, this is the same as ∂ cTab = Rk c (e^i)a (e^j)b Qij k ( J . 9 . 1 4 a ) or, using (J.1.3) that ( ∇T)ijk ≡ ∂kTij, (∇T)abc = Rk c (e^i)a (e^j)b Qij k . ( J . 9 . 1 5 ) Our coefficients of interest are the ( ∇T)'abc which are related to ( ∇T)abc as in (E.8.20), (∇ T)'abc = Ma AMb BMc C (∇T)ABC = Ma AMb BMc C { Rk C (e^i)A (e^j)B Qij k } . (J.9.16) Things will simplify. First, from (E.8.12) we know that ( e^i)A = NA i and ( e^j)B = NB j , so we continue the above (∇T)'abc = Ma AMb BMc CRk CNA iNB j Qij k = ( Ma ANA i)(Mb BNB j)Mc CRk C Qij k // reorder = ( δ a i)(δb j)Mc CRk C Qij k = Mc CRk C Qab k // M and N are inverses as in (E.8.9) = ( h ' c Rc C)Rk C Qab k = h'c(Rc CRk C)Qab k // (E.8.7) then reorder = h ' c(g'cdRdCRk C)Qab k = h'cg'cd(RdCRk C)Qab k // (7.5.9) with g = 1 then reorder = h ' c(g'cdδdk)Qab k = h'c(g'ck)Qab k // (7.6.4) orthog rule #4 = h ' c(h'k-2 δc,k)Qab k = h'c(h'c-2)Qab c // (5.11.9) for orthog coords = h ' c-1Qab c = h ' c-1 [∂'cT'ab + T'sbΓ^'a cs + T'asΓ^'b cs ] . // (J.9.10) (J.9.17) The final result with the Lai affine connection is then, Appendix J: Expansion of ( ∇T) 384 (∇T)'abc = (∇T)'abc = h'c-1 [∂'cT'ab + T'sbΓ^'a cs + T'asΓ^'b cs ] . (J.9.18) Now do an index shuffle abc → ijm to get, (∇T)'ijm = h'm-1 [∂'mT'ij + T'sjΓ^'i ms + T'isΓ^'j ms ] . (J.9.19) Expressing this result for Picture C1 of (F.1.1) removes the primes to give (∇ T)ijm = (∇T)ijm = hm-1 [∂mTij + T sjΓ^i ms + TisΓ^j ms ] . (J.9.20) The corresponding equations (J.9.4) and (J.9.5) become in Picture C1 (see (F.1.1) and following text) (∂ jq^n) = Γ^k jn q^k ⇒ (dq^n)i = Γ^k jn (q^k)i dxj (J.9.4) d q^n = Γ^k jn q^k dxj Γ^k jn = q^k • (∂jq^n) (J.9.5) d q^i = Γ^k ji q^k dxj . (J.9.21) where qn(x) are the tangent base vectors in the ξ-space of Picture C1 and q^n = qn(x)/hn. We now translate these results to the Lai et al. notation as follows: U s Lai q^n en // unit vectors in ξ-space Tij T ij Γ^i ms Γsmi // the Lai affine connectio n, note reverse index order (∇T)ijm M ijm ∇ T M . ( J . 9 . 2 2 ) The translation of (J.9.20) and the right side of (J.9.21) is then (∇ T)ijm hm = [∂mTij + T sjΓ^i ms + TisΓ^j ms ] // us, now take s →q Mijm hm = [∂mTij + TqjΓqmi + TiqΓqmj ] // Lai p 504 (8A.26) (J.9.23) d q^i = Γ^k ji dxj q^k // us d ei = Γijk dxj ek . // Lai p 502 (8A.12) (J.9.24) Lai et al. use (J.9.23) to compute the ( ∇T)ijm for cylindrical and spherical coordinates on p 504 and 505. Their results are in agreement with our calculations (J.8.1) and (J.7.9). Appendix K: Deformation Tensors 385 Appendix K: Deformation Tensors in Continuum Mechanics Tensor-like objects appear everywhere in contin uum mechanics. As was noted below (2.12.3), continuum mechanics texts generally refer to all objects having indices as being "tensors", whether or not these objects actually transform as tensors with resp ect to some underlying transformation. The most commonly appearing tensors have two indices and are just 3x3 matrices associated with 3D space. In this category, there are several kinds of stress tensors, a nd many kinds of strain a nd deformation tensors which describe how a tiny volume of continuous matter (perhaps a tiny cube near some point x) changes shape in response to some applied stress. For a fluid, a fixed stress pattern can cause a continuous ongoing change of shape which is measured then by a "rate of deformation tensor" often called D. An equation relating stress to strain/deformati on is called a constitutive e quation and describes the "response" of some physical system to "stimulus". The constitutive equation for a spring is F = -kΔx, for example, which is distinct from the equation of motion for a mass on a spring which is F = m a. For the spring, the stimulus is the force F ("stress"), and the response is the spring stretch Δx ("strain"). In this Appendix we shall study the tensor aspects of several kinds of deforma tion tensors appearing in continuum mechanics. In the book of Lai et al. this material is spread over several chapters, but here it will all be put in one place with Lai references provided. Section K. 3 below considers the form of a candidate constitutive equation for a continuous solid whose form is determined by the requirement that the equation be "covariant" as discussed in Secti on 7.15. Similarly, Sections K.4 and K.5 consider covariant constitutive equations for fluids K.1 A Preliminary Deformation Flow Picture Our flow version of Picture A was shown as (5.16.1), where F ≡ F-1 : (K.1.1) The upper arrow represents an unde rlying generally non-linear transformation between x-space and x'- space given by x' = F(x) while matrix S is the linearized-at-point- x version of the transformation which defines the notion of a vector with d x = S d x' as in (2.1.6). Since F will have another meaning below, we change the transformation name so that x = F(X) where F ≡ F-1 . Warning: The x' in Fig (K.1.2) below is unrelated to the x' appearing in (K.1.1) which is really X. Also, overloaded symbol S will be used both for reference frames and for linearized transformation matrices. Consider now the picture below in whic h appear two sequential transformations Fto and Ft. There are three spaces called X-space on the bottom, x-space in the middle, and x'-space on the top. The spaces are associated with frames of reference S 0, S and S'. Each frame has some set of basis vectors to be discussed Appendix K: Deformation Tensors 386 below. The linearized S matrices a ssociated with the two transformations are shown to the right and are given the names S t0 = F on the bottom and S t= Ft on the top. The transformation x' = Ft(x,τ) is a spatial coordinate transformation only, the time coordinate τ is a parameter. In the picture below, time increases in the upward direction, so τ > t > t0. (K.1.2) Consider for the moment just the bot tom transformation. The transformation Fto ≡ F describes the deformation of a particle of contin uous matter which starts at position X and time t 0 and ends up at position x at time t. If we look at a large cube of continuous matter, we might find that it deforms in a very complicated manner as determined by the non-linear transformation F applied to all the particles within this large cube. The cube ge ts stirred up and is probably no longer recognizable. But if instead we consider a differentially small starting cube at X and t0, we shall find that at time t that cube is at location x but has been transformed into a tiny rotated parallelepiped whose axes are no longer orthogonal. It is assumed that the flow is reasonable and smooth, we are not considering some kind of singular "explosion" here. We use the words flow and fluid, but the deformation concept applies to elastic solids as well as fluids since these deform in some way wh en they are stressed (think jello or even steel). Rather than think of the flow in terms of the e dges of this tiny cube, one can instead consider two very closely spaced points in the fluid close to X which are separated by spacing d X at time t 0, which we think of as "a little dumbbell". At time t, if one car efully tracks the "pathlines" of the ends of the dumbbell, one finds that the dumbbell tumb les and stretches and ends up as d x at time t and location x, (K.1.3) Appendix K: Deformation Tensors 387 This differential dumbbell can be regarded as a mathematical "probe" embedded in the continuous medium. The relationship between d x and d X is given by d x = F( X,t) dX dx i = FijdXj // Lai p 105 (3.18.3) (K.1.4) where the matrix F ij is called "the deformation gradient". Matrix F is also known as "the deformation gradient tensor" even though it is not a "tensorial te nsor" with respect to any identifiable transformation. In Section 5.16 the above equation d x = Fd X was identified with d x = Sd x' with S being F. Fig (K.1.3) is just Fig (2.1.2) with x'-space identified with X-space . Thus, the deformation gradient F is the linearized version (at point x) of some fancy non-linear (and unknown) "flow transformation" F. Since one can write dxi = (∂ xi/∂Xj)dXj , ( K . 1 . 5 ) one finds from (K.1.4) that Fij = (∂ xi/∂Xj) ≡ ∂jXi or F = (∇ x) // Lai p 105 (3.18.4) (K.1.6) where the gradient ∇ is with respect to X, so it is really ∇ = ∇(X). Thus the name "deformation gradient" for F. [ Notice that ( ∇x) is a matrix. In Appendix G the form of ( ∇v) for arbitrary vector v is found in arbitrary curvilinear coordinates. The index order reversal F ij = ∂jXi is mentioned there as well. ] The deformation gradient F(X,t) depends implicitly on the time t 0. At t = t 0+ε (with very small ε) no flow has yet taken place, so dx i = dXi and then F(X,t 0) = 1. Time t 0 is called the reference time and one could display it by writing F(X,t) = F t0(X,t), but normally this t 0 label is suppressed. Now consider the upper flow in Fig (K.1.2) above. It is entirely analogous to the lower flow, but names are changed. One gets from the lower flow to the upper flow by making these replacements : (t, x, dx, S) → (τ, x', dx', S') then (t0, X, dX, S0) → (t, x, dx, S) . (K.1.7) The equations corresponding to (K.1.4) are therefore (as shown in (K.1.2)), d x' = Ft(x,τ) dx dx' i = (Ft)ijdxj . // Lai p 457 (8.7.2) (K.1.8) Since one can write dx' i = (∂ x'i/∂xj)dxj , one finds that (F t)ij = (∂ x'i/∂xj) or Ft = (∇ x') // Lai p 457 (8.7.3) (K.1.9) where the gradient ∇ is with respect to x, so it is really ∇ = ∇(x). Appendix K: Deformation Tensors 388 In the lower flow, t 0 is the reference time, and t is the "curre nt time". In the upper flow, the current time t is the reference time, and τ is some time τ > t. The upper flow is relative to current time t as reference, and for that reason the word relative is pre-pended to the names of all related tensors. Thus, F t is called the "relative deformation gradient ", whereas F is just the "deformation gradient ". Why are relative tensors useful? The main motivation for use of the relative tensors c oncerns differentiation with respect to time in the vicinity of the current time t. One can write d tFt(x,t) ≡ [∂τFt(x,τ)]τ=t // x fixed (for example, x = x1) where ∂ τFt(x,τ) ≈ [ Ft(x,τ+dτ) - Ft(x,τ) ] / dτ . (K.1.10) Here d t = Dt = d/dt = D/Dt = the total time derivative, and ∂t = ∂/∂t = the partial time derivative. Since x is fixed, d x = 0 so d t = ∂t. We want to know the rate of deformation at some fixed current time t, and it is the τ argument of the function F t(x,τ) that lets this derivative be computed. One could in theory carry out this same differe ntiation using the "non-relative" tensors by doing d/dt 0 with t0 near t: d tFt(X,t) ≡ [∂t0Ft0(X,t)]t0=t // X fixed where ∂t0Ft0(X,t) ≈ [ Ft0+dt0 (X,t) - Ft0(X,t) ] / dt0 , (K.1.11) but this goes against the grain of the idea that X is a material coordinate at constant, fixed initial time t 0 which is earlier than t. And in the above, we end up with a statement about Lagrangian functions f( X,t) rather than Eulerian functions f( x,t), though one might argue that as t 0→ t, one has X → x. The relative tensor approach just makes the differentiation proce ss clearer, and will be used below for that purpose. [ In the Lagrangian or material picture, one tracks a blob that started at position X -- one's instrumentation so to speak flows with the blob. In the Eulerian picture the instrumentation is fixed in space and observes the flow passing by. ] The frames of reference and the cameraman Each of the three frames of reference S 0, S and S' in Fig (K.1.2) has its own set of orthonormal basis vectors which we are completely free to set in any manne r. As a construct it is helpful to imagine that, as the flow proceeds, it is observed by a cameraman who flies around on a camera platform which translates and rotates in some arbitrary manner. Since our main concern will be with the dumbbells like d X, dx and dx', the translational part of the camera platform motion is irrelevant since d X is invariant under translations. We allow the cameraman' s arbitrary orientation at times t 0, t and τ to determine the axes of the three frames S 0, S and S'. The cameraman is an "observer". The values of the deformation gradient matrix elements F ij depend on the choice of basis vectors in frames S 0 and S, so they in fact are dependent on how the cameraman flies his platform. Consider, Appendix K: Deformation Tensors 389 d x = dx1e^1(S) + dx2 e^2(S) + dx3 e^3(S) (K.1.12) d X = dX1e^1(S0) + dX2 e^2(S0) + dX3 e^3(S0) . (K.1.13) Once the axes are chosen, the value of the F ij are determined, for example, F 12 ≈ (dx1)/(dX2) . ( K . 1 . 1 4 ) If we were to rotate the basis vectors in frame S, for example, dx 1 would change, dX 2 would stay the same, and F 12 would change. Tensor expansions of the deformation gradients These two expansions won't be used below, they are ju st inserted here as an application of the work of Appendix E on tensor expansions. Th ese expansions use the cameraman basis vectors just defined above which are e^n(S0) for frame S 0 and e^n(S) for frame S. • Consider this candidate expansion for the deformation gradient F, F = Σij Fij e^i(S) ⊗ e^j(S0) = Σij Fij [e^i(S)] [e^j(S0)]T where F ij = [F(S,S0)]ij = [ e^i(S)]T F [e^j(S0)] = (∂xi/∂Xj) , (K.1.15) where we write the expansion in both direct product a nd matrix forms of Appendix E. This is a "mixed basis expansion" as discussed in Section E.10. C onsider the application of this expansion to d X : { Σij Fij [e^i(S)] [e^j(S0)]T } dX // ≡ {expansion} dX = Σ ij Fij [e^i(S)] [e^j(S0)]T { ΣkdXke^k(S0)} = Σ ij Fij ΣkdXk [e^i(S)] [e^j(S0)]T [ e^k(S0)] = Σ ij Fij ΣkdXk [e^i(S)] δj,k = [ Σ ij Fij dXj] e^i(S) = [ d x i ] e^i(S) // using the fact that F d X = dx = d x . ( K . 