tensor analysis and curvilinear coordinates

1 Tensor Analysis and Curvilinear Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: Feb 10, 2014 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.................................. 11 Example 1: Polar coordinates (N=2)............................................................................................ ......11 Example 2: Spherical coordinates (N=3)........................................................................................... 12 Cartesian Space and Qu asi-Cartesian Space...................................................................................... .14 Pictures A,B, C and D.......................................................................................................................... 14 Coordinate Lines................................................................................................................................. 15 Example 1: Polar coordinates, coordinate lines ................................................................................ .16 Example 2: Spherical coordinates, coordinate lines .......................................................................... 16 Level Surfaces..................................................................................................................................... 17 2. Linear Local Transformations associated with F : scalars and two kinds of vectors ................. 19 (a) Scalars.................................................................................................................... ........................ 20 (b) Contravariant vectors ...................................................................................................... .............. 21 (c) Covariant vectors.......................................................................................................... ................. 21 (d) Bar notation............................................................................................................... .................... 22 (e) Origin of the names co ntravariant and covariant........................................................................... 22 (f) Other vect or types? ........................................................................................................ ................ 23 (g) Linear transformations ..................................................................................................... ............. 23 (h) Vectors that are cont ravariant by definition ............................................................................... ...24 (i) Vector Fields .............................................................................................................. .................... 25 (j) Names an d symbols........................................................................................................................ 25 (k) Definition of the words "scalar", "vector" and "tensor" ................................................................ 26 3. Tangent Base Vectors e n and Inverse Tangent Base Vectors u' n.................................................. 28 (a) Definition of the e n ; the e n are the columns of S ......................................................................... 29 (b) en as a contrava riant vector ........................................................................................................... 30 (c) a semantic question: unit vectors......................................................................................... ......... 31 Example 1: Polar coordinates, tangent base vectors .......................................................................... 31 Example 2: Spherical Coordinates, tangent base vectors................................................................... 33 (d) The inverse tangent base vectors u' n and inverse coordinate lines................................................ 34 Example 1: Polar coordinates: inverse tange nt base vectors and inverse coordinate lines............... 35 4. Notions of length, distance and scalar product in Cartesian Space.............................................. 36 2 5. The Metric Tensor ........................................................................................................... ................. 38 (a) Definition of the metric tensor............................................................................................ ........... 38 (b) Inverse of the metric tensor............................................................................................... ............ 40 (c) A metric tensor is symmetric............................................................................................... .......... 41 (d) det(g) and g nn of a Cartesian-generated metric tensor are non-negative....................................... 41 (e) Definition of two ki nds of rank-2 tensors...................................................................................... 41 (f) Proof that the metric tensor and its inverse are both rank-2 tensors .............................................. 42 (g) Metric tensor c onverts vector types........................................................................................ ....... 44 (h) Vectors in Cartesian space ................................................................................................. ........... 44 (i) Metric tensor: covari ant scalar product and norm......................................................................... 45 (j) Metric tensor and tangent base vectors ..................................................................................... .....47 (k) The Jacobian J ............................................................................................................. .................. 49 (l) Some relations between g, R and S in Pictures B and C (Cartesian x-space). ............................. 52 Example 1: Polar coordinates: metric tensor and Jacobian............................................................... 53 Example 2: Spherical coordinates: metric tensor and Jacobian ........................................................ 54 (m) Special Relativity and its Metric Tensor: vectors and spinors .................................................... 54 (n) General Relativity an d its Metric Tensor ................................................................................... ...58 (o) Continuum Mechanics a nd its Metric Tensors.............................................................................. 59 6. Reciprocal Base Vectors E n and Inverse Reciprocal Base Vectors U' n........................................ 63 (a) Definition of the E n........................................................................................................................ 63 (b) The Dot Products and Reciprocity (Duality)................................................................................. 64 (c) Covariant partner for E n................................................................................................................ 66 (d) Summary of th e basic facts: ................................................................................................ .......... 66 (e) Repeat the above for the inverse transformation: definition of the U' n........................................ 67 (f) Expanding vectors on different sets of basis vectors ..................................................................... 68 (g) Another way to write the E n.......................................................................................................... 70 (h) Comparison of e ¯n and En.............................................................................................................. 71 (i) Handedness of coordinate systems: the e n , the sign of det(S), and Parity.................................... 72 7. Translation to the Standard Notation ........................................................................................ .....76 (a) Outer Products............................................................................................................. .................. 76 (b) Mixed Tensors and Notation Issues .......................................................................................... ....76 (c) The up/down bell goes off.................................................................................................. ........... 77 (d) Some Preliminary Translations: raisi ng and lowering indices on a vector with g ....................... 78 (e) Contraction of a Pair of Indices........................................................................................... .......... 79 (f) Dealing with the matrix R.................................................................................................. ............ 80 (g) Repeat the above section for S ............................................................................................. ......... 81 (h) About ε and δ ................................................................................................................................. 81 (i) Matrix Multiplication, the meaning of RT, and Rotations in Standard Notation............................ 82 1. Translation of determinants. ................................................................................................ ....... 82 2. Inverse of R and S.......................................................................................................... ............. 82 3. Tensor g raises and lowers any index. ........................................................................................ 83 4. Raising Lowering Rule: ...................................................................................................... ........ 84 5. Contraction Tilt Reversal Rule: ............................................................................................. .....85 6. The Diagonal g Rule: ........................................................................................................ .......... 85 7. Matrix Multiplication in the Standard Notation.......................................................................... 85 8. Transpose of a rank-2 tensor................................................................................................ ....... 89 3 9. Orthogonality rules. ........................................................................................................ ............ 91 10. Rotation matrices. ......................................................................................................... ............ 91 11. Variations on the relation between g and g'. ............................................................................. 94 (j) Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita........................ 94 (k) The Contraction Tilt-Reversal Rule ......................................................................................... .....97 (l) The Contraction Neutralization Rule .......................................................................................... ...99 (m) Raising and lowering indices on g .......................................................................................... ...101 (n) Other forms of R .......................................................................................................... .............. 101 (o) Summary of facts about R................................................................................................... ........ 102 (p) Repeat all the above for S................................................................................................. ........... 102 (q) Theorem: Sa b = Rba and Sab = Rb a ( reflect indices in vertical line between them) ................. 102 (r) Orthogonality Rules, the Inversion Rule, and the Cancellation Rule .......................................... 104 (s) The tangent and reciprocal b ase vectors and expansions on same .............................................. 105 (t) Comment on Covariant versus Contravariant .............................................................................. 109 (u) The Significance of Tensor Analysis ........................................................................................ ..110 (v) The Christoffel Business: covariant derivatives......................................................................... 113 (w) Expansions of higher order tensors ......................................................................................... ...114 8. Transformation of Differential Length, Area and Volume......................................................... 116 Overview........................................................................................................................................... 116 (a) The differential N-piped mapping ........................................................................................... ....117 (b) Properties of the finite N-piped spanned by the e n in x-space..................................................... 119 (c) Back to the differential N-piped mapping: how edges, areas and volume transform................. 121 1. The Setup. ................................................................................................................................. 121 2. Edge Transformation................................................................................................................. 122 3. Area Transformation......................................................................................................... ........ 122 4. Volume Transformation....................................................................................................... .....124 5. Covariant Magnitudes........................................................................................................ ....... 125 6. Two Theorems : g' / h' n2 = g'nn g' = cof(g' nn) and |(Πx i≠nei)| = cof(g'nn) ...................... 126 7. Cartesian-View Magnitude Ratios. ........................................................................................... 127 8. Nested Cofactor Formulas. ................................................................................................... ....128 9. Transformation of arbitrary differential vectors, areas and volume......................................... 128 10. Concatenation of Transformations.......................................................................................... 130 Examples of area magnitude transformation for N = 2,3,4............................................................... 131 Example 2: Spherical Coordinates: area patches ............................................................................ 132 (d) Transformation of Differential Volume applied to Integration................................................... 133 (e) Interpretations of the Jacobian............................................................................................ ......... 135 (f) Volume integration of a tensor field under linear transformations .............................................. 135 9. The Divergence in cu rvilinear coordinates ................................................................................... 137 (a) Geometric Derivation of the Curvilinear Divergence Formula................................................... 137 (b) Various expressions for div B .............................................................................................. ....... 140 (c) Translation from Pict ure B to Picture M&S................................................................................ 142 (d) Comparison of various authors' notations ................................................................................... 143 10. The Gradient in cu rvilinear coordinates.................................................................................... .145 (a) Expressions for grad f..................................................................................................... ............. 145 (b) Expressions for grad f • B........................................................................................................... 147 11. The Laplacian in curvilinear coordinates ................................................................................... 149 4 12. The Curl in curvilinear coordinates ........................................................................................ ....151 (a) Definition of curl B ....................................................................................................... .............. 151 (b) Computation of the line integral........................................................................................... ....... 152 (c) Solving for the curl....................................................................................................... ............... 154 (d) Various forms of the curl.................................................................................................. ........... 155 (e) The curl in orthogona l coordinate systems.................................................................................. 157 (f) The curl in N > 3 dimensions............................................................................................... ........ 157 13. The Vector Laplacian in curvilinear coordinates....................................................................... 159 (a) Derivation of the Vector Laplacian in general curvilinear coordinates....................................... 159 (b) The Vector Laplacian in or thogonal curvilinear coordinates ...................................................... 161 (c) The Vector Laplacian in Cartesian coordinates........................................................................... 163 14. Summary of Differential Opera tors in curvilinear coordinates ............................................... 165 (a) divergence................................................................................................................. ................... 166 (b) gradient and gradient dot vector........................................................................................... ....... 166 (c) Laplacian .................................................................................................................. ................... 167 (d) curl............................................................................................................................................... 167 (e) vector Laplacian ........................................................................................................... ............... 168 Example 1: Polar coordinates: a practical curvilinear notation....................................................... 169 15. Covariant derivation of all curvilinear differential operator expressions ............................... 171 (a) Review of Sections 9 through 13............................................................................................ .....171 (b) The Covariant Method....................................................................................................... .......... 172 (c) divergence (Section 9)..................................................................................................... ............ 174 (d) gradient and gradient dot vector (Section 10) ............................................................................. 174 (e) Laplacian (Section 11)..................................................................................................... ............ 174 (f) curl (Section 12).......................................................................................................... ................. 175 (g) vector Laplacian (Section 13).............................................................................................. ........ 175 Appendix A: Reciprocal Base Vectors the Hard Way.................................................................... 179 (a) Definition of E n........................................................................................................................... 179 (b) Simpler notation ........................................................................................................... ............... 180 (c) Generalized Cross Product of N-1 vectors of dimension N ........................................................ 180 (d) Missing Man Formation ...................................................................................................... ........ 