1 . 1 6 ) Since {expansion}d X = dx and since ( ∇(X)x) dX = dx by the chain rule, it seems reasonable to conclude that {expansion} = ( ∇(X)x) = F. Appendix K: Deformation Tensors 390 • If we agree to use the e^i(S) for both d x and d x', so that d x = dxie^i(S) and d x' = dx'ie^i(S), then the following is a viable expansion for the relative deformation gradient F t: F t = Σij (Ft)ij e^i(S) ⊗ e^j(S) = Σij (Ft)ij [e^i(S)] [e^j(S)]T where (F t)ij = [Ft(S,S)]ij = [ e^i(S)]T Ft [e^j(S)] = (∂x'i/∂xj) (K.1.17) The verification is similar to the above, { Σ ij (Ft)ij [e^i(S)] [e^j(S)]T } dx // ≡ {expansion} dX = Σ ij (Ft)ij [e^i(S)] [e^j(S)]T { Σkdxke^k(S)} = Σ ij (Ft)ij Σkdxk [e^i(S)] [e^j(S)]T [ e^k(S)] = Σ ij (Ft)ij Σkdxk [e^i(S)] δj,k = [ Σ ij (Ft)ij dxj] e^i(S) = [ dx' i ] e^i(S) // using the fact that F t dx = dx' = d x' . ( K . 1 . 1 8 ) Since {expansion}d x = dx' and since ( ∇(x)x') dx = dx' by the chain rule, it seems reasonable to conclude that {expansion} = ( ∇(x)x') = Ft. Appendix K: Deformation Tensors 391 K.2 A More Complicated Deformation Flow Picture Consider now this flow picture, (K.2.1) There is much to be said about this drawing. The left side is the same as in Fig (K.1.2) shown above. The picture is simplified in that it shows only the linearized S-type transformations like F and not the full transformations like F, and these S-type matrices are now la beled right on the transformation arrows. We only care about these F matrices becau se we only care about the dumbbells like d x. For example, on the lower left we have d x = F d X . The two sides of the picture represent observations of the same flow by two independent flying cameraman observers, call them C and C*. On each side the basis vectors of the various frames are set by the motions of these cameramen. The two came ramen have agreed to start off at time t 0 with their camera platforms in exact alignment, so there is no need for a frame S 0*. The two frames of reference S and S* are relate d by some Galilean transformation (rotation plus translation) which brings the two independent camera platforms into alignment at time t : x* = Q(t) ( x-x0) + c(t) ⇒ d x* = Q(t) d x . (K.2.2) Here x0 is a randomly selected origin for the rotation Q(t), and c(t) the corresponding translation. As above, we only care about the Q(t) part of this transformation, so in terms of d x objects, the frames S and S* are in effect related by the rotation Q(t). The arrows in Fig (K.2.1) correctly describe the transformations of the d x type objects in moving between frames. In the lower part, for example, we have d x* = Q(t) d x d x = F d X d x* = F* d X . (K.2.3) Appendix K: Deformation Tensors 392 Comparing the left and right equations one has Q(t) d x = F* d X and then the center equation can be used on the left side to get Q(t) F d X = F*d X . Since this has to be true for any d X, we have Q(t) F = F* as shown in the drawing. This result is trivially obtaine d just by looking at the alternate arrow paths from frame S0 to frame S*. So: F* = Q(t) F or F*( x*,t*) = Q(t) F( x,t) . t* = t (K.2.4) At time τ we have a similar situation, but there are four arrows instead of three. Comparing the arrow paths from frame S to frame S'* one finds Ft*Q(t) = Q( τ)Ft ⇒ Ft* = Q(τ)FtQ(t)T or F t*(x*,τ*) = Q(τ) Ft(x,τ)Q(t)T t* = t (K.2.5) The two Q's are rotations (reflections included) and are therefore orthogonal so Q -1 = QT. Does F transform as a tensor with respect to rotation Q(t) ? We can think of F( x,t) as a property of the continuous material at location x and current time t. F describes the "state of defo rmation". If F transformed as a tensor with respect to Q(t), one would need this to be true, F* = Q(t) F Q(t)T , // not true! (K.2.6) which is the matrix form for the transformation of a ra nk-2 tensor as shown in (5.7 .3). But we have just seen that F* = Q(t) F so the required Q(t)T on the right is missing. We conclude therefore that in fact F, although it is called a tensor, does not transform as a tensor under Q(t). One then says that F is a non- objective tensor with respect to Q(t). Equation F* = Q( t) F in fact says that the columns of matrix F transform as vectors under Q(t), which is very differe nt from saying F transforms as a rank-2 tensor under Q(t). If one is trying to construct a phenomenological eq uation modeling a continuous material at point x and time t, one must make sure that equation is "c ovariant" (frame-indifferent) with respect to rotation Q(t). The observers (cameramen) in frame S and frame S* must see equations which have exactly the same form, which means the elements in the equations must be objective with respect to Q(t). See Section 7.15 for a general discussion of "covariance". Since F is non-objective, it is not directly useful in the construction of covariant m odel equations. Do any of the usual "derived tensors" tran sform as tensors with respect to Q(t) ? By "the usual derived tensors" we mean B, C, U, V and associated R all defined as follows: B = FF T = the left Cauchy-Green deformation tensor = the Piola deformation tensor (K.2.7) C = F TF = the right Cauchy-Green deformation tensor = the Finger deformation tensor (K.2.8) F = RU = VR R = rotation U,V = symmetric positive definite (K.2.9) Appendix K: Deformation Tensors 393 Tensors B and C are defined as shown, and both are therefore symmetric tensors. The last line is a statement of the polar decomposition theorem which says that any (real) non-singular matrix (det ≠ 0) can be uniquely written in these two ways (we apply this theorem to the deformation tensor F) F = RU = VR ⇒ U = RTVR and V = RURT // Lai p 110 (3.21.1,2,4) (K.2.10) where R is a rotation matrix and V and U are symmetric positive definite matrices (meaning the eigenvalues are all positive) known as th e left and right stretch tensors. Note that R is the same matrix in both the RU and VR forms. The idea is that the R matrix takes into account the rotational part of the deformation F, while U or V take into account th e stretch component of the deformation. If the deformation is a pure rotation, U = V = 1, whereas if the deformation is a pure stretch then R = 1. A general deformation is a rotation/stretch/shear affair and one will find that none of R, U, V are unity. One can combine the three e quations above to find that B = FFT = (VR)(VR)T = VRRTVT = VVT = VV = V2 // Lai p 121 (3.25.1) (K.2.11) C = F TF = (RU)T(RU) = UTRTRU = UTU = U2 . // Lai p 115 (3.23.1,2) (K.2.12) So our task is to discover whether any of these deri ved tensors transform as a tensor relative to Q(t). If they do transform as tensors (if they are objective), th en they are candidates for use in constructing model equations for the continuous material. We start with B and C: B* = F*F* T = (QF)(QF)T = QF FTQT = QBQT ⇒ B* = Q(t)BQ(t)T (K.2.13) C* = F* TF* = (QF)T(QF) = FTQTQF = FTF = C ⇒ C* = C . (K.2.14) Thus, the left Cauchy-Green deformation tensor B act ually does transform as a rank-2 tensor with respect to Q(t), so it is a tensorial tensor, it is "objectiv e". In contrast, since C* = C, the right Cauchy-Green deformation tensor does not transform as a rank-2 tensor. In fact each element of matrix C transforms as a tensorial scalar with respect to Q(t). What about V and U as defined above, the left and right stretch tensors? F = RU = VR F* = R*U* = V*R* . (K.2.15) Consider, F* = QF = Q(RU) = (QR) (U) = R*U* . (K.2.16) Since U is positive definite symmetric, and since QR is a rotation, and since the polar decomposition is unique, it must be that R* = QR and U* = U . (K.2.17) Next write Appendix K: Deformation Tensors 394 F* = QF = Q(VR) = (QVQT)(QR) = V* R* . (K.2.18) Since the eigenvalues of symmetric V are determined by det(V- λI) = 0, and since this is the same as the equation det(QVQT-λI) = 0, QVQT has the same eigenvalues as V and so (QVQT) is symmetric and positive definite. Due to this fact and the fact that QR is a rotation, and the fact that the polar decomposition is unique, it must be that V* = QVQ T and QR = R* . (K.2.19) and thus V transforms as a true tens or. So here is a summary for our tensors of interest. Only two of the five deformation tensors actually transform as tensors. The references are to Lai page 336-337 : F * = Q ( t ) F / / L a i ( 5 . 5 6 . 2 1 ) B* = Q(t)BQ(t) T // rank-2 tensor with respect to Q(t) so objective // Lai (5.56.31) C * = C / / L a i ( 5 . 5 6 . 2 8 ) U * = U V* = Q(t)VQ(t) T // rank-2 tensor with respect to Q(t) so objective R * = Q ( t ) R ( K . 2 . 2 0 ) Comment: Recall that F = F( x,t) has a hidden parameter t 0 so in fact F = F t0(x,t). Similarly, all derived tensors have this same hidden parame ter. Thus, for example, one could write the transformation of B as Bt0*(x*,t*) = Q(t) B t0(x,t)Q(t)T t* = t x* = Q(t) ( x-x0) + c(t) dx* = Q(t) d x . (K.2.21) The parameter t 0 is treated as a fixed constant here and plays no role in the question of whether or not B transforms as a rank-2 tensor. The important time argumen t of B is the current time t, and the main idea is that B*(t) = Q(t) B(t)Q(t)T so that B(t) is objective with respect to the rotation Q(t). The transformation is valid for any value of t 0. In the limit that t 0 → t, the equation says 1 = Q(t) 1 Q(t)T which of course is true since rotation Q(t) is orthogonal. Do any of the usual relative derived tensors transform as tensors with respect to Q(t) ? Again we think of a relative tensor W t as being a property of the continuous material at current time t, a measure of the state of deformation. Such a tensor is objective only if W t* = Q(t)W tQ(t)T. With regard to the above Comment, in this new situation it is the t of W t which is the time variable of interest (the current time), and time τ is regarded as a fixed parameter, as was t 0 in the Comment. It just happens that the notational positions of the current time t and the parameter time τ are swapped in this case relative to the last, so now we have Wt*(x*,τ*) = Q(t)W t(x,τ) Q(t)T τ* = τ x* = Q(t) ( x-x0) + c(t) dx* = Q(t) d x . (K.2.22) Appendix K: Deformation Tensors 395 A tensor W t which transforms as a rank-2 tensor (is objectiv e) with respect to rotation Q(t) must satisfy the relation above, where the arguments of both Q ro tations are t. As in the Comment above, this transformation is valid for any value of parameter τ, and as τ→t, the equation says 1 = Q(t) 1 Q(t)T. Our study of the transformation properties of the relativ e tensors proceeds in a manner similar to that used for the regular tensors above. We start with B t ≡ FtFtT : Bt* = Ft*Ft*T = [Q(τ) Ft QT(t)] [Q(τ ) Ft QT(t)]T = Q(τ) Ft QT(t) Q(t) F tT Q(τ)T = Q(τ) Ft FtT Q(τ)T = Q(τ) Bt Q(τ)T // not a rank-2 tensor since t ≠ τ (K.2.23) Next comes C t ≡ FtTFt : C t* = Ft*TFt* = [Q(τ) Ft QT(t)]T [Q(τ) Ft QT(t)] = Q(t) F tT Q(τ)T Q(τ) Ft QT(t) = Q ( t ) F tT Ft QT(t) = Q(t) C tQT(t) // yes a rank-2 tensor with respect to Q(t) (K.2.24) What about the left and right relative stretch tensors V t and Ut? F t= RtUt = VtRt F t* = Rt*Ut* = Vt*Rt* (K.2.25) Consider, F t* = Q(τ) Ft QT(t) = Q(τ) RtUtQT(t) = [Q(τ ) RtQT(t)] [Q(t)U tQT(t)] = R t*Ut* . (K.2.26) Since [Q( τ) RtQT(t)] is a rotation and since [Q(t)U tQT(t)] is a symmetric positive definite matrix by the argument given in the previous section, and since the polar decomposition is unique, it must be that R t* = Q(τ) RtQT(t) and U t* = Q(t)U tQT(t) // U t is a rank-2 tensor (K.2.27) Finally, write F t* = Q(τ) FtQT(t) = Q(τ)VtRtQT(t) = [Q(τ )VtQT(τ)] [Q(τ) RtQT(t)] = V t* Rt* (K.2.28) By the same argument used severa l times above, we conclude that Rt* = Q(τ)RtQT(t) and V t* = Q(τ)VtQT(τ) // V t is not a rank-2 tensor, τ ≠ t (K.2.29) The rule for transforming R t is the same as found a few lines above. Appendix K: Deformation Tensors 396 Here then are the conclusions, w ith references to Lai page 472: Ft* = Q(τ)FtQT( t ) / / L a i ( 8 . 1 3 . 6 ) Bt* = Q(τ)BtQ(τ)T // Lai (8.13.12) Ct* = Q(t)C tQT(t) // rank-2 tensor with respect to Q(t) so objective // Lai (8.13.10) Ut* = Q(t)U tQT(t) // rank-2 tensor with respect to Q(t) so objective // Lai (8.13.9) Vt* = Q(τ)VtQT(τ) / / L a i ( 8 . 1 3 . 