182 (e) Apply this Notation to E.............................................................................................................. 182 (f) Compute E m • en........................................................................................................................... 183 (g) Compute E n • Em......................................................................................................................... 184 (h) Summary of relationship between the tangent and reciprocal base vectors ................................ 184 (i) Another Cross Product Notati on and another expression for E ................................................... 185 Appendix B: The Geometry of Pa rallelepipeds in N dimensions................................................... 186 (a) Preliminary: Equation of a plane in N dimensions..................................................................... 186 (b) N-pipeds and their F aces in Various Dimensions ....................................................................... 187 The 1-piped ................................................................................................................................... 187 The 2-piped ................................................................................................................................... 187 The 3-piped ................................................................................................................................... 189 The N-piped .................................................................................................................... .............. 190 (c) The question of inward versus outward facing normal vectors................................................... 191 (d) The Face Area and Volume of N-pipeds in Various Dimensions .............................................. 192 The 2-piped ................................................................................................................................... 192 5 The 3-piped ................................................................................................................................... 193 The 4-piped ................................................................................................................................... 195 The N-piped .................................................................................................................... .............. 197 (e) Summary of Main Resu lts of this Appendix ............................................................................... 199 Appendix C: Elliptical Polar Coordinates ( N=2, non-orthogonal).............................................. 201 (a) Elliptical polar coordinates............................................................................................... ........... 201 (b) Forward coordinate lines................................................................................................... .......... 202 (c) Inverse coordinate lines................................................................................................... ............ 202 (d) Drawing a contravariant vector V in x-space: the meaning of V' n ............................................ 203 (e) Drawing a contravariant vector V' in x'-space: two "Views" ..................................................... 204 (f) Drawing the specific contravariant vector dx in x-space and x'-space ........................................ 207 (g) Study of how dx transforms in th e mapping between x-space and x'-space ............................... 207 (h) Derivation of the Jaco bian Integration Rule................................................................................ 209 Appendix D: Tensor Densities and the ε tensor .............................................................................. 212 (a) Definition of a tensor density ............................................................................................. ......... 212 (b) A few facts about tensor densities............................................................................................... 213 (c) Theorem about Totally Antisymmetric Tensors: there is really only one: εabc....................... 215 (d) The contravariant ε tensor ........................................................................................................... 216 (e) Some facts about the ε tensor ...................................................................................................... 217 (f) The covariant ε tensor : repeat section (d) as if its weight were not known ............................... 219 (g) Generalized cross products................................................................................................. ......... 220 (h) The tensorial nature of curl B............................................................................................. ......... 220 (i) Tensor E as a weight 0 version of ε : three conventions.............................................................. 221 (j) Representation of ε, εε and contracted εε as determinants.......................................................... 224 (k) Covariant forms of th e previous section results .......................................................................... 230 Appendix E: Tensor Expansions: direct product, polyadic and operator notation.................... 232 (a) Direct Product Notation.................................................................................................... ........... 232 (b) Tensor Expans ions and Bases ..................................................................................................... 233 (c) Polyadic Notation ........................................................................................................................ 235 (d) Dyadic Products ............................................................................................................ .............. 236 (e) Transpose notation for dyadics............................................................................................. ....... 237 (f) Large and small dots used with dyadics..................................................................................... ..238 (g) Operators and Matrices for Rank-2 tensors................................................................................. 239 (h) Expansions of tensors on unit tangent base vectors .................................................................... 243 (i) Tensor expansions in a mixed basis ......................................................................................... ....251 (j) What is a tensor? .......................................................................................................... ................ 253 Appendix F: The Affine Connection Γc ab and Covariant Derivatives .......................................... 255 (a) Definition and Interpretation of Γ : Γc ab = ec • (∂aeb) = Rc i(∂aRbi) ...................................... 255 (b) Identities of the form ( ∂aRd n) = – Re n Rd m (∂aRem) .................................................................... 256 (c) Identities of the form ( ∂cgab) = – [gan Γ b cn + gbn Γa cn] .......................................................... 257 (d) Identity: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab].................................................................. 258 (e) Picture D1 Context ......................................................................................................... ............. 260 (f) Relations between Γ and Γ ' ......................................................................................................... 261 (g) Statement and Proof of the Covariant Derivative Theorem ........................................................ 262 (h) Rule for raising any index on a covarian t derivative of a covariant tensor density..................... 267 (i) Examples of covari ant derivative expressions ............................................................................. 268 6 (j) The Leibniz rule for the covariant deriva tive of the product of two tensor densities................... 271 Appendix G: Expansion of ( ∇v) in curvilinear coordinates (v = vector) ...................................... 275 (a) Continuum Mechanics motivation............................................................................................. ..275 (b) Expansion of ∇ v on ei⊗ej by Method 1: Use fact that v b;a is a tensor..................................... 275 (c) Expansion of ∇v on ei⊗ej by Method 2: Use brute force. ........................................................ 277 (d) Expansion on e i⊗ej and e ^i⊗e^j................................................................................................... 279 (e) Orthogonal coordinate systems .............................................................................................. .....280 (f) Maple evaluation of ( ∇v) in several coordinate systems ............................................................. 280 Appendix H: Expansion of div(T) in cu rvilinear coordinates (T = rank-2 tensor) ...................... 284 (a) Continuum Mechanics motivation............................................................................................. ..284 (b) Expansion of divT on e n by Method 1: Use fact that Tab ;α is a tensor....................................... 284 (c) Expansion of divT on e n by Method 2: Use brute force ............................................................. 285 (d) Adjustment for T expanded on ( e^i⊗e^j) and divT expanded on e^a............................................ 287 (e) Maple: divT in cylindri cal and spherical coordinates................................................................. 288 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates........................... 290 (a) The first met hod : a review............................................................................................... .......... 290 (b) The first method in spherical coordinates: Maple speaks .......................................................... 292 (c) The first method in spherical coordina tes: putting results in traditional form............................ 295 (d) The second method : Part I.......................................................................................................... 297 (e) The second method : Part II................................................................................................ ......... 298 (f) The second method in spherical coordinates: Maple speaks again............................................. 299 (g) Results for Cylindrical Coordinates from both methods............................................................. 302 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor).......................... 307 (a) Total time derivative as prototype equation ................................................................................ 307 (b) Computation of components ( ∇T)'ijk........................................................................................ 308 (c) Tensor expansions of ∇ T on the u n and en base vectors.............................................................. 309 (d) Tensor expansions of ∇T on the e ^n base vectors......................................................................... 309 (e) Total time derivative equation written in unit-base-vector curvilinear components ................... 311 (f) Shorthand notations and a con tinuum mechanics application ..................................................... 312 (g) Maple computation of the ( ∇T)'ijk components in spherical coordinates.................................. 314 (h) Maple computation of the ( ∇T)'ijk components in cylindrical coordinates ............................... 318 Appendix K: Deformation Tens ors in Continuum Mechanics ...................................................... 320 (a) A Preliminary Deformation Flow Picture.................................................................................... 320 (b) A More Complicated Deformation Flow Picture ........................................................................ 325 (c) Form of a solid constitutive equa tion involving the deformation tensor..................................... 330 (d) Some fluid constitutive equations ............................................................................................... 331 (e) Corotational and other objective time derivatives of the Cauchy stress tensor ........................... 332 References............................................................................................................................................ 340 Overview 7 Overview and Summary This paper develops elem entary tens or analysis (also known as tensor algebra or tensor calculus) starting from Square Zero which is an arbitrar y 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. The general mathematical classification of this subject is now called differential geometry. 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 sections develop the theory of tensor analysis in a simple developmental notation where all indices are subscripts, just as in normal college physics. After providing motivation, the seventh section translates this developmental notation to the Standard Notation in use today. The eighth section treats transformations of length, area and volume and then the curvilinear differential operator expressions are derived, one per section, with a summary in the penultimate section. The final section rederives all the same results using the notion of covarian ce and associated covariant derivatives. The information is presented informally as if it were a set of lectures. Little attention is paid to 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. Section 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. Section 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 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. Section 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. Section 4 is brief review of the notions of norm, metric and scalar product in Cartesian Space. Overview 8 Section 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 Section 3. The Jacobian J and associated functions are defined, though the significance of J is deferred to Sec tion 8. The last three subsections briefly discuss the connection between tensor algebra and special relativ ity (with a mention of spinor algebra), general relativity, and contin uum mechanics. Section 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 the expansion are in fact those of the contravariant vector in x'-space, V' i = RijVj. This fact proves useful in later sections 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. Section 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, confusing issues do ar ise. 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. The third last subsection discusses th e significance of tensor analysis with respect to physics in terms of covariant equa tions, and the second last broaches the topic of the covariant derivative of a vector field with its associated Christoffel sym bols. Finally, the last subsection describes how to expand tensors of any rank in various bases and notations. The focus then fully shifts to curvilinear coordinates as an application of tensor analysis. The final sections are all written in the Standard Notation. Section 8 shows how differential length, area and volume transform under x ' = F (x). This section considers the inverse mapping of a differential ortho gonal N-piped (N dimensiona l 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 volumes. The relationship between the vector areas of the N-pipeds is more comp licated, 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. Sections 9 through 13 use the information of Section 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 dimensions where the curl has a vector representation, but then the curl is gene ralized to N dimensions. Section 14 summarizes all the differential operator expressions in a set of tables, and revisits the polar coordinates example one last time to illustrate a re asonably clean and practical curvilinear notation. Section 15 rederives the general results of Sections 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). Overview 9 Much of our content is contained in a set of Appendices. Appendix A develops an alternative expression for the reciprocal base vector En 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 Section 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 Section 9 where 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 Section 8 with regard to th e 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 Section 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. 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 consitutive equa tions involving these tensor objects are listed. Overview 10 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), AT = 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^ = un = unit vector pointing along the nth positive axis of some coordinate system ∂i = ∂/∂xi and ∂t = ∂/∂t are partial deriviatives 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 Section 1: The Transformation F 11 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 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) 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 in an equivalent notation x' = x'(x) x = x(x') 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. The functions F i must be C 1 continuous to support the linearization derivatives appearing in Section 2, and they must be C 2 continuous to support some of the differential operators expressed in curvilinear coordinates in Sections 9-14 and the covariant derivative in Section 7 (v). 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') Section 1: The Transformation F 12 x' = F( x) ↔ r = x2+y2 x 2' = x12+x22 θ = tan-1(y/x) x 1' = tan-1(x2/x1) (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: The domain of x' = F( x) in x-space on the right is all of R2, but the range in x'-space is shown in gray. Imitating the language of complex variables, we can regard this gray range 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 rectangles. 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. (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 area map into the real axis in x-space, and that is fixed by rem oving the right edge. Thus, by doing a suitable trimming of the range, F can be made fully invertible. No one has ever 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') Section 1: The Transformation F 13 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) (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 The domain of x' = F( x) in x-space on the right is all of E3, 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 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 E3 of x-space, but we care only about the principle branch 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. Section 1: The Transformation F 14 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 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) as discussed in Section 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 was 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 where repeated indices are implicitly summed (sometimes called the Einstein 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 and of course this allows for the possibility of a negative distance squared (see Section 5 (i)). 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. 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 alway s 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. Section 1: The Transformation F 15 The matrices R and S are associated with transforma tion F as described in Section 2 below, while G and 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 Sections 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 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). Section 1: The Transformation F 16 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: The red circle is a θ-coordinate line, and the blue ray is an r-coordinate line 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. Section 1: The Transformation F 17 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 in 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 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. Section 1: The Transformation F 18 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. Section 2: Linear Transformations R and S 19 2. Linear Local Transformations associated with F : scalars and two kinds of vectors We now shift to the Picture A context, where x-spac e is not necessarily Cartesian. 