1 2 ) Rt* = Q(τ)RtQT(t) // Lai (8.13.8) (K.2.30) Notice that among the "normal" tensors, B and V are objective, whereas among the "relative tensors" it is Ct and Ut that are objective. All the other tensors are "non-objective". K.3 Covariant form of a solid constitutive equation involving the deformation tensor For a solid continuous material in frame S one can consider a stress/deformation relationship of the form T = f(B), where T is the Cauchy stress tensor, B is the left Cauchy-Green deformation tensor mentioned in (K.2.7) above, and f is "some function". In frame S*, there will be some covariant version of the equation T* = f*(B*). If the medium is isotropic (rotationally invariant in its properties), then f* = f and one will have T* = f(B*) in Frame S*. Two observers of the same system in frames relate d by a rotation cannot observe different functions f ≠ f* if the material is isotropic. Notice that there are two separate issues here: (1) equation must be covariant under rotations to be viable; (2) isotropic implies f = f*. If f is a polynomial, or a function which can be approximated by one (f is smooth), then T = f(B) with polynomial coefficients which are rotational scalars (with respect to Q) is a viable equation form for the following reason: since B is a rank-2 tensor by (K.2.20), so is any power of B, B* 2 = [QBQT][QBQT] = Q B2QT etc. ( K . 3 . 1 ) and if the polynomial coefficients are scalars, then f(B) is a rank-2 tensor. Just as a particle force F transforms as a rank-1 tensor under rotations, the Cauchy stress tensor T transforms as a rank-2 tensor under ro tations, and then both sides of T = f(B) transform in the same way -- as rank-2 tensors. Any candidate equation between T and a deformation tensor which did not have both sides transforming the same way would be invalid fro m the get-go (except perhaps as an approximation). The scalar coefficients must be functions of the B ij and there are three such scalars known as the principal scalar invariants of B (Lai p 40), one of wh ich is det(B), so the scalar coefficients can be any functions of these three scalar invariants. Fu rthermore, one can use the fact that B = FFT is symmetric along with the Cayley-Hamilton theorem (symmetric matrix B satisfies its own secular equation, whose coefficients by the way are those scalar invariants) to show that any powers of B in polynomial f(B) larger than degree 2 can be expressed as a linear combination of I, B and B 2. One ends up then with T = aI + bB + cB2 where a,b,c are functions of the three scalar invariants of tensor B. Since both sides of T = f(B) transform in the same way (rank-2 tensors), the equation T = f(B) is "covariant" as discussed in Section 7.15, meaning it has the same form in frame S* as it has in S. The equation T = f(B) is a relation between stress and strain in the form of deformation, and as such is called a constitutive equation for the continuous ma terial. One wants such equations to be covariant between frames of reference related by any Galilean tr ansformation (rotation + translation), even if one or Appendix K: Deformation Tensors 397 both of these frames are non-inertial. This is an extension of Hooke's Law for a spring, F = -k Δ x , which is covariant under rotations and translations. In contrast, equations of motion are covariant only if both frame S and S* are inertial frames. Notice that this entire discussion falls apart complete ly if one tries T = f(F) or T = f(C) as a candidate constitutive relation, since then the two sides of the equation don't transform the same way. This subject is discussed in Lai pp 334-342 and p 40 for the scalar invariants. The requirement of covariance for an isotropic material and the fact that B is symmetric and transforms as a tensor puts a severe restriction on the form of the constitutiv e equation and we end up with T = aI + bB + cB2. Since one can replace B3 = αB2 + βB + γI, if B is invertible (detB ≠ 0) one has B2 = αB + βI + γ B-1 and this allows the alternate form T = a'I + b'B + c'B-1 . This last equation is used to model large deformations of an isotropic elastic material. An example is the Mooney-Rivlin theory for rubber. K.4 Some fluid constitutive equations It was noted just above (K.1.10) that the relative deformation tensors are appropriate when one is interested in time derivatives of the tensors. It was also noted in (K.2.30) th at the relative deformation tensor C t is objective. One can expand C t(x,τ) in a Taylor series about current time t in this manner ( ∂τ ≡ ∂/∂τ) , Ct(x,τ) = Σn=0∞ [ ∂τnCt(x,τ)]τ=t (τ-t)n/n! = Σ n=0∞An(x,t) (τ-t)n/n! // Lai p 463 (8.10.1) An(x,t) ≡ [ ∂τnCt(x,τ)]τ=t , ( K . 4 . 1 ) where the coefficient derivatives are given the names A n(x,t) called Rivlin-Ericksen tensors. Each of these coefficient tensors is in fact objective, just as is C t, since (as usual, t = t*, τ = τ* ) Q(t) [ ∂ τnCt(x,τ)]τ=t QT(t) = { ∂ τn [Q(t) Ct(x,τ) QT(t)]}τ=t = { ∂τ*n Ct*(x*,τ*)}τ*=t ⇒ Q(t) A n(x,t) QT(t) = A*n(x*, t ) ( K . 4 . 2 ) These A n(x,t) tensors appear in various covariant models of "non-Newtonian" fluid behavior, the general study of which is called rheology, based on the Greek wo rd for a current flow (a rheostat controls electric current), // OED2 Here are a few covariant constitutive equations and the names assigned to them (Lai p 481). Note that for any normal fluid, there is always a -pI tensor term in the expression for stress T, where p is the fluid pressure and I is the identity matrix. The diagonal el ements of matrix -pI are the equal normal stresses of the surroundings of a tiny cube of fluid pulling out on the cube faces, hence the -p (p > 0) since we know the fluid actually pushes in on the cube. Appendix K: Deformation Tensors 398 T = -pI + functional of C t(τ), τ ≤ t // "simple" fluid, since ∇nFt not involved (C t=FtT Ft) T = -pI + ∫-∞ t dτ f1(τ) Ct(τ) // single-integral simple fluid. f 1(τ) = a memory weight function T = -pI + f(A 1, A2....AN) // Rivlin-Ericksen incompressible fluid of complexity N T = -pI + f(A 1,A2) // viscometric flow fluid (there are conditions on A 1 and A2) T = -pI + μ1A1 + μ2A12 + μ3A2 // second order fluid (paint, blood, polymers) T = -pI + μA 1 // incompressible Newtonian fluid (fluids like water) (K.4.3) It turns out that A 1 = 2D where D = [( ∇v) + (∇v)T] /2 ≡ (∇v)sym , so A1 is twice the rate of deformation tensor D. The other A n can then be found from this recursion relation, A n+1 = dtAn + An(∇v) + (∇v)TAn // Lai p 468 (8.11.2) (K.4.4) Here v is the fluid velocity vector and d t = d/dt = D/Dt. Again, ( ∇v) is the subject of Appendix G. K.5 Corotational and other objective time derivatives of the Cauchy stress tensor The Cauchy stress tensor T transforms as a tensor und er Q(t); it is objective. One can write therefore, T*( x*,t*) = Q(t) T( x,t) Q(t)T t* = t x* = Q(t) ( x-x0) + c(t) dx* = Q(t) d x . (K.5.1) Clarification of the above equation One can think of the above equation T* = QTQT as involving operators in Hilbert Space, as outlined in Section E.7. In the upper part of Fig (K.2.1) above we show four different frames of reference called S, S*, S' and S'* each of which has its own set of basis vectors we might call un, u*n, u'n and u*'n. It happens that the picture refers to S and S* at time t, and S' and S'* at time τ, but any basis vectors can be "used" at any time one wants. For example, he re are four expansions of the operator T( x,t) T( x,t) = Σab Tab(x,t) ua ⊗ ub = Σab T*ab(x*,t) u*a ⊗ u*b = Σ ab T'ab(x',t) u'a ⊗ u'b = Σab T'*ab(x'*,t) u'*a ⊗ u'*b (K.5.2) in which we see four different kinds of components T ab, T*ab, T'ab, T'*ab . The spatial arguments of each component are written as appropriate for that fram e of reference and of c ourse all "correspond" to each other (for example, x' = Ft(x,τ)). Recall from (2.5.1) the notion of the transformation of a contravariant vector field in developmental notation V'(x') = R V(x) contravariant R ik(x) ≡ (∂x'i/∂xk) R = S-1 (2.5.1) where the argument is appropriate to the space of interest. Appendix K: Deformation Tensors 399 The time argument t in the above four expansions of T can be set to any arbitrary value. The stress tensor at a point x is in general a function of time t. One could for example set t = τ in all the expansions. Having said this, we now decide that only the frame S basis vectors un shall be used in our expansions and components. Then for example ( these un were called e^n(S) earlier) T( x,t) = Σab Tab(x,t) ua ⊗ ub T*( x*,t) = Σab T*ab(x*,t) ua ⊗ ub Q(t) = Σab Qab(t) ua ⊗ ub . (K.5.3) Our operator statement of objectivity then become s the following when expressed in components, T*( x*,t*)ij = Q(t)ia T(x,t)ab Q(t)T bj t* = t . (K.5.4) Thus, there should be no confusion about the following two equations which we express back in operator notation with the position arguments su ppressed (but shown on the right) T*(t) = Q(t) T(t) Q(t)T // T*( x*, t) = Q(t) T( x,t) Q(t)T T*(τ) = Q(τ) T(τ ) Q(τ) T // T*( x*, τ) = Q(τ) T(x,τ) Q(τ )T . (K.5.5) Problem: The tensor dT/dt fails to transform as a rank-2 tensor, even though T does so transform. If one tries to construct covariant c onstitutive equations involving dT/dt, a problem arises because dT/dt is non-objective, T*(t) = Q(t) T(t) Q(t) T (dT*/dt) = Q (dT/dt) QT + [ (dQ/dt) T QT + Q T (dQ/dt)T ] , (K.5.6) so there are two extra unwanted terms. Just as B = FFT is constructed to provide an objective derived tensor from non-objective F, one can construct a derived version of (dT/dt) which is objective. In the next three sections, three different de rived versions are described. The corotational/Jaumann derivatives The first step is to define an adjusted stress tensor J t(τ) at time τ according to (see Lai p 483 (8.19.3). Lai does not have a t subscript on J). Jt(τ) ≡ RtT(τ) T(τ ) Rt(τ) // J t(x,τ) ≡ RtT(x,τ) T(x,τ) Rt(x,τ) (K.5.7) where R t(τ) is the rotation which appears above in (K.2.25), where we had (showing τ arguments), Appendix K: Deformation Tensors 400 Rt*(τ) = Q(τ) Rt(τ)QT(t) . (K.2.27) (K.5.8) The tensor R t(τ) is non-objective due to appearance of Q( τ) instead of Q(t) on the left (see comments near (K.2.22) on W t ). Recall that this rotation R t(τ) is unique and is determined from the deformation tensor by the polar decomposition F t(τ) = Rt(τ)Ut(τ) = Vt(τ) Rt(τ). Thus, in some sense J t(τ) knows about the stress tensor T( τ), and it knows something about the deformation tensor through R t(τ). [ The meaning of the term "corotational" is explained far below. ] The claim now is that the time derivative of this corotating stress tensor J t is objective, meaning that tensor dtJt transforms as a rank-2 tensor under th e rotation Q(t). Here is a proof : We first assemble the following facts, T*(τ) = Q(τ) T(τ ) Q(τ) T // transformation of stress tensor T at time τ, (K.5.5) Rt*(τ) = Q(τ) Rt(τ)QT(t) // how R t(τ) transforms, where F t(τ) = Rt(τ)Ut(τ), (K.5.8) Jt(τ) ≡ RtT(τ) T(τ ) Rt(τ) // definition of J t(τ) in frame S, (K.5.7) Jt*(τ) ≡ Rt*T(τ) T*(τ ) R*t(τ) // corresponding J t* in frame S* (K.5.9) and then we combine these ingredients to obtain a transformation rule for J t : J t*(τ) ≡ Rt*T(τ) T*(τ ) R*t(τ) = [Q( τ) Rt(τ)QT(t)]T [Q(τ) T(τ ) Q(τ)T] [Q(τ) Rt(τ)QT(t)] = [Q(t) R tT(τ)QT(τ)] [Q(τ) T(τ ) Q(τ)T] [Q(τ) Rt(τ)QT(t)] = Q ( t ) R tT(τ) [QT(τ)Q(τ)] T(τ) [Q(τ )TQ(τ)] Rt(τ)QT(t) = Q ( t ) [ R tT(τ)T(τ) Rt(τ)] QT(t) = Q ( t ) J t(τ) QT( t ) . ( K . 5 . 1 0 ) Since this equation J t*(τ) = Q(t) J t(τ) QT(t) fulfills the condition described earlier for W t to be objective, we conclude that the corotating stress transforms as a rank-2 tensor, where τ is treated as a parameter. Consider now the limit of (K.5.10) as τ → t. One finds, Jt(τ) ≡ RtT(τ) T(τ ) Rt(τ) / / (K.5.7) Jt(t) ≡ RtT(t) T(t) R t(t) = 1 T(t) 1 = T(t) (K.5.11) and J* t(τ) ≡ R*tT(τ) T*(τ ) R*t(τ) // (K.5.9) J*t(t) ≡ R*tT(t) T*(t) R* t(t) = 1 T*(t) 1 = T*(t) . (K.5.12) In this limit, the corotation R t-1(τ) has come to a halt, and (K.5.10) becomes a statement that T is objective. Appendix K: Deformation Tensors 401 More interestingly, we can apply ∂τn = ∂n/∂τn to both sides of(K.5.10) to get dτn Jt*(τ) = Q(t)[ d τn Jt(τ) ] QT(t) . (K.5.13) Taking the limit τ→t then gives [dtn Jt*](t) = Q(t) [d tn Jt](t) QT( t ) ( K . 5 . 1 4 ) which says that d tnJt are all objective tensors. And in particular, for n = 1, (d tJt)* = Q(t) (d tJt) QT( t ) , ( K . 5 . 1 5 ) and this concludes our proof that dJ t/dt is objective, whereas dT/dt is not objective. The above objective tensor time derivatives are so metimes written using the following strange notation To n ≡ [dnJt(t)/dtn], n = 1,2,3... To ≡ To 1 J t(τ) ≡ RtT(τ) T(τ ) Rt(τ) (K.5.16) and these are called corotational or Jaumann derivatives (Lai p 484) [J aumann-Zaremba]. It can be shown that To = dtT + TW-WT where W = [( ∇v) – (∇v)T]/2 = "the spin tensor" // Lai p 484 (8.19.10) ( K . 5 . 1 7 ) The Oldroyd Lower convected derivatives An alternative solution to the same problem uses a different adjusted stress tensor, J L(τ) ≡ FtT(τ) T(τ ) Ft(τ) // Lai p 484 (8.19.12) (K.5.18) We suppress the t subscript on J L just to avoid having to write (J L)t(τ). In the table (K.2.30) one sees that Ft transforms the same way R t does, so one can repeat the above anal ysis to conclude that the derivatives [dnJL(t)/dtn] are all objective tensors (just replace R t→ Ft everywhere), so TΔ n ≡ [dnJL(t)/dtn] , n = 1,2,3... TΔ ≡ TΔ 1, JL(τ) ≡ FtT(τ) T(τ ) Ft(τ) // = T∪ n (K.5.19) and these are the "Oldroyd lower convected derivatives" (Lai p 485 uses T ∪ n) . TΔ is sometimes called the Cotter-Rivlin stress rate. It can be shown that TΔ = T∪ = dtT + T(∇ v) + (∇v)TT . // Lai p 485 (8.19.21) (K.5.20) The Oldroyd Upper convected derivatives Finally, consider again from (K.2.30) the non-objective way that F t transforms, Appendix K: Deformation Tensors 402 Ft*(τ) = Q(τ)Ft(τ)QT(t) ⇒ (Ft-1)*(τ) = Q(t) (F t-1(τ)) QT(τ) // inverted ⇒ (F t-1)T*(τ) = Q(τ ) (Ft-1)T(τ) QT(t) . // then transposed (K.5.21) This object F t-1,T therefore transforms the same way R t and Ft transform, so we obtain a third set of objective time derivatives called the Oldroyd upper convected derivatives (Lai p 486 uses T^n) T∇ n ≡ [dnJU(t)/dtn] , n = 1,2,3... T∇ ≡ T∇ 1, JU(τ) ≡ Ft-1(τ) T(τ ) Ft-1,T(τ) // = T^n (K.5.22) The meaning of the term "convected" is explained below. It can be shown that T∇ = T^ = dtT – (∇v)T – T(∇v)T // Lai p 486 (8.19.26) (K.5.23) Covariant constitutive equations Constitutive equations involving an objective time de rivative of the stress tensor are called "rate type constitutive equations". Here are some models for incompressible fluids : T = -pI + S where S + λSo = 2μD // a convected Maxwell fluid T = -pI + S where S + λ(∂ S/∂t) = 2μD // linear Maxwell fluid, see below (non-covariant) T = -pI + S where S = 2 μD // Newtonian fluid T = -pI + S where S + λ 1So = 2μ(D + λ2Do) // a corotational Jeffrey fluid T = -pI + S where S + λ 1T∇ = 2μ(D + λ2T∇ ) // Oldroyd fluid A (K.5.24) Fluids with stress time derivatives in their constitutive equations exhibit both elas tic and viscous behavior at the same time. Pull on a chunk of such a fluid and th e pull is initially resisted by an elastic force, but after a while the internal stress fiel d damps out (molasses, honey) and that elastic force goes away, as if the fluid were microscopically constructed of little springs and dragging dashpots. When a constitutive equation includes a time derivative of stress, the "response" (in this case D = [( ∇v) + (∇v)T]/2 ) to the "stimulus" (T or S) includes factors of the form e-t/c where the c are decay time constants which are functions of the fluid parameters λi. In this case, the fluid has memory of its past over a time period less than these time constants, as with the honey example. For flow that is very slow relative to these time constants, the time derivative term may be neglected. In the moderately slow flow case, it can be shown that the distinction between the corotational time derivative So and (dS/dt) can be neglected and then the Appendix K: Deformation Tensors 403 convected Maxwell fluid shown above becomes the tr aditional linear Maxwell fluid which is modeled on those springs and dashpots with S + λ (∂S/∂t) = 2μD. (The time derivatives here are meant to act only on the second argument of S( x,τ) so may be regarded as partial derivatives. ) If λ = 0, the linear Maxwell fluid becomes an (incompressible) Newtonian fluid like water which has no memory. Comment: The linear Maxwell fluid equation S + λ (∂S/∂t) = 2μD can be solved for S using the standard Green's Function method and the solution is S(t) = 2 ∫-∞ t dt' [ (μ/λ)e-(t-t')/ λ] D(t') where the bracketed quantity (the Green's Function or kernel) is called the stress relaxation function φ(t-t'). One can see in this solution the notion of memory (history) with time constant λ: the stress of the present is a function of the rate of deformation D going on in the entire past histor y. This solution fits into the "simple fluid" form shown earlier, where recall that D = (1/2)A 1 and A1 = [ ∂τCt(x,τ)]τ=t. Our main point is to demonstrate the construction of constitutive equations which are covariant with respect to rotations, and which therefore can contain only tensors which in fact transform as tensors under rotations. In continuum mechanics, such te nsors are said to be objective tensors. Interpretation of the adjusted stress tensors discussed above. In Chapter 2 we discuss the notion of the transformation of a contravariant vector V' = R V in developmental notation. In x'-spa ce, the vector components are V' i = RijVi where V i are the components in x-space. If R is a rotation matrix, then the unit basis vectors in the two spaces can be taken as Cartesian, call them u'n in x'-space and un in x-space. We have these two expansions of V: V = Σn Vn un = Σn V'n u'n where V n = V • un V'n = V • u'n . (K.5.25) In the "active view" of things, we can think of V' = R V as creating a new vector V' in x-space from the old vector V by rotating the vector V by R. In the "passive view", we think of the V' i as the components of the original vector V projected onto the backwards-rotated basis vectors u'n = R-1 un. To verify this relation between the basis vectors, we can write V' n = V • u'n = V • R-1un = R V • RR-1un = RV • un = V'• un = V'n . (K.5.26) So one can think either of V being rotated forward in x-space into V' where V' has x-space components V'n , or one can think of the V' n as the components of V one measures in frame that is backwards rotated by R-1 , that is, u'n = R-1 un. Consider then a rank-2 tensor M that transforms as shown in (5.7.1) according to M' = R M RT. The passive interpretation is that the components M' ij are those one observes in a frame of reference whose basis vectors are rotated by R-1 relative to the basis vectors of the unpr imed frame, just as in the vector case of the last paragraph. If we now set R = R-1, then M' = R-1 M (R-1)T tells us that the components M'ij of tensor M are those measured in a frame whose basis vectors are rotated forward by R relative to the unprimed frame. If it happens that R = R(t), we would say that M' ij are the components of M which are observed in a frame of reference which is rotating by R(t) relative to the frame of the unprimed components M ij. The basis vectors of the primed frame are then u'n = R un . Appendix K: Deformation Tensors 404 With this long-winded introduction, we now consider the corotational stress tenser J t(τ) shown in (K.5.7), J t(τ) ≡ RtT(τ) T(τ ) Rt(τ) . (K.5.7) (K.5.27) Since Rt(τ) is a rotation, R tT(τ) = Rt-1(τ), so we have, suppressing τ, Jt = Rt-1T Rt . ( K . 5 . 2 8 ) Therefore, we can regard (J t)ij as T'ij, the components of stress T measured in a frame which is rotating by Rt relative to the frame in which the T ij are measured. Since this primed frame rotates by R t relative to the unprimed frame, it is called a corotating frame, and J t is then called the corotational stress, and its time derivative is called the corotational stress rate. We next consider the upper Oldroyd stress defined above in (K.5.22), J U ≡ Ft-1 T (Ft-1)T . (K.5.22) ( K . 5 . 