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, 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 few 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 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 R ik ≡ (∂x'i/∂xk) Doing the same operation in the other direction gives dx i = Σk( ∂xi/∂x'k) dx'k = Σk Sik dxk' S ik ≡ (∂xi/∂x'k) One can regard R ik and Sik as elements of NxN matrices R and S. In vector notation then, Section 2: Linear Transformations R and S 20 d x' = R( x) dx R ik(x) ≡ (∂x'i/∂xk) R = S-1 // dx' i = Rij dxj d x = S( x') dx' S ik(x') ≡ (∂xi/∂x'k) S = R-1 // dx i = Sij dx'j 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 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 in Section 6 (f), 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) 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 leng th, as will be demonstrated later. 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 Section 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. (a) 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 that φ is evaluated at x, while φ' is evaluated at x' = F(x). As noted in section (k) below, one could be more precise by referring to the objects described here as a "tensorial scalar" and a "tensorial scalar field". Section 2: Linear Transformations R and S 21 (b) Contravariant vectors If transfor mation F (possibly non-linear) transforms x-sp ace to x'-space without affecting time, and if F is time independent so F = F( x), then consider the familiar velocity vector, v i = dxi/dt => v = dx/dt 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 . 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 (which would be the case only if F were a linear transformation) 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 . (c) Covariant vectors Much of physics i s described by differential equations involving the gradient operator ( the reason for the overbar is given in the next section) ∇¯ i = ∂¯i = ∂/∂xi 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) , ∇¯ 'i = ∂¯ 'i = ∂ ∂x'i = ∂xk ∂x'i ∂ ∂xk = Ski∂¯k = ST ik ∂¯k = ST ik ∇¯k => ∇¯' = ST ∇¯ 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) 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. Section 2: Linear Transformations R and S 22 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 (d) 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 Section 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') One could imagine replacing S with some Q T 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 that R → Ri j and S → Si j = Rji and S can be removed from the picture (see Section 7 (q) ) . (e) Origin of the names contravariant and covariant A justificatio n of the terms covariant and contravariant is presented at the end of Section 7 (t), 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 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 contravariant. In Section 7 (t) this idea is applied to the transformation of two "things", where one thing is the component of a vector like V n and the other thing is a basis vector onto which a vector is expande d. 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 Section 2: Linear Transformations R and S 23 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, 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. (f) 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 anal ysis" (for rank-n tensors, see Section 7 (j)). Some authors refer to the rank of a tensor as the order of a tensor.) 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 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 (m). (g) Linear transformations For a linear transfor mation F, the matrix elements of R and S are constants and don't depend on x or x '. The reason is fairly obvious. For linear x' = F(x) (an added constant w ould make F non-linear ) F(αx' + βy') = α F(x') + β F(y') => x' i = Fi1x1 + Fi2x2 + .... FiN xN // = F i(x) where the F ij are constants independent of the coordinates, in which case Section 2: Linear Transformations R and S 24 dx' i = Fi1 dx1 + Fi2 dx2 + .... FiN dxN = Σk Fik dxk so R = F and S = F -1 This is the situation with rotations and Lorentz transformations. (h) 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 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 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. 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. 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 e n. Section 2: Linear Transformations R and S 25 (i) 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). 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) 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 ) 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') Similar transformation rules apply to tensors of any rank. For example, the metric tensor g ab (developmental notation) is a rank-2 contravariant te nsor field and the transformation rule is this g' ab(x') = Raa'Rbb'ga'b'(x) or g' ab = Raa'Rbb'ga'b' 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 (j) 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 (e) to have a certain significance, and that determinant is called " the Jacobian " = det(S( x')) ≡ J(x'). Section 2: Linear Transformations R and S 26 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 always used for di fferential operators, 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 confused with the full transformation F, or Cauchy stress T. Our chosen notation R makes one think perhaps 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 ). (k) Definition of the words "scalar", "vector" and "tensor" In section (a) 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 Section 5 (e) certain "rank-2" tensors will appear -- they are matrices that transform in a certain way under a transformation F. In Section 7 (j) 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 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 expr essions. No transformation rule is implied. • 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: Section 2: Linear Transformations R and S 27 • 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. 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 (u) 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 equation which is covariant is said to be frame-indifferent. Possible definitions of "ten sor" are examined further in Appendix E (j). 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. 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). Section 3: Tangent Base Vectors 28 3. Tangent Base Vectors e n and Inverse Tangent Base Vectors u 'n This entire section is in the context of Picture A , In the previous picture showing d x and dx ', one has much freedom to "try out" different differential vectors. For any d x one picks at point x, one gets some dx ' according to d x' = R( x) dx. Consider this slightly enhanced version of the previous drawing (red curves added) 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, Section 3: Tangent Base Vectors 29 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. (a) 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 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 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 where this last equality serves as the definition of e n , e n ≡ Se'n Vector en = 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 the above in components, d x = en dx'n => dxi = (en)i dx'n . But of course dx i = Sin dx'n , and therefore ( e n)i = Sin = ∂xi/∂x'n or en = ∂x/∂x'n = ∂'nx => ( en)i = ∂xi/∂x'n = Sin This says that the vectors en are the columns of the matrix S: S = [ e 1, e2, e3 .... eN ] matrix = N columns We shall call these e n 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. Section 3: Tangent Base Vectors 30 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 was 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: 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, we are using u n with ( un)i = δn,i as "axis-aligned basis vectors" in x-space. If g ¯= 1 for x-space, then these are the usual Cartesian unit vectors (see section (c) below). Many authors use the notation en for these vectors which then conflicts with our use of en 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 un = n^ . Often the notation en is used to represent some generic arbitrary set of basis vectors. For this purpose, we use the notation bn. (b) en as a contravariant vector The situation described above was this, Section 3: Tangent Base Vectors 31 d x' = e 'n dx'n x'-space // no implied sum on n d x = en dx'n x-space // no implied sum on n 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 e'n dx'n = R(x ) en dx'n => e'n = R( x) en 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 = Rij (en)j => δn,i= RijSjn recovering the fact that RS = 1. This is an exampl e of a vector being "contravariant by definition", as discussed in Section 2 (h). These two expansions are easy to show just by verify ing that components of both sides are the same: e n ≡ Se'n = Σi Sin e'i since (e n)j = Σi Sin (e'i)j = Σi Sin δi,j = Sjn = (en)j e' n ≡ Ren = Σi Rin ei since (e' n)j = Σi Rin (ei)j = Σi Rin Sji = (SR)jn = δj,n = (e'n)j (c) a semantic question: unit vectors Above it was noted that e'1 = (1,0,0....). Should this be called "a un it vector" ? It will be seen below 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 single 1 in its column vector definition, but it is not a unit vector in the sense that it does not (in general) 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 Appendix C (e). 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) x = F -1(x') ↔ x = rcos( θ) x 1 = x2' cos(x1') y = r s i n ( θ) x 2 = x2' sin(x1') Section 3: Tangent Base Vectors 32 So S11 = (∂ x/∂θ) = -rsinθ S12 = (∂ x/∂r) = cosθ S ik ≡ ( ∂xi/∂x'k) S21 = (∂ y/∂θ) = rcosθ S22 = (∂ y/∂r) = sinθ 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. ] The tangent base vectors en can be read off as the columns of S e1 = r(-sinθ,cosθ) = eθ = r e^θ // = r θ^ e2 = (cosθ,sinθ) = er = e^r // = r^ 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. 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 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, | 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 Appendix C (e) for further discussion of this topic with a specific 2D non-orthogonal coordinate system. Section 3: Tangent Base Vectors 33 (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 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 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 . The tangent base vectors are the columns of S, so e r = (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 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^ Section 3: Tangent Base Vectors 34 The unit vectors can be displayed in this standard picture, Notice that ( r^, θ^, φ^) = (e^1, e^2, e^3) form a right-handed coordinate system at the point x = r. (d) 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 S = [ e1, e2, e3 .... eN ] // are the columns of S ( 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 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 below and another in Appendix C. In section (b) above the vector en transformed as a contravariant vector into an axis-aligned basis vector e'n in x'-space e' n = R en ( e'n)i = Rij (en )j (en)i = Sin (e'n)i = δn,i The same thing happens here, only in reverse : u'n = S un ( u'n)i = Sij (un)j (u'n)i = Rin (un)i = δn,i Section 3: Tangent Base Vectors 35 where now the un 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 Section 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 so the inverse tangent base vectors are given by the columns of R, 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 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). Looking at the small θ region of the plot on the left below, one sees slopes as just described above. For the polar coordinates mapping discussed above, horizont al (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 coordinate lines in x'-space, th e picture is a bit more complicated. Since y = rsin θ, 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 yi 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: x'-space ( θ,r) x-space (x,y) Another example is given in Appendix C. Section 4: Cartesian Space 36 4. Notions of length, distance and scalar product in Cartesian Space This section can be interpreted in either Picture B or Picture D wh ere 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 Section 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 | 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 . 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 // = Σi,j δi,j xi yj which of course implies this special case, (x,x) = x • x = Σ ixi2 = ||x||2 = | x |2 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 To avoid dealing with the square root, one usually writes (ds) 2 ≡ || dx ||2 = Σi(dxi)2 = (dx1)2 + (dx2)2 + ... + (dx N)2 = Σi dxi dxi = Σi,j δi,j dxi dxj As shown in the next section, one can interpret δ i,j as the metric tensor in Cartesian Space. Section 4: Cartesian Space 37 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. Section 5: The Metric Tensor 38 5. The Metric Tensor The metric tensor is the heart of the machine of tensor analysis and we shall have a lot to say about it in this section. Each subsection is best presented in the c ontext 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 subsections below will give the reader some e xperience 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. 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: 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) The unprimed S,R can be expressed in terms of the primed objects this way (chain rule) R ik ≡ (∂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" (a) Definition of the metric tensor The metric or distance between vectors x and x+dx can be specified as done in Section 4 in terms of the norm of differential vector d x, metric(x +dx, x) = norm( [x +dx] - x) = norm(d x) ≡ ds with the caveat that this is not an official nor m, see section (i) 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) Section 5: The Metric Tensor 39 (ds)2 = Σi=1N Σj=1N [ metric tensor ] ij dxi dxj This is a bit of chicken and egg because one is really defining "distance" and "m etric tensor" at the same time. Each selection of a metric tensor defines th e meaning of distance ds in the space of interest. Suppose the length of a small vector d x in a Quasi-Cartesian x-space is known to be ds. Recall from Section 1 that such a space has a diagonal metric tensor G whose diagonal elements are independently either +1 or -1. How might one express this same ds in terms of the other spaces' coordinates x' and x" ? (see Picture D) Going to x'-space one finds, since d x = S'( x') dx', (ds)2 = ΣiGiidxidxi = Σi Gii (ΣkS'ik dx'k) (ΣmS'im dx'm) = Σ kΣm { Σi Gii S'ikS'im } dx'k dx'm Defining the metric tensor in x'-sp ace to be (comment on the bar below) g¯'km ≡ ΣiGiiS'ikS'im = Σij S'T kiGijS'jm => g ¯' = S'TG S' one then has, with implied summation on the right, (ds) 2 = ΣkΣm g¯'km dx'k dx'm = g¯'km dx'k dx'm For the transformation from x-space to x"-space in Picture D, a similar result is obtained, g¯"km ≡ Σi GiiS"ikS"im = > g ¯" = S"TG S" (ds) 2 = g¯"km dx"k dx"m Since (ds) 2 is a number which is the same in all three systems (that number is the distance between two points in x-space), the quantity g ¯'km dxk' dxm' is a tensorial scalar. The metric tensor is specific to a space; it is a propert y of the space; it is part of the space's definition. We have placed bars over the g's anticipating what wi ll soon be shown, that these matrices are "covariant" matrices. Then we won't have to go back and fix things up. 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" Concerning the invariance of (ds). In the above discussion, it was assumed that distance (ds)2 is the same in x'-space as it is in x-space. As will be seen soon, this is equivalent to saying that the covariant dot product of any two vectors gives the same number re gardless of which space is used to compute the dot product: A • B = A ' • B'. This in turn implies that | A| = |A'| . In other words, it was assumed above that the dot product of two tensorial vectors is a scalar with respect to the underlying transformation F. Section 5: The Metric Tensor 40 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 physical vector is the same no matter how one chooses to describe that vector. 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 one again wants dot pr oducts to be scalars and the notion that (ds)2 is a scalar under Lorentz transformations (that is, d x•dx = dx'•dx') is a key assumption/requirement of the theory. There are, however, applications of transformations where the scalarity of (ds)2 is not required and in fact it is crucial that (ds)2 can change under a transformation. For example, in continuum mechanics one can consider x-space to be a space describing a fl ow of continuous matter at some initial time t 0 and x'- space to be the same flow at a later time t. A general flow has x' = F( x) where x is the position of a continuum "particle" at time t 0 and x' is the position of that same partic le at time t. In general F is non- linear. The distance between two differentially spaced particles at the two times is d x and d x', and one has dx' = R d x where R is not simply a rotation. The whole point here is that during the flow, the distance vector between two close particles rotates and stretc hes in some manner, and in general (due to this stretch) |d x| ≠ |dx'|, so (ds)2 is definitely not invariant under the flow (ie, under the transformation F) . In Lai (p 105), the idea d x' = Rd x translates to d x = Fd X, and F is called the deformation gradient and is written F = ( ∇x) which is a dyadic like notation discussed in Appendices E and G. In Section (i) below we show that in this flow situation there are in fact two distinct metric tens ors involved, each having its own role to play. The continuum mechanics applicati on is discussed further in Section (o) below, and the deformation tensor is considered in some detail in Appendix K. Except when dealing with continuous mechanics flows, we shall treat (ds)2 = (ds')2 as a scalar. (b) Inverse of the metric tensor The inverses of the three metric tensors 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 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 Comment: In the Picture C context but with a Quasi-Cartesian x(0)-space, one could take the second column above and write it this way, Section 5: The Metric Tensor 41 g = RGRT g¯ = STGS g¯g = 1 (ds)2 = g¯km dxk dxm where now the clutter of primes is gone. If x-space is Cartesian so G = 1, then g = RRT and g ¯ = STS. But we continue with Picture D. (c) A metric tensor is symmetric Any matrix of the form M = ATDA where D is a diagonal matrix (so D=DT) is symmetric: MT = (ATDA)T = ATDA = M // and similarly with A → AT Since all metric tensors shown above match this form, they are all symmetric: g ab = gba for any g (with or without an overbar). (d) 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 . Any matrix of either of these forms has positive diagonal elements and positive determinant: (ATA)aa = Σb (AT)abAba = Σb (A)baAba = Σb (Aba)2 ≥ 0 // diagonal elements ≥ 0 det(A TA) = det(AT) det(A) = det(A) det(A) = [ det(A) ]2 ≥ 0 // det ≥ 0 To show these results for the AAT 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. (e) Definition of two kinds of rank-2 tensors We now switch to Picture A, Section 5: The Metric Tensor 42 Recall the vector transformation rules from Section 2 (d), 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') which can be written out in components 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') 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 and again we put a bar over the covariant objects. Digression : Proof that (A-1)T = (AT)-1 for any invertible matrix A: • 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. (f) Proof that the metric tensor and its inverse are both rank-2 tensors The above rank-2 tensor tr ansformation rules 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 where recall But we now switch these rules to the Picture D context where F maps x'-space to x"-space, Section 5: The Metric Tensor 43 M" = R M' RT // contravariant rank-2 tensor 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 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 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 g" = (R" S') g' (S'T R"T) // regroup g" = (R" S') g' (R" S')T // (AB)T = BTAT g" = R g' RT // expressions in section (b) above for R and S 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 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. 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, Section 5: The Metric Tensor 44 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 Since RS = 1, the equations can be inverted to get g = S g' S T 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' Further variations of the above are obtained using RS = 1, gg ¯ = g'g¯' = 1 and g = gT (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 (g) 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 ( see Section 7 (u) concerning "covariant equations") W = g ¯ V x-space W' = g ¯' V' x'-space 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 Therefore this new vector W is a covariant vector under F, so it should have an overbar, W¯ ≡ g¯ V 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¯ V¯ i = g¯ij Vj Vi= gijV¯j (h) Vectors in Cartesian space Theore m: There is no distinction between a contravariant and a covariant vector in Cartesian space. Section 5: The Metric Tensor 45 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 also covariant and vice versa. In other words, if g = 1, every vector is the same as its covariant partner vector. The transformation rules in this case are V' = R V V¯' = S T V¯ = ST V Although the vectors V and V¯ are the same, eliminating V shows that V¯' and V ' are not the same. One finds that V¯' = (STS) V' = g¯' V', so V¯' = g¯' V' ≠ V'. (i) Metric tensor: covariant scalar product and norm For a Cartesi an space, Section 4 defined the norm as the 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 exte nsions of these three concepts can result in all three quantities being negative. Nevertheless, we sh all use the term "covariant 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 scalar 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 bottom 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), 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 scal ar product" of two contravariant vectors (a new and different use of the word "covariant", but th e same as appears in Section 7 (u) ) as: A • B ≡ g¯abAaBb = g¯abBbAa = g¯baBaAb = g¯abBaAb = B • A This covariant scalar (or dot) product is more interesting and useful than the object A aBa because the covariant scalar product of two contravariant vectors is a tensorial scalar, as we now show (Picture A) A' • B' = g¯'abA'aB'b = g¯'ab(Raa'Aa') (Rbb'Bb') = g¯'ab Raa' Rbb' Aa' Bb' = [ ( RT)a'a g¯'ab Rbb' ] Aa' Bb' = [RT g¯' R]a'b' Aa' Bb' = g¯a'b' Aa' Bb' = g ¯ab Aa Bb = A • B Section 5: The Metric Tensor 46 Recall that for any contravariant vector B, there is a partner covariant vector B¯a = g¯abBb. Using this partner B¯ one can restate the above covariant scalar product as A • B = g¯abAaBb = Aa B¯a or, taking instead A¯b = g¯baAa , A • B = A¯ b Bb = A¯a Ba And finally, if in A • B = A a B¯a we write A a = gabA¯b , we get A • B = A a B¯a = gabA¯bB¯a = gabA¯aB¯b where the scalar product is now expressed in terms of the covariant partner vectors A¯ and B¯. To summarize, there are four different ways to write this covariant scalar product : A • B = g¯abAaBb = Aa B¯a = A¯a Ba = gabA¯aB¯b = B • A Using the appropriate expressions on the above line, one may conclude that the covariant dot product of any two tensorial vectors is a tensorial scalar. In the special case that A = B, we use the shorthand notation (with caveat as noted above) |A| 2 ≡ A • A 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 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 = dxμdxμ = dx'• dx' = dx • dx = a Lorentz scalar = (dt)2 - dx • dx , xμ = (t, x) 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. ] Going back to Section 3 and the vectors e' n and en, a claim made there can now be verified: Section 5: The Metric Tensor 47 |e'n|2 = e'n • e'n = en • en = |en|2 => | e'n| = |en| . In applications in which (ds)2 is NOT regarded as a scalar with respect to transformation F, which includes continuum mechanics flows, thi ngs are a little different. We have here (ds)2 ≡ dx • dx = dx' • dx' ≠ (ds')2 if (ds') ≡ physical distance in x'-space If x'-space has Cartesian coordinates, we have (ds')2 = dx'idx'i = δi,j dx'i dx'j and this is different from the quantity d x' • dx' = g¯'ij dx'i dx'j . One might go so far as to define two different • symbols: d x' •g dx' ≡ g¯'ij dx'i dx'j ≠ (ds')2 d x' • 1 dx' ≡ δi,j dx'i dx'j = dx'i dx'i = (ds')2 = physical distance squared By default, we have • = • g, so then d x' • dx' ≠ (ds')2. In this situation, x'-space in effect has two different "metric tensors". The first is the g ¯'km that appears in the entire discussion of this Section 5 and which is involved in raising and lowering indices on a tensor and satisfies g ¯' = ST g¯ S (the fact that g transforms as a rank-2 tensor), and so on. The second we might call g ¯'km and this is what determines physical distance in x'-space, (ds')2 = g¯'km dx'k dx'm . In continuum mechanics with Cartesian x'-space, we then have g¯'km = δk,m. There are two different "metric tensors" because there are two different "metrics" which are of interest in x'-space : dg2(x,y) = ( x-y) •g (x-y) = g¯km (x-y)k(x-y)m // raising/lowering metric tensor d g2(x,y) = ( x-y) •g (x-y) = g¯km (x-y)k(x-y)m // physical distance metric tensor For Cartesian coordinates in x'-space, g¯'km = δk,m, and our two metric tensor s may be identified with those of "Cartesian View x'-space" and "Curvilin ear View x'-space" as discussed in Section 8 and Appendix C (e). (j) Metric tensor and tangent base vectors The context of Picture A continues, Recall this fact from Section 3, S = [ e 1, e2, e3 .... eN ] Section 5: The Metric Tensor 48 where the columns of S are the tangent base vectors. It follows that (see end of section (f)) g¯' = ST g¯ S = [ e1, e2, e3 .... eN ]T g¯ [e1, e2, e3 .... eN ] so 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 ........ 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 using the covariant scalar product defined in the prev ious section. Taking the n,m component of the above matrix equation, one gets g¯' mn = em • en or g ¯'mn = ∂'mx • ∂'nx 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 is g¯'nm = (STg¯S)nm = ST na g¯ab Sbm = g¯ab San Sbm = g¯ab (en)a(eb)n ≡ en • em . Therefore, the tangent base vectors will only be mutually orthogonal when the metric tensor g ¯' of x'-space is a diagonal matrix . We refer to the coordinates of an x' -space having a diagona l metric tensor as comprising an orthogonal coordinate system . At any point x in x-space, the tangents en to the N coordinate lines passing through that point are or thogonal. Most examples below will involve such systems, with Appendix C providing a non-orthogonal example. In particular, g ¯' mn = em • en lets us write the length of a tangent base vector in terms of the corresponding diagonal element of g ¯', |en|2 = en • en = g¯'nn => | en| = g¯'nn => e^n = en / g¯'nn The quantities | e n| = g¯'nn are called scale factors and are sometimes written h' n or Q'n or H'n. h'n ≡ Q'n ≡ |en| = g¯'nn As a reminder, had we called x'-space something like ξ-space, there would be no primes on these symbols, but then if g ¯ ≠1 there would be confusion as to which space the symbols applied. Section 5: The Metric Tensor 49 Section (d) 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 : 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) (k) The Jacobian J The context of Picture A continues, First of all, note that since RS = 1, det(S) = 1/det(R) The Jacobian J( x') is defined as follows, J( x') ≡ det(S( x')) = det( ∂x i/∂x'k) = 1/det(R( x(x')) = 1/ det( ∂x'i/∂xk) 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) as if x and x' were independent variables. In our presentation, x' = F(x) is not an independent variable but is determined by F(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 ( ε abc.. is the permutation tensor, Section 7 (h)), det(S) = εabc...x Sa1 Sb2 ... SxN Section 5: The Metric Tensor 50 For our particular S with S in = (en)i this becomes det(S) = εabc...x (e1)a(e2)b....... ( eN)x so J is related to the tangent base vectors by J = εabc...x (e1)a(e2)b....... ( eN)x . It was shown in section (f) 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-2det(g) 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 So that J 2(x') = det(g ¯'(x')) / det(g ¯(x)) = g'( x') / g( x) Normally the argument dependence is suppressed and one then writes J 2 = det(g ¯')/ det(g ¯) = g'/g As explained in Appendix D (a), the equation g' = J 2 g says that g, instead of being a tensorial scalar, is a scalar density of weight -2. One must be a little careful to distinguish the scalars g and g' from the tensors gij and g'ij expressed in matrix notation as g and g'. It is convenient to make the following definition, called the signature of the metric tensor, s = sign[det(g ¯)] Since g g ¯ = 1, one has det(g)det(g ¯) = 1 so that sign[det(g ¯)] = sign[det(g)] . Since det(g ¯') / det(g ¯) = [det(S)] 2, one has sign[det(g ¯')] = [det(g ¯)]. Therefore: s = sign[det(g ¯)] = sign[det(g)] = sign[det(g ¯')] = sign[det(g')] = sign(g) = sign(g') 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¯), if we assume det(g ¯) vanishes Section 5: The Metric Tensor 51 nowhere 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 J2 = g'/g one can write |J| = sg' / sg = | det(S) | = g'/g For the curvilinear coordinates app lication, x-space is Cartesian, det(g ¯) = 1, and thus s = 1 and then |J| = g' = | det(S) | // curvilinear For the relativity application, x-space is Minkowski space with det(g ¯) = -1 so s = -1 and |J| = -g' = | det(S) | // relativity Here then is a summary of the results of this section: J( x') ≡ det(S( x')) = det( ∂x i/∂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 s ≡ sign[det(g ¯)] = sign[det(g)] = sign[det(g ¯')] = sign[det(g')] = sign(g) = sign(g') |J| = sg' / sg = | det(S) | = g'/g 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. Section 5: The Metric Tensor 52 (l) Some r elations between g, R and S in Pictures B and C (Cartesian x-space). In Picture B, which is Picture A with g = 1, the statement of the rank-2 tensor transformation of g' and g ¯' becomes (from section (f) above) g' = RRT g¯' = STS which can be written in a variety of ways, RT = (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 . In summary: g' = RR T RT = S g' R = g' ST 1 = S g' ST g¯' = STS ST = g¯' R S = RT g¯' 1 = RT g¯' R . 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 If the x' i are orthogonal coordinates, then g ¯'nm = h'n2δnm and g'nm = h'n-2δnm where the h' n are the scale factors mentioned above in section (j). These scale f actors may then be expressed as ( M&F p 23 1.3.4) h'n2 = g¯'nn = Σi (∂xi/∂x'n)2 h ' n-2 = g'nn = Σi (∂x'n/∂xi)2 In Picture C (sometimes x(0)-space is called ξ-space and g(0) is called η ) the above equations are identical ex cept all the g's are now unprimed, so Section 5: The Metric Tensor 53 g = RRT g¯ = STS g = RRT 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 Example 1: Polar coordinates: metric tensor and Jacobian Picture C continues (so now θ = x 1 and r = x 2) and the metric tensor for polar coordinates will be computed in two ways. On the last visit to this example ( end of Section 3), it was shown that S = ⎝⎛ ⎠⎞-rsinθ cosθ rcosθ sinθ = [ e1, e2 ] e1 = r(-sinθ, cosθ) e2 = (cosθ, sinθ) One way to compute g ¯ is this: ( 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 way is this: 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 (ds)2 = g¯km dxk dxm = g¯θθ dθ dθ + g¯rr dr dr = r2 (dθ)2 + (dr)2 The Jacobian is given by J(r,θ) = det(S(r, θ)) = det ⎝⎛ ⎠⎞-rsinθ rcosθ cosθ sinθ = -r so |J| = r and g = J2 = r2, g = r Section 5: The Metric Tensor 54 Example 2: Spherical coordinates: metric tensor and Jacobian As with Exam ple 1, Picture C is used, wherein (x 1, x2, x3) = (r,θ,φ) . In our last visit to this example (end of Section 3) it was found that S = ⎝⎜⎛ ⎠⎟⎞ sinθ cosφ rcosθcosφ -rsinθsinφ sinθ sinφ rcosθsinφ rsinθ cosφ cosθ -rsinθ 0 The metric tensor is then given by Maple as g¯ = S TS = ⎝⎜⎜⎛ ⎠⎟⎟⎞ 1 0 0 0 r2 0 0 0 r2sin2θ det(g ¯) = r4sin2θ 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 The Jacobian is found by Maple to be, J(r,θ,φ) = det(S) = r 2sinθ Differential distance is then (ds) 2 = g¯km dxk dxm = (dr)2 + r2(dθ)2 + r2sin2(dφ)2 and if dφ = 0, this agrees with the polar coordinates result. (m) Special Relativity and its Metric Tensor: vectors and spinors In this section the Standard Notation introduced be low in Section 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 Greek letters for 4-vector indices and Latin le tters 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 Section 5: The Metric Tensor 55 x'μ = Fμ ν xν = Rμ ν xν = > xν is a contravariant vector 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. 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 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 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 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, [ J i, Jj] = +i εijkJk // [ A,B ] ≡ AB - BA [ Ji, Kj] = +i ε ijk Kk [ Ki, Kj] = -i εijkJk . 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) ) (Jμν)α β = i ( gμαδν β – gναδμ β) (J 1)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 where (-i)(Jμν)αβ = ( gμαgνβ – gναgμβ) is a rank-4 tensor, antisymmetric under μ↔ν and α ↔ β. Section 5: The Metric Tensor 56 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 shown above ( all J an d 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. One can also consider two other representations of the Lorentz group which are pretty obvious: 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 analys is edifice based on 2-vectors. Just as with the 4- vectors, one has contravariant and covarian t 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 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, X' ab• = Ra a'Rb• b•' Xa'b•' 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 : Section 5: The Metric Tensor 57 Section 5: The Metric Tensor 58 (n) General Relativity and its Metric Tensor In general relativity a Picture of interest is Picture C but the x(0) space is replaced by a Quasi-Cartesian space with coordinates ξi with the metric tensor of special relativity. This ξ-space represents a "freely-falling" coordinate sy stem in which the laws of special relativity apply and the metric tensor is taken to be 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. Using the Standard Notation introduced in Section 7 below, this is usually written as g¯ = STηS // result from Section 5 (b) g dn = ST ηdnS // where (η dn)ab = ηab gμν = (ST)μ α ηαβ Sβ ν = Sα μ ηαβ Sβ ν // Standard Notation as in Section 7 gμν = (∂ξα/∂xμ) ηαβ (∂ξβ/∂xν) g μν = (∂ξα/∂xμ) (∂ξβ/∂xν) ηαβ // Weinberg p 71 (3.2.7) 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) 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. Γ μ νλ = (1/2) gμσ( ∂νgλσ + ∂λgνσ – ∂σgνλ ) // p 75 (3.3.7) These brief comments are only meant to convince the read er that the equations of general relativity also have their place under the general umbre lla of tensor analysis as discusse d in this document. The fact that Γμ νλ is not a mixed rank-3 tensor is discussed in Appendix F (f). Section 5: The Metric Tensor 59 (o) 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 them 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 (picture 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 dX → dx during the flow. [ This flow is fu rther described in Appendix K. ] The finite-time 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) as in Section 1 above. Recall from Section 1 that a general transformation was x' = F(x) and the linearized transformation was d x' = R d x. To be compatible with Lai notation which uses symbol F for the deforma tion gradient, we have renamed the Section 1 transformation F to be F, and we shall see below that R will in fact become deformation gradient F. 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. As discussed at the end of Section 5 (i), there are really two metric tensors in this situation called g ¯'km and g¯'km. The former is the metric tensor which raises and lowers tensor indices in the general tensor formalism, while the latter is the metric tensor whic h determines physical distance in x'-space. For the usual Cartesian coordi nates in x'-space, g¯'km = δk,m which is to say g¯' = 1. The other metric tensor is given by g ¯' = ST g¯ S where g ¯ is the metric tensor in x-space. Using Cartesian coordinates there means g ¯ = 1 and then g ¯' = ST S. In order to put this flow into the notation of this document, let X → x and x → x' so that continuum mechanics this document ( Forward Flow X → x [ x → x'] ) x, X ↔ x', x x = x(X,t) = F(X) ↔ x' = F(x) // Lai p70 (3.1.4) d x = F d X ↔ d x' = R d x // as in Section 2 // Lai p86 (3.7.6), p105 (3.18.3) F ↔ R F-1 ↔ S X = Cartesian ↔ g = 1 x = Cartesian ↔ g' = 1 B = FFT ↔ g' = RRT // as in Section 5 ( l) // Lai p121 (3.25.2) B-1 = (F-1)T (F-1) ↔ g ¯' = STS // since g ¯' = g'-1 Thus, the deformation gradient F is just the R matrix of the forward transformation x = x(X,t) = F(X). The raising/lowering metric tensor g' = RRT appears as B = FFT which is known as the left Cauchy-Green deformation tensor (manifestly symmetri c, so a viable metric tensor). Alternately, we can consider the above flow going backwards in time and then x = x is the starting position and x' = X is the ending position. For this inverse flow, we let F have the same meaning as in the forward flow, d x = F d X, and thus end up with this translation table where we now x → x and X → x' : Section 5: The Metric Tensor 60 continuum mechanics this document ( Inverse Flow x → X [x → x'] ) X, x ↔ x', x X = X(x,t) = F(x) ↔ x' = F(x) // this F is the inverse of the forward flow F d X = F-1 dx ↔ d x' = R d x // as in Section 2, same F as in forward flow F-1 ↔ R F ↔ S // S = R-1 and Sik = (∂xi/∂x'k) ↔ Fik = (∂ xi/∂Xk) X = Cartesian ↔ g = 1 x = Cartesian ↔ g' = 1 C = FTF ↔ g ¯' = STS // as in Section 5 ( l) // Lai p114 (3.23.2) C-1 = F-1(F-1)T ↔ g' = RRT // since g' = (g ¯')-1 To avoid confusion, in both tables g goes with flow X-space and g' and g' go with flow x-space. For the inverse flow, the raising/lowering metric tensor g ¯' = STS appears as C = FTF which is the right Cauchy-Green deformation tensor (again manifestly symmetric, so a viable metric tensor). Given the above flow situation, it is then possible to add two more transformations F 1 and F2 which take X-space and x-space to independent se ts of curvilinear coordinates X' and x' : and we then have an interesting trip le application of the notions of Section 1 to a real-world situation. This drawing is the implicit subject of Section 3.29 (p131) of Lai. In (reverse) dyadic notation the de formation gradient is written F = ( ∇x) where ∇ means ∇(X)so that dx = F d X = (∇x) dX F ij = (∇x)ij = ∂j(X)xi = ∂xi/∂Xj The (∇x) notation is explained in Appendix E, and in Appendix G the object ( ∇v) for an arbitrary vector field v(x) is expressed in general cu rvilinear coordinates. Consider now this picture taken from Section 8 below, Section 5: The Metric Tensor 61 Flow X-space F l o w x - s p a c e We can identify the mapping shown in this picture with our Inverse Flow situation (table above). Section 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 Section 8 (c) 7 and converting 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 We can then translate these three lines into our Inverse Flow context: | d x(n)| / | d X(n)| = h'n = [g¯'nn]1/2 = [(FTF)nn]1/2 = [Cnn]1/2 // Lai p114 (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 p129 (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) where edge area volume X-space : d X(n) dA0n dV0 time t 0 x-space : d x(n) dAn dV time t Section 5: The Metric Tensor 62 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 above 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 which seems a far cry from our result [cof(FTF)nn)]1/2. But consider, using the Inverse Flow table, | (F-1)T un |2 = | RT un |2 = [RTun]i[RTun]i = Rni Rni = (RRT)nn = g'nn so det(F) | (F-1)T un | = g'1/2 g'nn1/2 = [cof(g ¯'nn)]1/2 = [cof(FTF)nn)]1/2 where we use the Section 8 (c) 6 theorem convert ed to developmental notation, stating that g' g' nn = [cof(g ¯'nn)] . // more generally, (detA) A-1 = cof(A) if A = AT 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 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 Appendix E (g). 