2 9 ) In analogy with the above discussion, we can regard (J U)ij as T'ij, the components of stress T measured in a frame which is deforming by Ft relative to the unprimed frame. That is to say, the basis vectors of the primed frame are given by u'n = Ft un. In this case, since F t is not a rotation, if the un start as unit vectors, then the u'n are not unit vectors. One can think of each basis vector un as being aligned with its own dumbbell d x(n) and then we have d x'(n)= Ft dx(n). What this says is that the basis vectors are embedded in the fluid which defo rms as it flows according to F t. The basis vectors "convect" with the fluid, so this upper Oldroyd stress is called the upper convective stress tensor. For the lower Oldroyd stress in (K.5.19) we have J L ≡ FtT T Ft (K.5.19) ( K . 5 . 3 0 ) and this cannot be written in the form J L = R-1 M (R-1)T so this does not fit into our interpretive template. But in the next section we show that J L is the covariant partner to the contravariant tensor J U so they are both the same animal and we are happy to have the interpretation above for J U. The Oldroyd convected stresses in developmental and standard notation In the previous section two adjust ed stress tensors were introduced, J U ≡ Ft-1 T Ft-1T // upper JL ≡ FtT T Ft . / / l o w e r ( K . 5 . 3 1 ) In developmental notation, a covari ant tensor gets an overbar while a contravariant one does not. If we assume that frame S is Cartesian, then T = T ¯ as was discussed for vectors in Section 5.9. We can interpret the above two equati ons in this manner Appendix K: Deformation Tensors 405 J ≡ Ft-1 T Ft-1T // upper J¯ ≡ FtT T¯ Ft / / l o w e r ( K . 5 . 3 2 ) which we compare with (5.7.1) where we change generic matrix name M to T, T' = R T RT // contravariant rank-2 tensor transforms this way T¯' = ST T¯ S // covariant rank-2 tensor transforms this way . (K.5.33) Setting R = F t-1 and S = R-1 = Ft gives T' = F t-1 T Ft-1T // contravariant rank-2 tensor T¯' = FtT T¯ Ft // covariant rank-2 tensor (K.5.34) Therefore we identify J U = J = T' = contravariant stress tensor T viewed from a frame convecting at F t-1 JL = J¯ = T¯' = covariant stress tensor T viewed from a frame convecting at F t-1 (K.5.35) In Standard Notation the two equations J ≡ F t-1 T Ft-1T // upper J¯ ≡ FtT T¯ Ft / / l o w e r ( K . 5 . 3 6 ) become J ij = (Ft-1)i a (Ft-1)j b Tab ( J U)ij = Jij = contravariant Jij = (Ft-1)ia (Ft-1)jb Tab ( J L)ij = Jij = covariant (K.5.37) and this explains the meaning of the words "upper" and "lower" in respect to the Oldroyd objects. See footnote on Lai page 485. References 406 References Listed in alphabetical order by last name of firs t author. A quick web search on article title can usually locate documents with broken web links. Thos e below were last checked May 18, 2016. L. A. Ahlfors, Complex Analysis, 2nd Ed. ( McGraw-Hill, New York, 1966). G. Backus, Continuum Mechanics (Samizdat Press, Golden Colo., 1997). J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964). E. B. Christoffel, "Ueber die Transformation der homogenen Di fferentialausdrücke zweiten Grades", Journal für die reine and angewandte Mathemarik , 70 (1869), 46–70, 241–245. This paper may be found in Christoffel's Collected Mathematical papers , Gesammelte Mathematisch e Abhandlungen, 2 vols. (Tuebner, Leipzig-Berlin, 1910), downloadable from Google books. R. Hermann, Ricci and Levi-Civita's Tensor Analysis Paper (English translation with comments) (Math Sci Press, Brookline, MA, 1975, perhaps on-line) First, with much praise, Hermann provides an English translation of the French Ricci & Levi-Civita pa per referenced below, updating words, phrases and symbols to the current day. Second, he inserts perhap s 180 pages of inline italicized text which translates the ideas of the paper into mathematical frameworks not known or not connected to by the authors (eg, fiber bundles, direct product spaces, Killing vector s, moving frames, group theory, etc.). Hermann presents material that was more simply understood by later authors (eg, E. Cartan). Earlier in 1966 Hermann wrote a book Lie Groups for Physicists which contains the group theory chunk of this added material. One realizes that differential geometry is a very large field touching upon many areas of Mathematics (a house with many mansions). W.M. Lai, D. Rubin and E. Krempl, Introduction to Continuum Mechanics, 4th Ed. (Butterworth- Heinemann/Elsevier, Amsterdam, 2010) . This book has had the same three authors since its first edition in 1974, and it is with apologies to the last two authors that we reference the book just as Lai when we should be saying Lai et.al . or perhaps LRK. P. Lucht, Tensor Products, Wedge Produc ts and Differential Forms (2016, http://user.xmission.com/~rimrock ). H. Margenau and G.M. Murphy, The Mathematics of Physics and Chemistry, 2nd Ed. (D. van Nostrand, London. 1956). A. Messiah, Quantum Mechanics (John Wiley, New York, 1958). Refe rence is made to page 878 (Vol II) of the North-Holland 1966 fifth printing paperb ack two-volume set, Chapter XX paragraph 2. P. Moon and D.E. Spencer, Field Theory Handbook, Including Coor dinate Systems, Differential Equations and their Solutions (Springer-Verlag, Berlin, 1961). Th is is the place to find explicit expressions for differential operators in many speci fic curvilinear coordinate systems. The book writes the Laplace and Helmholtz equations in various systems and discusses separability and the special functions that arise in the separated equations. References 407 P.M. Morse and H. Feshbach, Methods of Theoretical Physics ( McGraw-Hill, New York, 1953). M.M.G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen (Springer) 54 (1–2): 1 25–201 (March 1900). This huge 77 page paper is sometimes referred to as "the bible of tensor analysis". Levi-Civita was a student of Ricci and they worked together on this paper and later elaborations . Their work was instrumental in Einstein's later discovery of general relativity. The title is "Met hods of absolute differential calculus and their applications". Absolute differential calculus was the authors' phrase for what is now called Tensor Analysis/Calculus/Algebra. The word absolute referred to the idea of equations being covariant (see Section 7.15 above). R. Sjamaar, Manifolds and Differential Forms (2015, http://www.math.cornell .edu/~sjamaar/manifolds ). I. Stakgold, Boundary Value Problems of Mathematical Physics, Volumes 1 and 2 (Macmillan, London, 1967). J.J. Sylvester, "On the General Theory of Associat ed Algebraical Forms" (Cambridge and Dublin Math. Journal, VI, pp 289-293, 1851). This pa per appears in H.F. Baker, Ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge University Press, 1901). S. Weinberg, Gravitation and Cosmology: Principles and App lications of the General Theory of Relativity (John Wiley & Sons, New York, 1972). E. B. Wilson (notes of J.W. Gibbs), Vector Analysis (Dover, New York, 1960)