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. Section 6: Reciprocal Base Vectors 63 6. Reciprocal Base Vectors E n and Inverse Reciprocal Base Vectors U' n This entire Section uses the Picture A context, (a) Definition of the E n Although various definitions are possible, we shall define the reciprocal tangent vectors En in the following manner (implied sum on i) En ≡ g'ni ei = g'ni∂'ix => en = g¯'niEi // since g ¯' = g'-1 Comment : Notice how this differs in structure from th e rule for forming a covariant vector from a contravariant one, V ¯n = g¯'niVi In the last equation, the right side is a linear combination of vector components Vi, while in the previous equation 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 (Section 3 (b)), 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. One can express the components of En in two ways, ( En)k ≡ g'ni (ei)k = Skig'ni // since ( ei)k ≡ Ski, Section 3 = g' niSki = Rnigik // since R abgbc = g'abScb, end of Section 5 (f) so that ( En)i = g'ncSic = gicRnc // sum on second indices Applying R to both sides of En ≡ g'ni ei 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 Section 6: Reciprocal Base Vectors 64 Using the axis-aligned basis vectors un in x-space, (u n)m = δn,m, we also have this dot product, En • um = Rnm . Proof : En • um = g¯ab(En)a(um)b = g¯ab (gacRnc)(δm,b) = (g¯magac) Rnc = δm,c Rnc = Rnm (b) The Dot Products and Reciprocity (Duality) Three covariant dot products are of great interest. The first is this (Section 5 (f) 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 The second is En • em = g¯ij (En)i (em)j = g¯ij gia Rna Sjm = δj,a Rna Sjm = Rnj Sjm = (RS)nm = δn,m and the third is En • Em = g¯ij (En)i (Em)j = g¯ij gia Rna gjb Rmb = δj,a Rna gjb Rmb = Rnj gjb Rmb = R nj gjb RT bm = (R g RT)nm = g'nm To summarize, en • em = g¯'nm => | en| = g¯'nn = h'n (scale factor) En • em = δn,m En • Em = g'nm => | En| = g'nn Using the transformed E'n defined above, one finds 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 which is consistent with the fact that this is a covariant dot product of tensorial vectors: E'n • e'm = En • em = δn,m The other two dot products above work this same way, so e'n • e'm = g¯'nm E'n • e'm = δn,m E'n • E'm = g'nm Section 6: Reciprocal Base Vectors 65 Notes on Reciprocity (Duality) 1. Suppose the vectors bn form a complete basis for x-space. Can one find a set of vectors Bn that has the property Bm • bn = δm,n? 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 . 2. Above it was shown 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. Solving for the Bn in terms of the bn. Each B n 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 N2 variables. Since the bn form a complete basis, one can expand Bm = wmnbn . Then δm,k = Bm • bk = wmn bn • bk // bn • bk = g¯ij(bn)i(bk)j Define W nk ≡ bn • bk and note that W nk is symmetric. Then δm,k = wmnWnk or wW = 1. Assuming for the moment that detW ≠ 0, the solution is given by w = W-1 . The Section k (e) "Digression" showed that (A-1)T = (AT)-1 for invertible A, so wT = (W-1)T = (WT)-1 = W-1 = w and therefore w is symmetric. Since w is known, the Bm = wmnbn have been found. Finally, Bm • Bn = wmibi • wnjbj = wmiwnj bi • bj = wmiwnjWji = wmiδni = wnm 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 = wmnbn says En = g'mn en which agrees with our original definition of the En. Moreover, det(W) = det( g ¯'ab) = g' of Section 5 (k), so as long as g' ≠0, one has detW ≠ 0. 5. For the general bn case, if one imagines that the bn are the tangent base vectors for some transformation F b with some R b , 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', and the assumed invertibility of transformation F means that J ≠ 0 everywhere, so g' ≠ 0 and then detW ≠ 0. In all cases considered in this document, detW ≠ 0. 6. If the bm are true tensorial vectors, then the Bm = wmnbn will be as well 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 (e) 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'). Section 6: Reciprocal Base Vectors 66 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 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 (f) below for the two dual sets En, en and Un, un. 9. In standard notation (Section 7) the equations relating to bn will become: bm • bn = δm,n bn • bk = wnk bn • bk = wnk bm = wmnbn bm = wmnbn (c) Covariant partner for E n The covariant partner for En is given by ( E¯n)i = g¯ij (En)j so that ( E¯n)i = g¯ij (En)j = g¯ij gja Rna = δi,a Rna = Rni Thus, one can regard the covariant vectors E¯n as being the rows of matrix R R = ⎣⎢⎡ ⎦⎥⎤ E¯1 E¯2 E¯3 ... E¯N = [ E¯1, E¯2, E¯3 .... E¯N ]T which compare to S = [ e1, e2, e3 .... eN ] (d) Summary of the basic facts: (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 ( E¯n)i = Rni e'n = R en where ( e'n)i = δn,i // from Section 3 (b) In general, neither set of base vectors -- the tangent {.. en .. } or the reciprocal {.. En .. } -- is orthogonal, since the metric tensor g' in general is not diagonal . And in general none of these vectors is a unit vector. Section 6: Reciprocal Base Vectors 67 (e) Repeat the above for the inverse transformation: definition of the U 'n Section 3 (d) introduced the inverse tangent base vectors called u'n. The prime indicates that these vectors exist in x'-space. In analogy with what was done above , one can define the inverse reciprocal base vectors U'n according to U'n ≡ gni u'i => u'n = g¯ni U'i // since g ¯ = g-1 Everything goes along as in the previous sections, but with these changes: g'↔ g R ↔ S e n → u'n e'n → un En → U'n E'n → Un Here are the key results, translated from above, U'n ≡ gni u'i (S U'n) ≡ gni (Su'i) => Un = gni ui S = R-1 ( U'n)i = gnaRia = g'iaSna // sum on second indices ( un)k = Ski(u'n)i = δn,k ( Un)k = Ski(U'n)i = Ski gnaRia = SkiRiagna = δk,agna = gnk ( U'¯n)i = g¯'ij (U'n)j (u'n)k = Rkn u'n • u'm = g¯nm |u'n| = g¯nn = hn R = [ u'1, u'2, u'3 .... u'N ] ( U'n)i = g'ia Sna U'n • U'm = gnm | U'n| = gnn S = [ U'¯1, U'¯2, U'¯3 .... U'¯N ]T = g na Ria u'n • U'm = δn,m U'n ≡ gni u'i u'n = g¯ni U'n (U¯'n)i = Sni un • um = g¯nm Un • um = δn,m Un • Um = gnm un = S u'n where ( un)i = δn,i // from Section 3 (translated) It is helpful to keep all these eight vector sym bol names in mind (and each has a covariant partner) x'-space x-space 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 and recall that An • am = δn,m for each of the four dual pairs (two primed, two unprimed). Section 6: Reciprocal Base Vectors 68 (f) Expanding vectors on diff erent 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 Un = gni ui The components V n are Un • V because Un • um = δn,m. From Section 5 (g), one can write V n = gnmV¯m ( regarded here as a definition of the V ¯m) so one finds that V = ΣnVn un = Σn gnm V¯m un = ΣnV¯m gmn un = ΣnV¯m Um and thus another expansion for V is this V = V¯1 U1 + V¯2 U2 +... = ΣnV¯nUn where un • V = V¯n 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 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 Section 6: Reciprocal Base Vectors 69 Therefore, the expansion is V = V'1e1 + V'2e2 +... = Σn V'n en where En • V = V'n . 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 V = Σn V'n en = Σn,m g'nmV¯'m en = Σm V¯'m Σng'mn en = Σm V¯'m Em so V = V¯'1E1 + V¯'2E2 +... = Σ n V¯'n En where en • V = V¯'n 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 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 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 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 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 Section 6: Reciprocal Base Vectors 70 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 and here then are the x'-space expansions: 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¯'m 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 (g) 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 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 This is a generalized cross product (Appendix A) 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 Here ε is the totally antisymmetric tensor with N indices. If κ is the k th 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 Section 6: Reciprocal Base Vectors 71 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 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 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. 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. 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 (h) Comparison of e ¯n and En One could create a covariant partner to en which would be e¯n = g¯ en as described in Section 5 (g). This e¯n is not the same as the reciprocal base vector En ≡ g'nk ek. The comparison is interesting: ( e¯n)i ≡ g¯ik(en)k // matrix acts on vector index ( En)i ≡ g'nk (ek)i // matrix acts on ek label Section 6: Reciprocal Base Vectors 72 If x-space is Cartesian, then e¯n = en as usual, but of course En ≠ en in this case since g' ≠1. We mention this to head off a possible confusion when the Standa rd Notation is introduced in the next Section and the above two equations become ( en)i ≡ gik(en)k // Standard Notation, g lowers an index ( en)i ≡ g'nk (ek)i // Standard Notation, k is a label on ek, not an index 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). (i) 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] where the generalized cross product is discussed in Appendix A. This basis bn defines a "coordinate system" in that we can expand a position vector as follows x = Σn x(b) n bn 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 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] 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. Section 6: Reciprocal Base Vectors 73 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 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 Section 3 (a). This system is "right handed" because B = ε abc..x (e'1)a (e'2)b.... (e'N) = εabc..x δ1aδ2b....δxN = ε123...N = +1 > 0 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 orthonorma l basis in x-space. See Section 3 (c) 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 discussed in Section 3 (a). One has B = ε abc..x (e1)a (e2)b.... (eN) = εabc..x Sa1 Sb2.... SxN = det(S) Therefore, using σ ≡ sign ( detS ) and J being the Jacobian as in Section 5 (k), 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 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 Section 6: Reciprocal Base Vectors 74 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 transformed object under this transformation. 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. 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), de t(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 transf ormation 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. Section 6: Reciprocal Base Vectors 75 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. Section 7: Standard Notation 76 7. Translation to the Standard Notation In this Section 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. (a) Outer Products It is possible to form larger tensors from smaller ones using the "outer product" method. For example, consider, T ab ≡ UaVb 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' so in this way a contravariant rank-2 tensor (Section 5 (e)) has been successfully constructed from two contravariant vectors. Similarly, T ¯ab ≡ U¯aV¯b => T ¯'ab = ST aa' ST bb' T¯a'b' so the outer product of two covariant vectors transforms as a covariant rank-2 tensor. (b) Mixed Tensors and Notation Issues Suppose we take the "outer product" of a contravariant vector with a covariant vector, [ ... ] ab ≡ UaV¯b 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 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 Section 7: Standard Notation 77 [.....] abcd = UaV¯b XcY¯d 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 ob ject transforms, looking at the right hand side of the equation, 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 i mmediately available for inspection. Worse, there may be no right hand side for a mixed tensor, because not a ll mixed 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 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- 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 ( see Section 5 (m) ), dots are placed on certain indices 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. (c) 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 section (j) 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 w ould then have for our ongoing example, Section 7: Standard Notation 78 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 and then we get this translation W ab-cd- = UaVb- XcYd- → Wa bc d = UaVbXcYd // the path taken 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 Sections into this new notation. There are some subtle details that will be discussed in the following sections. (d) Some Preliminary Translations: raising and lowering indices on a vector with g In the entire rest of this entire Section, anything to the left of a → arrow is in "developmental notation", while anything to the right of → is in "Standard Notation". 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 g ab → gab // the contravariant rank-2 metric tensor g¯ab → gab // the covariant rank-2 metric tensor g is inverse of g ¯ → gab is inverse of g ab 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, Section 7: Standard Notation 79 "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 "representa tions" and the notation makes this very clear. Still, one cannot just write det(g) because det(g) is repres entation dependent, so one must say something like det(gab) or det(g **) to denote a particular determinant. As for converting a vector from one type to the other, V ¯a = g¯abVb → Va = gabVb // g ab "lowers" a contravariant index Va = gab V¯b → Va = gab Vb // gab "raises" a covariant index , and so in this new notation, the covariant metric tensor g ab becomes an "index lowering operator" and the contravariant 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 accounting (credits and debits), where the notation itself serves as a check on the accuracy of bookk eeping entries, as will be seen below. As for bolded vectors, the translation rule is, V → V V¯ → V 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. The invariant distance and covariant dot products: dx i → dxi (ds)2 = g¯ab dxa dxb → gabdxadxb A•B = g¯ab Aa Bb → A•B = gab Aa Bb = AbBb = AaBa = gabAaBb 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 sp ace 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 for a short while. (e) Contraction of a Pair of Indices When two indices are summed together in a tensor e xpression and one is up and the other down, one says that the two indices are contracted . Here is an example, where the index b is contracted, Section 7: Standard Notation 80 Va = gabVb It is shown below that contracted indices neutraliz e each other in terms of how an object transforms. Thus, for example, the RHS above g abVb transforms as a covariant vector, which conveniently matches the LHS. Similarly, AaBa transforms as a scalar. (f) Dealing with the matrix R 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) In terms of "existence", this object has one leg in each space of Picture A. The gradient operator ∂/∂x k is an x-space thing, while x'i is an x'-space thing. Since this object 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 that the gradient is a covariant vector, so we regard k as a covariant index. And of course dx' i is a contravariant vector, so i is a contravariant index. Here then is the proper translation: Rik ≡ (∂x'i/∂xk) → Ri k ≡ (∂x'i/∂xk) 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 very typical for an object to have up and down indices but the object is not a tensor. The canonical example is that for a non-linear transformation x' = F(x), xi has a contravariant index but is not a contravariant vector. Consider now the translation of the tran sformation rule for a contravariant vector V' a = RabVb → V'a = Ra bVb // contravariant 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 co urse then matches the LHS. So 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) one cannot ask how the equation V'a = Ra bVb "transforms" under a transformation. That question can only be asked about equations constructed of objects all of which are tensors in the same space. Here V and half of R are in one space, and V' and the other half of R are in a different space. There is no object called R', as if R were in x-space and R' were in x'-space. Section 7: Standard Notation 81 (g) Repeat the above section for S We omit the words and just show the translations ( ∂xi/∂x'k) → ( ∂xi/∂x'k) S ik ≡ (∂xi/∂x'k) → Si k ≡ (∂xi/∂x'k) V¯'a = ST abV¯b = Sba V¯b → V' a = Sb aVb // covariant (h) About ε and δ The Kronecker δ is sometimes written in different wa ys to make things "look nice", δab = δa b = δba = δb a = δa,b Section (m) will show that one can regard the above sequence of equalities as saying g ab = ga b = gba = gb a = δa,b 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 tensor, but tradition 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 det(M) = εabc...x M1aM2b.....MNx = εabc...x Ma1Mb2.....MxN or in an ordinary cross product A a = εabcBbCc . This permutation tensor has 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 (d) just as with any te nsor. Therefore, in Cartesian space with g = 1, indices on ε are raised and lowered with no consequence, 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 te nsor described above and these properties Section 7: Standard Notation 82 are the same in x-space as in x'-space. Then for general g ≠1, εabc..x is NOT the permutation tensor. The bottom line is that one must be aware of the sp ace in which one is working (the Picture). The ε appearing above in the determinant expansion is always just th e permutation tensor, but in the cross product that is not the case, and one would properly write A a = εabcBbCc and conclude that the cross product of two ordinary contravariant vectors is a covariant vector density (Appendix D (g)). 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 (u), 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... 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 Appendix D (c), there IS only one antisymmetric te nsor 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 shown in Appendix D, the covari ant 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. (i) Matrix Multiplication, the meaning of RT, and Rotations in Standard Notation. After reading this section, one will understand why ma trix multiplication notation is often avoided in the standard notation. Since both indices of rank-2 te nsors and tensor-like objects can be up or down, the meaning of matrix multiplication requires some clar ification. Part of the complication is defining a reasonable transpose for the various up and down matrix forms. The smaller subsections below are presented here in a "short form" to allow for a reason able logic flow since certain "facts" have not yet been introduced. The "long form" presentations of th ese facts occur in later sections of Section 7. 1. Translation of determinants. Section (f) showed that R ab → Ra b, so one translates from old to new notation, det(R) = ε abc... R1aR2b....RNx → det(Ri j) = εabc... R1 aR2 b....RN x det(S) = εabc... S1aS2b.....SNx → det(Si j) = εabc... S1 aS2 b....SN x det(R) = ε abc... Ra1Rb2....RxN → det(Ri j) = εabc... Ra 1Rb 2....Rx N det(S) = εabc... Sa1Sb2.....SxN → det(Si j) = εabc... Sa 1Sb 2....Sx N where ε is the bookkeeping permutation tens or discussed in section (h). 2. Inverse of R and S. Again, section (f) showed that 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 Section 7: Standard Notation 83 (∂xc/∂x'a) (∂x'a/∂xb) = δc b or Sc a Ra b = δc b and therefore it must be that (R -1)c a = Sc a. A similar argument shows that (S-1)c a = Rc a. Using the results of the next section which allow us to raise and lower indices on both sides of an equation, this relationships R-1 = S is valid for all four matrix position possibilities, (R-1)ik = Sik (R-1)i k = Si k (R-1)ik = Sik (R-1)ik = Sik and of course the same is true for S -1 = R. Thus arise these translations from old to new notation: R -1 = S → (R-1)i k = Si k and all other index combinations S-1 = R → (S-1)i k = Ri k and all other index combinations RR-1 = RS = 1 etc → Ri k(R-1)k a = Ri kSk a = δi a etc (1) We have silently used a certain "down tilt" matrix multiplication here which w ill be explained later. 3. Tensor g raises and lowers any index. So far the following translation rules have been established: gab → gab g ¯ab → gab Rik → Ri k Sik → Si k . It was shown in developmental notation Section 5 (e) how a rank-2 contravariant tensor transforms. Here then is how that statement translates to the new notation M' ab = Raa'Rbb'Ma'b' → M'ab = Ra a'Rb b'Ma'b' contravariant rank-2 tensor ( 2 ) M¯'ab = Sa'aSb'bM¯a'b' → M' ab = Sa' aSb' bMa'b' covariant rank-2 tensor Since g itself is such a rank-2 tensor, replace M by g to get g'ab = Raa'Rbb'ga'b' → g'ab = Ra a'Rb b'ga'b' ( 3 ) g ¯'ab = Sa'aSb'b g¯a'b' → g'ab = Sa' aSb' b ga'b' It was shown in section (d) that V a = gaa'Va' and Va = gaa' Va' so that g aa' lowers a vector index and gaa' raises a vector index. That is to say, g aa' converts a contravariant vector index into a covariant one, and gaa' does the reverse. What does g aa' do to the index of a rank-2 tensor? Consider the following definition: M a b ≡ gaa'Ma'b ( 4 ) Section 7: Standard Notation 84 Since gab and gab are inverses, it follows that M ab = gaa' Ma' b ( 5 ) How does this new object Ma b transform? The claim is that it transforms as a mixed rank-2 tensor, which would mean that M' a b = Ra a' Sb' b Ma' b' The upper index gets a factor R a a' and the lower index gets a factor Sb' b , consistent with (*) above. It is not hard to prove this claim: (4) (3) (2) (5) M'a b = g'aa'M'a'b = ( Ra cRa' d gcd ) ( Se a'Sf bMef) = ( Ra cRa' d gcd ) ( Se a'Sf b gei Mi f ) = [ R a cRa' d gcd Se a'Sf b gei] Mi f = [ R a c (Se a'Ra' d) gcd Sf b gei] Mi f = [Ra c (SR)e d gcd Sf b gei] Mi f = [ R a c δe d gcd Sf b gei] Mi f = [Ra c gcd Sf b gdi] Mi f (1) = [ R a c (gcd gdi) Sf b] Mi f = [Ra c δc i Sf b] Mi f = [Ra i Sf b] Mi f inverses = Ra a' Sb' b Ma' b' QED Similarly one could define M ab ≡ gaa'Ma'b and one would find that M' ab = Sa' a Rb b' Ma'b' so M ab is then another member of the family of rank-2 tensors. Finally were one to define M ab ≡ gbb'Mab' one would find that M ab transforms as in (2). To summa rize the four transformation results M' ab = Ra a' Rb b' Ma'b' M'a b = Ra a' Sb' b Ma' b' M'ab = Sa' a Rb b' Ma'b' M'ab = Sa' a Sb' b Ma'b' One sees then a family of four tensors associated with M. One is contravariant, one is covariant, and the other two are mixed. [Later we will show that Si j = Rji and this allows one to write the above equations in a manner that is easier to remember. ] 4. Raising Lowering Rule: g aa' [----a'---] = [---- a---] a n d gaa' [----a'---] = [----a---] Section 7: Standard Notation 85 Here [----i---] represent a tensor with a certain contrava riant index i and dashes indicate other indices which could be up or down. Similarly [---- i---] is another tensor in the same family where the index i that was up is now down. The notion of higher rank tensors is coming soon, but we just want to establish the general idea that ANY index on ANY tensor can be raised or lowered by an appropriate g tensor. For the rank-2 tensors this was demonstrated explicitly ab ove, and section (d) showed it was va lid for rank-1 tensors (vectors), g aa'Va' = Va gaa' Va' = Va Comment : Notice that in every equation shown above, the summed indices always occur in the contracted form discussed in section (e) above, which is to say, one index is up and the other is down. 5. Contraction Tilt Reversal Rule: [-----a---------a----] = [----- a---------a----] This is proved in section (k) below, but since we are going to need it right now, here is a preview of that proof: ( note that g ab gac = gba gac = δb c ) [-----a---------a----] = gab gac [-----b---------c----] = δb c [-----b---------c----] = [----- b---------b----] The upshot is that one can "reverse the tilt" on any pair of contracted indices. 6. The Diagonal g Rule: ga b = δa b and g ab = δab // and same for g' This is proved in section (m) below, and here a preview: g a b = gaa' ga'b // gaa'raises the first index of tensor g a'b = δa b // because g ij and gij are inverses of each other 7. Matrix Multiplication in the Standard Notation. Although there are various forms of matrix multiplication, the most standard form is that obt ained when all matrices have a "down-tilt" form. Consider this example, Ca b = Aa cBc b where it is assumed that all three objects are down-tilt rank-2 tensors. Although these are "split level" matrices, one can see that the index c has the correct "adjacency" property to justify matrix multiplication. A second requirement is that any matrix summed inde x should be a genuine contraction with one index up and the other down. The above tensor transformation rule can then be written in this more compact matrix notation ( SN means Standard Notation, dt means down-tilt) : C = AB // all down-tilt [C = AB] SN,dt . By application of suitable g tensors to the first equa tion above (or by lowering index a and raising index b on both sides), one gets C ab = AacBcb . Application of the Tilt Reversal Rule on index c then gives Section 7: Standard Notation 86 Cab = AacBcb . Again the adjacency and contraction rules are met, so this equation can also be represented by C = AB // all up-tilt [C = AB] SN,ut where ut means up-tilt. It has just been shown that, [C = AB] SN,dt ⇔ [C = AB] SN,ut The conclusion is that in Standard Notation, matrix notation can be used if all matrices in the equation being represented are either all down-tilt or all up-tilt. As will be shown later in section (u), such matrix equations are all "covariant" in that both sides of the equation have the same tensor transformation property, and this is due to the fact that the matrix summation index is a contraction. One might wonder about the conversion from developmental notati on (DN) to standard notation (SN) of the following matrix equation, where A a nd B are contravariant rank-2 tensors, [ AB = C ] DN ⇔ AacBcb = Cab → AacBcb = [???]ab . It is not hard to show entirely in developmental notation that, unless STS = 1 (meaning S is a rotation), the object C ab is not a tensor. So for genera l underlying F, C is not a tensor, even though A and B are tensors. This fact is reflected in the LHS of th e partially translated equation, where AacBcb is seen to not be a tensor because the c index is not a contraction (it is not tilted). The rules for translation from developmental to standard notation specify what to do with R, S, and with tensors. Since C is none of these objects, we cannot really "translate" the equation [ AB = C ] DN into standard notation. However, we could define Cab ≡ AacBcb and then write [ AB = C ] DN ⇔ AacBcb = Cab → AacBcb = Cab ⇔ [ AB = C ] SN,up with the understanding that Cab is not a contravariant rank-2 tensor, and we now have yet a third kind of matrix multiplication in the Standard Notation, qualifie d by "up" meaning all indices are up on all objects. One could of course do the same thi ng with covariant matrices and write [A¯ B¯ = C¯ ]DN ⇔ A¯acB¯cb = C¯ab → AacBcb = Cab ⇔ [ AB = C ] SN,dn The situation can be summarized in this picture: Section 7: Standard Notation 87 In standard notation, there are four different kinds of "matrix multiplication" and only the down-tilt and up-tilt equations are the same equation, and only they are covariant The moral here is that if one chooses to use matrix multiplication notation in the Standa rd Notation, one needs to be very clear which multiplication form one is dealing with. What about the special case [ AB = 1 ] DN where A is a contravariant tensor ? In this case, it must be that B = A-1 and then one has, [ AB = 1 ] DN ⇔ Aac(A-1)cb = δab → Aac(A-1)cb = (AA-1)ab = (1)ab = δab t h a t i s t o s a y : [ A ( A-1) = 1 ]SM,up and in this case all four Sta ndard Notation equations are valid, the other three being: [ AB = 1 ] SM,dn ⇔ Aac(A-1)cb = (AA-1)ab = (1)ab = δab [ AB = 1 ] SM,ut ⇔ Aac(A-1)cb = (AA-1)ab = (1)ab = δab [ AB = 1 ] SM,dt ⇔ Aa c(A-1)c b = (AA-1)a b = (1)a b = δa b so that [ AB = 1 ] DN ⇔ [ AB = 1 ] SN,up ⇔ [ AB = 1 ] SN,dn ⇔ [ AB = 1 ] SN,ut ⇔ [ AB = 1 ] SN,dt Now what about the special case [ RS = 1 ] DN ? Since R is not a contravariant tensor the above logic does not apply in the sequence stated. But we ha ve rules for translating R and S, so [ RS = 1 ] DN ⇔ RacScb = δab → Ra cSc b = δa b ⇔ [ RS = 1] SN, dt and since the four SN forms are equivalent (S = R-1), we end up with the same conclusion as above, [ RS = 1 ] DN ⇔ [ RS = 1 ] SN,up ⇔ [ RS = 1 ] SN,dn ⇔ [ RS = 1 ] SN,ut⇔ [ RS = 1 ] SN,dt confirming what was stated in a preliminary way in item 2 above. Finally, what about the special case [ STS = 1 ]DN ? Again using the rule that S ij → Si j, [STS = 1 ]DN ⇔ ScaScb = δab → Sc aSc b = δab ⇔ ??? Section 7: Standard Notation 88 The combination Sc aSc b does not have the proper adjacency to be matrix multiplication. In order to achieve adjacency, we must have some notion of "tra nspose" in the Standard Notation. That notion is the subject of the next section, and the result is going to be this: (A T)ab = Aba (AT)a b = Aba (AT)ab = Ab a (AT)ab = Aba Therefore Sc a = (ST)ac and we then have [STS = 1 ]DN ⇔ ScaScb = δab → (ST)ac Sc b = δab ⇔ ??? Although adjacency of the c index is now obtained, the form matches none of the Standard Notation forms for matrix multiplication defined above. Thus, no contact is being made with any of these standard forms by this translation. Since S TS = 1, one must have S = (ST)-1 and therefore all four of our stated SN forms are equivalent. So we then have this interesting situation: [STS = 1 ]DN ⇔ (ST)ac Sc b = δab ⇔ / [STS = 1 ]SN,dt ⇔ the other three SN forms What this says is that [STS = 1 ]DN and [STS = 1 ]SN,dt are completely different equations . Specifically, [STS = 1 ]DN ⇔ ScaScb = δab → (ST)ac Sc b = δab or Sc a Sc b = δab [S TS = 1 ]SN,dt ⇔ (ST)a cSc b = δa b or S ca Sc b = δa b . The very last equation on the right can be written (as will be seen below) Ra c Rbc = δab and this is one of the standard orthogonality conditions on R and this is always true for any underlying transformation F. Therefore, [S TS = 1 ]SN,dt is always true, and this means that in terms of Standard Notation down-tilt matrix multiplication, the matrix S is always real orthogonal. In contrast, in terms of Developmental Notation matrix multiplication, S is only real orthogonal if it is a rotation matrix! That is to say, in the general case: [S T = S-1 ]DN ⇔ / [ST = S-1 ]SN,dt So the very meaning of the equation ST = S-1 is dependent on which notion of matrix multiplication is involved. The operator S-1 is really defined in terms of SS-1 = S-1S = 1 and this involves matrix multiplication. One final thing to note: (S ab)DN = (Sa b)SN ≠ (Sab)SN = gac(Sc b)SN // unless it happens that g = 1 so clearly when one writes S ab or Rab one should be clear which notation is involved. Section 7: Standard Notation 89 8. Transpose of a rank-2 tensor. If A is a contravariant rank-2 tensor, the translation mapping (AT)ab = Aba → (AT)ab = Aba seems obvious, and the object A T therefore also transforms as a rank-2 tensor. Once (AT)ab = Aba is established in standard notation, one can apply the metric tensor g to lower either or both of the indices of this equation, to get (AT)a b = Aba , (AT)ab = Ab a, and (AT)ab = Aba. Notice in all four equations that the indices on the two sides of the equation are reflected in a vertical axis passing between the indices. This causes a left index to become a right index and vice ve rsa, as one would expect for transposing a matrix. Moreover, on each side of all four equations, each index has the same contravariant/covariant sense. This same argument also applies to R and S even though they are not tensors. The only difference is that the first index of Rba is lowered by g' while the second by g. For example, (RT)a b = Rba where b is a g' type index and a is a g type index, as will be elaborated in section (o) below. Similarly (ST)a b = Sba . To summarize the situation with transposes in Standard Notation: (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba (AT)a b = Aba (RT)a b = Rba (ST)a b = Sba (AT)ab = Ab a (RT)ab = Rb a (ST)ab = Sb a (AT)ab = Aba (RT)ab = Rba (ST)ab = Sba . Notice that the rule is not (A T)a b = Ab a which would be a straight swap of indices (and would result in AT not being a tensor). The straight swap idea works fo r the pure contravariant and pure covariant forms of A, but not for the tilted forms! As shown in Theo rem 4 below, this tilted transpose form has some interesting implications. Theorem 1: In the Standard Notation, S is a real-orthogonal matrix. This means all of the following: [S-1 = ST]SN [SST = 1]SN [STS = 1]SN where [...] SN can mean any of the four Standard Notati on matrix multiplication forms given in the previous section. This theorem does not apply to the matrix S in the developmental notation as also explained in the previous section. That is to say, the equations [SST = 1]DN and [SST = 1]SN are different. Proof of theorem : start with (3) above which says g ij is a covariant rank-2 tensor, g' ab = Sa' aSb' b ga'b' // next, apply gb'b to both sides (or just raise index b on both sides) g'ab = Sa' aSb'b ga'b' // next, reverse the tilt of the b' index g'ab = Sa' aSb'b ga'b' // next, use the Diagona l g rule in two places δab = Sa' aSb'b δa'b' = Sa' aSa'b // next, use (ST)aa' = Sa' a δab = (ST)aa'Sa'b // next use matrix notation as per above [1 = STS]SN,ut => ST = S-1 => 1 = SST Section 7: Standard Notation 90 Theorem 2: In the Standard Notation, R is a real-orthogonal matrix. This means all of the following: [R-1 = RT]SN [RRT = 1]SN [RTR= 1]SN In other words, the previous theorem also applies to R. Again, this fact is not true of the developmental notation matrix R ab. The proof is very similar but just different enough to warrant showing it : Proof of theorem : start with (3) above which says gij is a covariant rank-2 tensor, g' ab = Ra a'Rb b'ga'b' // next, apply g b'b to both sides (or just lower index b on both sides) g'a b = Ra a'Rbb'ga'b' // next, reverse the tilt of the b' index g'a b = Ra a'Rbb'ga' b' // next, use the Diagona l g rule in two places δa b = Ra a'Rbb'δa' b' = Ra a'Rba' // next, use (RT)a' b = Rba' δa b = Ra a'(RT)a' b // next use matrix notation as per above [1 = RRT]SN,dt => RT = R-1 => 1 = RTR Comment: In the developmental notation, neither R nor S is a real-orthogonal matrix, unless by accident, but in the standard notation R and S are always real-orthogonal. Theorem 3: (a) [S = RT]SN and [R = ST ]SN (b) Sa b = Rba and Sab = Rb a ( reflect indices in vertical line between them) Proof of theorem: (a) As shown in the previous section, S = R-1 for any index positions in the standard notation. From Theorem 2, RT = R-1 in standard notation. Therefore S = RT (and so ST = RTT = R) . (b) From (a) , S = R T => S ab = (RT)ab = Rb a S = RT => Sa b = (RT)a b = Rba QED An implication of this theorem is that one can complete ly eliminate references matrix S in tensor analysis and that is what is usually done . This is like replacing S with R-1 in the developmental notation. Theorem 4: For a standard notation tilted matrix A, det(A) ≠ det(AT). This surprising result points out a potential hazard of using tilted matrices in th e standard notation, and perhaps is an indication of why people avoid matrix notation and just write out all the components. For a traditional matrix A ab the determinant is given by either of these forms (figure on rows or columns) det(A) = εab..A1aA2b.... = ε ab..Aa1Ab2 ... det(A T) = εab..AT 1aAT 2b.... = ε ab..Aa1Ab2...... = det(A) Section 7: Standard Notation 91 In the tilted standard notation, however, one has det[A i j] = εab..A1 aA1 b.... = ε ab..Aa 1Ab 2 ... det[(AT)i j] = εab.. (AT)1 a(AT)1 b.... = εab... Aa1Ab2 but this is not the same as either of the det[Ai j] forms! Here is a simple example: det(A) = εab..A1 aA1 b... = ⎪⎪ ⎪⎪ A1 1 A1 2 A2 1 A2 2 = A1 1 A2 2 – A2 1 A2 2 det(AT) = εab... Aa1Ab2 = ⎪⎪ ⎪⎪ A11 A21 A12 A22 = A11 A22 – A21 A12 ≠ det(A) The point is that the standard notation transpose rule (AT)i j = Aij doesn't just swap the indices, it also changes the tilt. That means that the columns of one de terminant matrix are not the same as the rows of the other and that is why det(A) ≠ det(AT). Corollary : [RTR = 1]SN does not imply that det(R) = ± 1 for a tilted R matrix. The usual proof goes that det(R TR) = det(RT)det(R) = det(R)det(R) = [ det(R) ]2 = 1, but of course the part saying det(RT) = det(R) is no longer valid for a tilted R ma trix in standard notation. Therefore, [RTR = 1]SN does not lead us to the conclusion that the Jacobian of Section 5 (k) is forced to be ± 1 (it is always the tilted R and S matrix that appears in equations like 1/J = det(Ri j) ). 9. Orthogonality rules. The rules are easy to show: [RTR = 1]SN ,dt => (RT)a bRb c = δa c => R baRb c = δa c [RRT = 1]SN ,dt => Ra b(RT)b c = δa c => Ra bRcb = δa c [RTR = 1]SN ,ut => (RT)abRbc = δac => Rb aRbc = δac [RRT = 1]SN ,ut => R ab(RT)bc = δac => R abRc b = δac These orthogonality rules are derived in a slightly different manner and order in section (r) below. 10. Rotation matrices. In developmental notation, a real orthogonal matrix R has the property that RRT = 1 ⇔ R-1 = RT ⇔ RTR = 1 The determinant of R must be ± 1 since RR T = 1 => (detR)2 = det(1) = 1 => detR = ± 1 Section 7: Standard Notation 92 If detR = +1, then R can be a rotation matrix (in an y number of dimensions N). If detR = -1, then R can be that same rotation matrix combined in any ma nner with an odd number of axis reflection matrices, each represented by a unit matrix in which one of the diagonal 1's is replaced by a -1. For example, for N = 2 one can write, R = R = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ = rotation matrix detR = +1 rx = ⎝⎛ ⎠⎞-1 0 0 1 = reflect the x-axis ry = ⎝⎛ ⎠⎞1 0 0 -1 = reflect the y-axis R = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ ⎝⎛ ⎠⎞-1 0 0 1 = ⎝⎛ ⎠⎞ -cosθ sinθ sinθ cosθ = R rx detR = -1 For N = 3 one might write R = R = rotation matrix (Euler angles) detR = +1 rx = ⎝⎜⎛ ⎠⎟⎞ -1 0 0 0 1 0 0 0 1 ry = ⎝⎜⎛ ⎠⎟⎞ 1 0 0 0 -1 0 0 0 1 r z = ⎝⎜⎛ ⎠⎟⎞ 1 0 0 0 1 0 0 0 -1 P = r xryrz = ⎝⎜⎛ ⎠⎟⎞ -1 0 0 0 -1 0 0 0 -1 A = R r x or R ry or rxR or RP etc all have: detA = -1 The parity P operation is included only when N = odd. In general, then, a real orthogonal matrix R satisfying RRT = 1 is a rotation matrix combined with any even number (including 0) of axis re flection matrices (detR = +1), or it is a rotation matrix combined with any odd number of axis reflections (detR = -1). If one is interested in real-orthogonal transformations that can be arrived at in a continuous morphing manner from the identity transformation, then one would include only the rotation matrices in the set of real orthogonal matrices. So, our developmental notation characteri zation of a rotation matrix is this: [ RR T = 1 ]DN where we now use the notation described in item 7 above. Up to this point R has stood for a "rotation matrix", but now we shall assume that this R is also the matrix which represents the linearized F transformation as in Section 2. In standard notation one finds RR T = 1 ⇔ RabRcb = δa,c → Ra bRc b = δa,c . The equation Ra bRc b = δa,c cannot be written in any of the four standard notation matrix multiplication forms discussed in item 7, so we just leave it "as is". There are several other characterizations of R bei ng rotation matrix. First consider this sequence: Section 7: Standard Notation 93 Ra bRc b = δa,c // the above Ra bSbc = δa,c // Rc b = Sbc as shown above in Theorem 3 Ra bSbc Rcd = δa,c Rcd // apply R cd to both sides and sum on c Ra b (SR)bd = Rad // SR = 1 Ra b δbd= Rad Ra d = Rad Sda = Sd a where on each of the last two equations the two si des are related by reflection of the indices in a horizontal line running through the indices. Since it has now been shown that for a rotation Ra b = Rab, it follows that Ra bRc b = δa,c => RabRcb = δa,c The following theorem has now been proven: Theorem 5 (Rotations): A general transformation F can be represented at a point x by the linearized matrices R and S as discussed in Section 2. In develo pmental notation if R is a rotation matrix then it can be characterized by any of these equivalent statements in developmental notation (DN) : [RR T = 1]DN [RTR = 1]DN [R-1 = RT]DN . In standard notation, the fact that R is a rotation matrix can be characterized by any of the following equivalent statements: Ra bRc b = δa,c RabRcb = δa,c Ra b = Rab . Of course if R is a rotation, so is S = R -1. Since Ri j = Sji, the above are the same as S baSbc = δa,c Sb aSb c = δa,c Sba = Sb a . In principle, R = R( x) might be a different rotation matrix at different points x in x-space, so that F in that case would be a non-linear transformation. This woul d happen in 3D space if the Euler angles which parameterize a general rotation were arbitrar y (but reasonable) functions of position x. What can be said about a rotation matrix Q which is not the R matr ix which linearizes F? The translation from developmental to standard notation could be defined this way (same as for R and S) QQT = 1 ⇔ QabQcb = δa,c → Qa bQc b = δa,c QTQ = 1 ⇔ QbaQbc = δa,c → Qb aQb c = δa,c Since Qa b(Q-1)b c = δa,c comparison shows that one must have (Q-1)b c = Qc b which is a straight swap of the indices. Since we have no information regarding th e possible tensor or tensor-like nature of Q, we don't know how to raise or lower its indices , and nothing much more can be said. Section 7: Standard Notation 94 11. Variations on the relation between g and g'. The third subsection above gave the basic statement of the transformation properties of tensor g. These can be inverted as follows, g'ab = Ra a'Rb b'ga'b' => gab = (R-1)a a'(R-1)b b'g' a'b' = Sa a' Sb b' g' a'b' g'ab = Sa' a Sb' b ga'b' => g ab = (S-1)a' a (S-1)b' b g'a'b' = Ra' a Rb' b g'a'b' and here then is a summary, g' ab = Ra a'Rb b' ga'b' gab = Sa a'Sb b' g'a'b' g'ab = Sa' a Sb' b ga'b' g ab = Ra' a Rb' b g'a'b' If x-space is Cartesian with g = 1, the first column above simplifies to g' ab = Ra cRb c // g = 1 g'ab = Sc a Sc b = Rac Rbc // g = 1 Notice that the summation index c is not a contraction here. Also, although R and S are not tensors, the sums shown produce the true tensors g' ab and g'ab. In the previous section is was shown that if R is a rotation matrix, then [RRT = 1]DN and in standard notation on gets Ra cRb c = δa,b . If x-space is Cartesian so metric tensor g = 1, then it was just shown above that for any R, g'ab = Ra cRb c . Therefore, when R is a rotation, g' = 1 as well as g = 1, just as one would expect. (j) 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' as shown below in section (q)) 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' Note that for Ra a' and Raa' the second index is the summation index, but for Sa' a it is the first index. Section 7: Standard Notation 95 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 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. 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 (i). 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' and one can consider the operator Ra 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¯ One could then define ( Rdp)abc de ; a'b'c'd'e' ≡ Ra a' Rb b' Rc c' Rdd' Ree' so that A' aB'bC'cD'dE'e = ( Rdp)abc de ; a'b'c'd'e' Aa'Bb'Cc'Dd'Ee' 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' 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. Section 7: Standard Notation 96 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 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. Section 7: Standard Notation 97 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 A'abc... = Raa'Rbb'....... Aa'b'c'... = Σprimed Aa'b'c'... Raa'Rbb'....... (6) A' abc... = Ra a' Rb b'....... Aa'b'c'... = Σprimed Aa'b'c'... Ra a' Rb b''....... (7) (k) 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 dow n. As noted above, such a sum is called a contraction . The contracted indices could be on the same object or th ey could be on different objects. We depict this situation with the following symbolic notation, [-----a---------a----] 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, Section 7: Standard Notation 98 [-----a---------a----] = gab gac [-----b---------c----] 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, gab gac = gba gac = δa,c, which at once gives the desired result [-----a---------a----] = [----- a---------a----] // the Contraction Tilt-Reversal Rule A notable example of course is this: A aYa = AaYa = " A • Y " // or perhaps " A .Y " as noted in Section 5 (i) When is index tilt-reversal allowe d and when is it not allowed? It is always allowed when both indices of the tilted contraction are valid tensor indices. Consider these four exam ples to be discussed below: R abAb = RabAb Proof: R abAb = Racgcb gbdAd = gcbgbdRacAd = δc dRacAd = RacAc AaRab ≠ 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 ∂ aAa ≠ ∂aAa Proof: ∂aAa = (gac∂c)(gadAd) = gacgad(∂cAd) + gac(∂cgad)Ad = δ cd(∂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 rule of the next section also does not apply for this same reason, so ∂ aAa does not transform as a scalar under general F. Section (v) below continues this topic. Here is one more example along th e lines of the fourth example above that shows the potential danger of reversing a tilt when it is not ju stified. Consider the equation, V a = εabcBb;c (1) // valid where B b = a tensorial vector B b;c = ∂cBb – Γn bcBn = the covariant derivative of vector B b = a rank-2 tensor Section 7: Standard Notation 99 εabc = a rank-3 tensor density (weight -1) (the Levi-Civita tensor) Va = a vector density (weight -1) 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 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 Were one to blindly lower a and reve rse the b and c tilts, one would get V a = εabcBb,c (4) // NOT valid 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") => V a = 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" = ε abc(∂cBb) + gbb' εab'c (∂c gbb") Bb" = ε abcBb,c + gbb' εab'c (∂c gbb") Bb" = εabcBb,c + extra term! 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) (l) The Contraction Neutralization Rule A contracted index pair plays no role in how an ob ject transforms, the two indices neutralize each other, as we now show. First, recall that the indices on a general rank- n tensor (perhaps formed from several tensors) transform the same way an outer product of n vect ors transforms, where the vector index types match those of the tensor. The vectors transform this way: Section 7: Standard Notation 100 V'a = Ra bVb V ' a = Sb aVb So, we take our same "big object" a bove and now ask how it transforms. In the following, the X's represent either R or S factor s 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----] = δ c b XXXXX XXXXXXXXX XXXX [-----b---------c----] = XXXXX XXXXXXXXX XXXX [-----a---------a----] where now the only X's left are for the other indices. Again we look at our canonical example, A aYa = AaYa = A.Y The contracted vector indices cancel each other ou t 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 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 Dbc e ≡ Tabc ae = a mixed rank-3 tensor Ec = Tabc ab = a contravariant vector (rank-1 tensor) Q = A aYa = a scalar (rank-0 tensor) It is similarly possible to build larger tensors from smaller ones, for example Z abc de = Va We gab Lc which goes under the same rubric "outer product" mentioned earlier. Section 7: Standard Notation 101 (m) Raising and lowering indices on g On the one hand, since g ab and gab are inverses of each other (formerly g ¯ and g) , one has gabgbc = δa,c = δac where the above-mentioned "look-nice" form of δ a,c makes indices match. On the other hand, g abgbc = gbc // left g lowers the left index of the right g, or the converse Comparison shows that g bc = δbc As a sanity check, consider Va = gabVb Applying our Contraction Tilt-reve rsal Rule, this can be written Va = gabVb but this is = δabVb = Va There are many ways to write things, here is a collection (g ab = gba !) g abgbc = δac = gac // start out gabgbc = δac = gac // tilt reversal of the line above; just says δabδbc = δac g abgbc = δa c = ga c ga bgb c = δa c = ga c (n) Other forms of R 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 Section 7: Standard Notation 102 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. In each case, examination of the corresponding partial derivative shows that that the index sense matches on both sides. For example, in R ab = (∂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. (o) Summary of facts about R Rik ≡ (∂x'i/∂xk) → Ri k ≡ (∂x'i/∂xk) V'a = RabVb → V'a = Ra bVb Va = Sa bV'b [= RbaV'b] R a b(x) = ( ∂x'a/∂xb) // original object (formerly R ab) Rab = Ra b' gb'b = ( ∂x'a/∂xb) // g pulls up the second index Rab = g'aa'Ra' b = ( ∂x'a/∂xb) // g' pulls down the first index Rab = g'aa'Ra' b' gb'b = (∂x'a/∂xb) // both actions at once R a b = g'aa'Ra'b' gb'b // the inverse of the previous line (using gabgbc = δa c twice) (p) Repeat all the above for S S ik ≡ (∂xi/∂x'k) → Si k ≡ (∂xi/∂x'k) V¯'a = ST ab V¯b = Sba V¯b → V'a = Sb aVb V a = Rb aV'b [= SabV'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 S a b ≡ gaa'Sa'b' g' b'b // the inverse of the previous line (using gabgbc = δa c twice) (q) Theorem: Sa b = Rba and Sab = Rb a ( reflect indices in vertical line between them) This theorem has already been proven in section (i) Th eorem 3 as part of the di scussion there of the fact that, when tilted matrix forms are consider for R and S, one has all of the following matrix results: S-1 = ST SST = 1 STS= 1 S = RT S = R-1 R-1 = RT RRT = 1 RTR= 1 R = ST R = S-1 Here two slightly different lower-level proofs of this theorem that S a b = Rba . Once this is established, one ran raise and lower indices on either side to get all of the following Sa b = Rba S ab = Rb a Sab = Rba Sab = Rba Section 7: Standard Notation 103 As noted earlier, an implication is that one can comp letely eliminate references to matrix S in tensor analysis and that is what is usually done! Proof of Theorem: This proof is a bit long-winded, but brings in many earlier results: δ ba" 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 of each other. g' bb' δb' b" Sa a" g' a"b"= Sa b // reorder and introduce another δ g' bb' (Rb' a' Sa' b") Sa a" g'a"b"= Sa b // 1 = RS so δb' b" = (Rb' a' Sa' b") g' bb' Rb' a' (Sa a"Sa' b" g'a"b") = Sa b // regroup g' bb' Rb' a' (gaa') = Sa b // use gaa' = Sa a"Sa' b" g'a"b", see end of (i) above (g' bb' Rb' a' gaa') = Sa b // regroup R ba = Sa b // g and g' raise and lower R's indices, see (n) above Notice that the above theorem says S a b = (∂ xa/∂x'b) = (∂x'b/∂xa) = Rba A faster way to derive this result is to differentiate dx cdxc = dx'cdx'c and use the chain rule: dx' b = ( ∂( dx'cdx'c)/∂x'b ) = (∂(dxcdxc)/∂xa) (∂xa/∂x'b) = dxa (∂xa/∂x'b) => (∂x' b/∂xa) = (∂ xa/∂x'b) => R ba = Sa b 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) Here index a is always in x-space, while index b is in x'-space. The two vector transformation rules V' a = Ra bVb V' a = Sb aVb Section 7: Standard Notation 104 can now be written V'a = Ra bVb V' a = RabVb which has the advantage that the indices are prope rly arranged for matrix multiplication in both cases. Here then is a restatement of the transf ormation of the example given in section ( l) , T 'abc de = Ra a' Rb b' Rc c' Sd' d Se' e Ta'b'c' d'e' // no pairs contracted becomes T 'abc de = Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' // no pairs contracted It is easy to remember since th e second index is always the summed index and the other index has to match (up or down) the left side of the equation. (r) Orthogonality Rules, the Inversion Rule, and the Cancellation Rule The above theorem Sa b = Rba can be used to eliminate S in various forms of RS = 1: SR = 1 S a b Rb c = δa c R ba Rb c = δa c R ba Rb c = δa c Σ 1st RTST = 1 Rb a Sc b = δac Rb aRbc = δac Rb a Rbc = δac Σ 1st RS = 1 Ra b Sb c = δa c Ra b Rcb = δa c R cb Ra b = δca Σ 2nd STRT = 1 Sb a Rc b = δac Rab Rc b = δac Rc b Rab = δc a Σ 2nd The four results in the right column are called orthogonality rules for R. The first pair is summed on the first index, the second on the second. In section (i) it wa s shown that these rules are just statements of the fact that in up or down tilted standard notation R is a real-orthogonal matrix so RRT = RTR= 1. Inversion Rule. Consider now an equation which one wants to invert for the object on the right, [---- a-----] = Ra b [--------b------] (*) // before The inversion rule for moving R to the other side of the equation is to reflect R's two indices in the horizontal index plane, so the result will be Rab [----a-----] = [--------b------] // after Proof : Rename b →b' in (*), apply R ab to both sides and sum on a, then use an orthogonality rule: Rab [----a-----] = R ab Ra b' [--------b'------] = δb b' [--------b'------] = [--------b------] QED Recall from the last section that reflection in the vertical index plane has a different implication, Section 7: Standard Notation 105 Sa b = Rba or (R-1)a b = Rba or (RT)a b = Rba Cancellation Rule. Next, consider a different generic equation, Ra b [----b-----] = Ra b [--------b------] (**) // before The cancellation rule says the equation is still valid if identical contracted R factors are canceled on both sides such that the contraction index becomes a free index, [---- b-----] = [--------b------] // after Proof: Rename b →b' in (**), apply R ab to both sides and sum on a, then use an orthogonality rule, R abRa b' [----b'-----] = R abRa b' [--------b'------] δ b b' [----b'-----] = δb b' [--------b'------] [---- b-----] = [--------b------] QED (s) The tangent and reciprocal base vectors and expansions on same Tangent and reciprocal base vectors Here are some basic translations: ( 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 → ( en)i = Si n = Rni // contravariant index i ( E¯n)i = Rni → ( en)i = Rn i // covariant index i ( En)i = Rnkgki → ( en)i = Rn kgki = Rni // contravariant index i As noted earlier, writing a vector in bold such as en is not enough to say whether the vector is contravariant or covariant. If one form or the other is intended, one must show an index up or down, even if it is just a dummy placeholder index. As examples, S = [ e1, e2, e3 .... eN ] → Si j = [(e1)k, (e2)k, (e3)k .... (eN)k] R = [ E¯1, E¯2, E¯3 .... E¯N ]T → Ri j = [( e1)k, (e2)k, (e3)k .... (eN)k]T Section 7: Standard Notation 106 The relationship between en and en is very simple, En ≡ g'ni ei → en = g'ni ei and en = g'ni ei 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 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 The dot products are 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 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 at the end of Section 6 (a) becomes En • um = Rnm → en • um = Rn m // = < en | um> in bra-ket notation (App E (g)) 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, g'↔ g R ↔ S e n → u'n e'n → un En → U'n E'n → Un we can obtain the corresponding results for the i nverse tangent and reciprocal base vectors: ( u'n)i → ( un)i // contravariant index i ( u¯'n)i → ( un)i // covariant index i (U'n)i → ( un)i // contravariant index i (U¯'n)i → ( un)i // covariant index i ( u'n)i = Rin → ( u'n)i = Ri n = Sni // contravariant index i Section 7: Standard Notation 107 ( U¯')i = Sni → ( un)i = Sn i // covariant index i ( U'n)i = Snkg'ki → ( un)i = Sn kg'ki = Sni // contravariant index i R = [ u'1, u'2, u'3 .... u'N ] → Ri j = [(u'1)k, (u'2)k, (u'3)k .... (u'N)k] S = [ U¯'1, U¯'2, U¯'3 .... U¯'N ]T → Si j = [( U¯'1)k, (U¯'2)k, (U¯'3)k .... (U¯'N)k]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 Summary table. The summary table given at the end of Section 6 (e) was this x'-space x-space 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 which translates into this → : x'-space x-space 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 x-space expansions The x-space expansions of Section 6 (f) were 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 Section 7: Standard Notation 108 and they now become → : 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 x'-space expansions Similarly, the x'-space expa nsions of Section 6 (f) were 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¯'m 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 and they now become → : 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'm 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 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 = hnV'n V' = V'n e'n = V'n e'n = Vn u'n = Vn u'n // x'-space expansions 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. 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 (w) 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 Section 5 (k). Section 7: Standard Notation 109 (t) Comment on Covariant versus Contravariant Consider this expansion for a vector V in x-space, V = Vnbn Vn = V • bn where bn is some basis having dual basis bn where as usual bn • bm = δnm. 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 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 as in section (r)) V'nb'n = [Rn m Vm][ Rnj bj] = (Rn m Rnj) Vm bj = δmj Vm bj = Vj bj = Vnbn Compare then the transformation of V n with that of the basis bn: V 'n = Rn m Vm bn' = Rnm bm 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 one finds not surprisingly that the dual basis must vary as bn' = Rn m bm and then one has V' n = Rnm Vm bn' = Rn m bm 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 Section 7: Standard Notation 110 If the basis bm is varied as shown here, then the dual basis bm varies as shown above and V remains invariant. Comparing now the way the V n transform with the way the basis vectors bm transform, 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 . (u) 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) 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') 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", the objects on the two sides of the equation have the same indices which are of the same type, so both sides are 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 transformations for all the pieces, one ends up with the unprimed equation. 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 form s of the R matrix. Recall from section (q) that S b a = Rab and that SR = 1 is replaced by the vari ous orthogonality rules of section (r). 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 [R aa'Rd d'Rcc'Qa'd' c'(x)] = [Raa'Rbb' Ha'b'(x)] [Rb b"Rcc' Tb" c'(x) ] [Rd d'Bd'(x)] Section 7: Standard Notation 111 = R aa' Rd d' Rcc'(Rbb' Rb b") Ha'b'(x) Tb" c'(x)Bd'(x) Using one of the orthogonality rules of section (r), = 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) so that, using the fact that Q is a tensor to replace Q' on the left side of (*), (Raa'Rd d'Rcc') Qa'd' c'(x) = (Raa' Rd d' Rcc') Ha'b'(x) Tb' c'(x) Bd'(x) Now apply the Cancellation Rule of sec tion (r) three times to conclude that Qa'd' c'(x) = Ha'b'(x) Tb' c'(x) Bd'(x) and then remove all primes to get Q ad c(x) = Hab(x)Tb c(x) Bd(x) // x-space equation 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) 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) 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 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') . 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. Section 7: Standard Notation 112 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 (m)). 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" Aμ: Fμν ≡ ∂μAν - ∂νAμ where ∂ μ means gμα∂α, the contravariant form of the gradie nt operator. The components are then 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) while the two homogeneous equations become ∂αFμν + ∂μFνα + ∂νFαμ = 0 or ∂αFμν + cyclic = 0 . One can see that each of these equa tions involves only tensors and we expect that in x'-space these equations will take the form ∂' νF 'μν = μ0 J'μ with J'μ = (cρ',J') ∂'αF 'μν + ∂'μF 'να + ∂'νF 'αμ = 0 or ∂'αF 'μν + cyclic = 0 Objects like ∂νFμν and ∂αFμν are true rank-3 tensors because the transformation F is linear. Covariance of tensor equations involving derivatives with non-linear F. A tensor equation which involves derivatives of tensors is non-covariant unde r transformations 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 derivati ves by covariant derivatives (which are indicated by a semicolon). In general relativity, th is is known as the Principle of General Covariance (Weinberg p 106). A simple exam ple is the tensor equation g ab;c = 0 (Appendix F (i)). Examples relating to the transformation from Cartesian to curv ilinear coordinates appear in Section 15. Section 7: Standard Notation 113 (v) 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 This second term did not arise earlier when we looked at ∂ a on a scalar field φ'(x') = φ (x) , (∂' aφ') = (Rad∂d) φ = Rad (∂d φ). 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 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 V b;a ≡ ∂aVb – Γk ab Vk => V d;c ≡ ∂cVd – Γk cd Vk where Γ c ab is a certain function of the metric tensor g. One then finds that 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] (*) 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 Γ c 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 Γ dab ≡ [ab,d] ≡ ½ ( ∂agbd + ∂bgad – ∂dgab ) // Christoffel 1st kind Section 7: Standard Notation 114 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(n). 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') = Rbb'Rdd'(∂d gb'd') + other terms 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. (w) 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 (s) ab ove. 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 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 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 . In the case of rank-2 tensors, a product like ui⊗uj = uiuj is called a dyadic (see Appendix E). 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 . Section 7: Standard Notation 115 Appendix E (g) promotes the interpre tation of a rank-2 tensor 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 As Appendix E shows, these two matrices are rela ted to each other by a similarity transformation A' = R A R-1. 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). Section 8: Length Area and Volume 116 8. Transformation of Differential Length, Area and Volume This Section and all remaining Sections use th e Standard Notation introduced in Section 7. The term N-piped is short for N dimensional parallelepiped. The context is Picture B: Since this Section is quite lengthy, a brief overview is in order: Overview 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 Section 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. Section (a): The differential N-piped mapping The differential N-piped mapping is descri bed and various symbols are defined. Section (b): Properties of the fi nite 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. Section (c): Back to the differential N-piped mapping: how edges, areas and volume transform 1. 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). 2. Edge Transformation. The edges en of the x-space N-piped map into axis-aligned edges e'n in x'-space. 3. Area Transformation. Tensor density notions as presen ted in Appendix D are used here. 4. Volume Transformation. The volume transformation is computed several different ways. 5. Covariant Magnitudes . These are | d x'(n)|, | dA'n | and | dV' | in the Curv ilinear View of x'-space. 6. Two Theorems. (1) g' / h' n2 = g'nn g' = cof(g' nn) and (2) |( Πx i≠nei)| = cof(g'nn) . 7. Cartesian-View Magnitude Ratios. Appropriate for the continuum mechanics application. 8. Nested Cofactor Formulas: Transformation of N-piped areas of dimension N-2 and lower. 9. 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. Section 8: Length Area and Volume 117 10. Concatenation of Transformations. What happens to the transformati on of vectors, areas and volumes when two transformations are concat enated? One result is that J = J 1J2. Examples of area magnitude transformation for N = 2,3,4 Example 2: Spherical Coordinates: area patches Section (d): Transformation of Differential Volume applied to Integration The volume transformation obtained in Section (c) 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. Section (e): Interpretations of the Jacobian (a) The differential N-piped mapping Section 3 (a) above considered the following situation (dx' n > 0) : d x'(n) = e'n dx'n x'-space axis-aligne d differential vector, and ( e'n)i = δni 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) A superscript (n) on the differentials makes clear there is no implied sum on n. 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 (a), the x'-sp ace 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. 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 Appendix C (e). 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: Section 8: Length Area and Volume 118 The orthogonal Cartesian-View N-piped allows visualization of these curvilinear coordinate variations, all dx'k > 0, d L'n ≡ dx'n d A'n ≡ Πi≠ndx'i d V' ≡ Πidx'i = d An dLn For example, for N=3 one would have d L'1 ≡ dx'1 d A'3 = dx'1dx'2 d A'1 = dx'2dx'3 d A'2 = dx'3dx'1 d V' = dx'1dx'2dx'3 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 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 associated with that edge (such as er → erdr in spherical coordinates), a nd that will bring us back to the differential N-piped picture above. Section 8: Length Area and Volume 119 (b) 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 ). • There are 2 N vertices. • 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 poi nt (the far face). • 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. • A face's vector area An is spanned by all the ei except en and is labeled by this missing en vector. • 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. • 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 • The volume of the N-piped is given by (see Section 5 (k) concerning J) V = | det [ e1, e2, e3 ... eN] | = | det(Sa b) | = g'1/2 = |J| 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 (Section 1 Example 2), 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 which 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 Section 8: Length Area and Volume 120 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 matrix R. Conventions for defining the area vector and volume . In Appendix B (and as shown above) 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 Section 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 one sees 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: 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 situation, we will redefine A3 = e1xe2 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 = e1xe2 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 (i). Here then are the expressions for An with this new definition, where the new forms are obtained from the previous ones by multiplying by σ = sign ( det(S) ) : An = det(Sa b) en = J en An = (-1)n-1 e1 x e2 ... x eN // en missing ( An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // e n missing The second line shows that pseudovectors can only exist for an odd number of dimensions (such as N=3). Section 8: Length Area and Volume 121 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 (i). So: V = det [ e1, e2, e3 ... eN] = det(Sa b) = J (c) Back to the differential N-piped mapping : how edges, areas and volume transform 1. 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 whic h has the properties listed above with the following edges and areas and volume: d x(n) = en dx'n / / e d g e s d An = J en (Πi≠ndx'i) // areas dAn = (-1)n-1 (dx'1e1) x (dx'2e2) ... x (dx' NeN) // 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 of Appendix A (i) (d An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) // en factor and index missing dV = det [dx' e1, dx'e2, dx'e3 ... dx' eN] = det [ e1, e2, e3 ... eN] (Πidx'i) = J det(Sa b) = ε abc..x (dx'e1)a(dx'e2)b..... (dx' eN)x where d An and dV obtained from An and V according to the new definitions described above. These equations apply to the right side of the N-pi ped mapping picture which is replicated here Section 8: Length Area and Volume 122 For spherical coordinates, the N-piped on the right would be spanned by these vectors (Sec. 3, Ex. 2 ) erdr = r^dr, eθdθ= rθ^dθ eφdφ = rsinθ φ^dφ 2. Edge Transformation. The edge d x(n) we know transforms as a tensorial vector under transformation F, so d x'(n) = R d x(n) where d x(n) = en dx'n 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 so this contravariant edge points in the n axis direction in x'-space. 3. Area Transformation. How does d An transform under F? Looking at the component form stated above (d An)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) // en factor and index missing 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 is that one adds the weights of the objects being combined. Therefore, one may conclude that the object d An is a vector density of weight -1. This has the immediate implication that dAn transforms under F according to the rule Section 8: Length Area and Volume 123 d A'n = J R d An or (d A'n)i = J Ri j (dAn)j // J-W = J-(-1) = J where J = det(S) is the Jacobian of Section 5 (k). Moreover, the above equation for (d An)i is a "valid tensor density equation" as per Section 7 (u) 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 Insertion of ε' iabc..x = J2 εiabc..x ( Appendix D (e) 3) 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) 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 εi123..N = (-1)n. The conclusion then is that d A'n = J2(Πi≠ndx'i) e'n since ( e'n)i = δn i // Section 7 (s) and the covariant vector area d A'n points in the n-axis direction in x'-space. For the reader dubious of the claim that d A'n = J R d An, consider: (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 If i ≠ n, one has a determinant with two columns the same since S κ n is missing so the result is 0 so the result is proportional to δn i. Continuing, = δ n i J { (-1)n-1 εnabc..x Sj n Sa 1 Sb 2.... Sx N (Πi≠ndx'i) } // Sκ n missing in group = δ n i J { (-1)n-1 εnabc..x Sa 1 Sb 2.... Sj n.... Sx N (Πi≠ndx'i) } = δn i J { εabc..n...x Sa 1 Sb 2.... Sj n.... Sx N (Πi≠ndx'i) } = δn i J { det(S) ( Πi≠ndx'i) } = δ n i J2 (Πi≠ndx'i) Section 8: Length Area and Volume 124 which agrees with the result just obtaine d from the covariant x'-space equation. 4. 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 Installing our expression for d An and d x(n) gives dV = J en (Πi≠ndx'i) • (en dx'n) = J (Π idx'i) en • en = J (Πidx'i) which is seen to agree with the modified dV definition stated above. Looking at dV = (d An)i [dx(n)]i, dV is seen to be a tensor combination of a vector de nsity of weight -1 with a vector density of weight 0 (an ordinary vector d x(n)), so according to Appendix D, dV must be scalar density of weight -1. This then tells us that dV' = J dV Since dV = J ( Π idx'i) one gets dV' = J 2 (Πidx'i) . 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) since e'n• e'n = en•en = 1. An alternate derivation of the dV transform rule dV' = J dV comes from just staring at dV = ε abc..x (dx'e1)a(dx'e2)b..... (dx' eN)x which by the argument above is seen directly to transf orm as a tensor density of weight -1. To complete the circle, we can verify for a second time the claim made above 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) = det(S) (Π kdx'i) = dV 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: Section 8: Length Area and Volume 125 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 V' = J dV // scalar density W = -1 5. 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)| 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 | where we note that d A'n • dA'n, 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 dV => | dV' | = |J| | dV | . Moreover it has been shown above that d x(n) = en dx'n => | d x(n)| = | en| dx'n = h'n dx'n d x'(n) = e'n dx'n => | d x'(n)| = | e'n| dx'n = h'n dx'n d An = J (Πi≠ndx'i) en => | d An| = |J| | en| (Πi≠ndx'i) = (|J| /h' n) (Πi≠ndx'i) d A'n = J2(Πi≠ndx'i) e'n => | d A'n| = J2 |e'n| (Πi≠ndx'i) = (|J|2/h'n) (Πi≠ndx'i) dV = J (Π idx'i) => |dV| = |J| ( Πidx'i) dV' = J2 (Πidx'i) => |dV'| = |J2 (Πidx'i) which can be written more compactly using the Ca rtesian-View coordinate variation groupings, 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| /h' n) dA'n dA'n ≡ Πi≠ndx'i d A'n = J2 dA'n e'n => | d A'n| = (|J|2/h'n) dA'n dV = J d V' => |dV| = |J| d V' d V' ≡ Πidx'i dV' = J2 dV' => |dV'| = |J|2 dV' 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 Section 8: Length Area and Volume 126 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 (i). 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. 6. Two Theorems : g' / h' n2 = 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' / h'n2 = g'nn g' = cof(g' nn) where g' = det(g' ij). The left equality is obvious since g'nn = 1/h'n2 (orthogonal coordinates) . Th e right equality can be shown as follows: (g'up)ab ≡ g'ab (g'dn)ab ≡ g'ab g' up = (g'dn)-1 = cof(g' dnT)/det(g'dn) = cof(g' dn)/det(g'dn) => (g' up)nn = cof[(g' dn)nn]/det(g'dn) or g' nn = cof[g' nn] / g' QED Theorem 2: |(Πx i≠nei)| = cof(g'nn) The quantity on the left is this |(Π x i≠nei)| ≡ | e1 x e2 ... x eN | where en is missing from cross product. We shall give three quick proofs of Theorem 2, the last being valid only for N=3. • First, at the start of section (c) above, one form for d An is given by 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 Comparison of this last result with the third line of the table above shows that the following must be true |(Πx i≠nei)| = (|J| /h' n) = g'1/2/h'n = (g' / h' n2)1/2 = cof(g'nn) QED where the last step follows from Theorem 1. • Here is a more direct proof: | Π x i≠n (ei)|2 = Πx i≠n (ei) • Πx j≠n (ej) = [ Πx i≠n (ei)]k [ Πx j≠n (ej)]k Section 8: Length Area and Volume 127 = [ 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 // 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). QED. • 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 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). and the cases n = 2 and 3 are similar. 7. 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' Then from the three x-space equations in the above table (just above Two Theorems) one obtains the following three ratios of x-space objects divided by th eir 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 = (1/h'n) |J| = (1/h' n) 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 It is convenient to make the definition dAn ≡ | dA(n)| Section 8: Length Area and Volume 128 and then the above area magnitude ratio relation may be written dAn = cof(g'nn) dA'n and d A'n = Πi≠ndx'i is just a product of the appropriate curvilinear coordinate variations. 8. Nested Cofactor Formulas. The object d An is the "area" of a face on an N-piped. 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 determined by equations similar to that above. For example, the mth face of face n of an N-piped has area ratio cof[cof(g' nn)]mm . This at least makes some sense since the matrix cof(g' nn) has dimension N-1, so its cofactor matrix has dimension N-2, a nd so on. For N=3, these faces would be line segments and one would have 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 and h'1 is in fact the edge length ratio given above. 9. 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 associated 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]) (d A)i = εiabc..x (dx[1])a(dx[2])b.... (d x[N-1])x , 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 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 Section 8: Length Area and Volume 129 differential vectors d x[i] ( arbitrary as long as they are linearly independent), and these will form a volume in x-space, dV = det (d x[1], dx[2], ..... d x[N] ) = εabc..y (dx[1])a(dx[2])b.... (d x[N])y 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' Review and covariant form of the d A and dV equations To review, then, 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]] . Written out in terms of covariant vector components 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 , where ε is the usual permutation tensor involved in cross products and determinants. As noted earlier, these are both "valid tensor density e quations" (end of Section 7 (u)) , so they 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 . where d x'[i]= Rd x[i]. From Appendix D, 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 . 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]] . 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" . Section 8: Length Area and Volume 130 Going back to x-space, we could have wr itten the equations there using g = det(g' ij) = det(δij) = 1 : 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]) . 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 Sections 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]] . (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 , (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 . d x' = R d x |dx'| = |d x| dA¯' = J ST dA¯ |dA'| = |J| |dA| dV' = JdV |dV'| = |J| |dV| J = det(S) J2 = g' Recall from Section 2 (c) 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. 10. Concatenation 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 F is R = R 2R1, and then S = R-1 = R1-1R2-1 = S1S2. Each Section 8: Length Area and Volume 131 transformation will have an associated Jacobian: J 1 = det(S 1) and J2 = det(S2). The concatenated transformation then has J = det(S) = det(S 1S2) = det(S 1)det(S2) = J1J2. The implication is that when one concatenates 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, 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. Examples of area magnitude transformation for N = 2,3,4 In the previous section 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 shown above. For N=2 the area magnitude transformation results ar e (for a general non-orthogonal x'-space system) d A 1 = g'22 dA'1 = h'2 dA'1 d A'1 = dx'1 = dL'1 d A2 = g'11 dA'2 = h'1 dA'2 d A'2 = dx'2 = dL'2 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 Section 8: Length Area and Volume 132 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 For an N=4 orthogonal system, dA 1 = 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 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 Section 5 showed 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 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 so that d Ar = r2sinθ dθdφr^ ρdφ rdθ ρ = rsinθ d Aθ = rsinθ drdφ θ^ ρdφ dr d Aφ = rdrdθ φ^ r d θ dr where all three vectors are seen to have the correct dimensions L2. 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, Section 8: Length Area and Volume 133 (d) Transformation of Differential Volume applied to Integration As discussed in Appendix C (h), the integral ∫D dV h( x) is the same regardle ss of the way the dV elements are chosen, as long as those elemen ts exactly 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. Moreover, the shape of the differential volume N-piped varies over the region of integration. Nevertheless, this "rag-tag band" of differential volum es, as noted in Appendix C 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, and of course this set of differential 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 integral as the limit of a sum, lim Σi dVa(xi) f(xi) = lim Σi dVb(xi) f(xi) that ∫D dVa h(x) = ∫D dVb h(x) 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 one might have to require h( x) to be a "test function" φ(x). Certainly one would require that both integrals converge. ] What has been shown in the previous section, regarding the Jacobian, is that dV a = |J( x')| dV' = |J( x')| ( Πi=1N dx'i) |J( x')| = g'(x) // g = +1 Combining this with the distributional symbolic equation dV a = dVb gives Section 8: Length Area and Volume 134 d V a = dVb | J ( x')| ( Πi=1N dx'i) = ( Πi=1N dxi) or | J ( x')| dV' = dVb Now overriding our previous notation, we can make these new commonly used definitions dV ≡ ( Πi=1N dxi) dV' ≡ ( Πi=1N dx'i) and express the distributional result as |J( x')| dV' = dV We refer to this distributional e quality in Appendix C as the "Jacobian Integration Rule". The symbolic equation is a shorthand for this equation ∫D dV h( x) = ∫D' dV' |J( x')| h( x) 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')) and finally, using Section 5 (k), ∫D ( Πi=1N dxi) h(x) = ∫D' ( Πi=1N dx'i) [ det(g'ab) ] h( x(x')) 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) = r ∫D dxdydz h( x) = ∫D' drdθdφ [ r2sinθ ] h(x(r,θ,φ)) det(g'ab) = r2 sinθ 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). Section 8: Length Area and Volume 135 (e) Interpretations of the Jacobian Using Section 5 (k) facts (in standard notation) and 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) // Section 5 (k) |J(x')| = det(g'ab(x')) = g'(x') / / S e c t i o n 5 ( k ) |J( x')| = the volume of the N-piped in x-space spanned by the en(x), where x = F-1(x') |J( x')| = dVN-piped /dV' = ratio of differential x-space N-piped volume / ( Πi=1N dx'i) |J( x')| = dV/dV' = ( Πi=1N dxi)/ ( Πi=1N dx'i) // distributional Jacobian Integration Rule As discussed in Section 6 (i), if the curv ilinear 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. (f) Volume integration of a tensor field under linear transformations In section (d) above it was shown that, with its interpretation as a distribution, dV' = J -1 dV . 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 g 1/2 transforms as a scalar density of weight -1, according to App. D (b) item 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) (*) 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) ) = ∫D g1/2dV Ri i'Rj j'..... Ai'j'k'...(x) If the underlying transformation x' = F(x) is linear, then the Ra b are independent of position and one has Section 8: Length Area and Volume 136 = Ri i'Rj j'..... ∫D g1/2dV Ai'j'k'...(x) = Ri i'Rj j'..... Ti'j'k'... Therefore, for linear transformations x' = F(x), an integral of the form (*) of a tensor field is itself a tensor of the same type. Examples: If g = 1 and x' = R x where R is a rotation (independent of position), then g' = 1 as well and one may conclude that the volume integral of a tensor fi eld 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] 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 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 Section 3. Back in x-space, th e 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 . Section 9: Divergence 137 9. The Divergence in curvilinear coordinates Note: Covariant derivations of all the curvilinear differen tial operator expressions appear in Section 15. In Sections 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 Sections 9 through 13. (a) Geometric Derivation of th e 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. This derivation is an exercise in using the transf ormation results obtained in Section 8 above, and in understanding the meaning of the components of a vector, as discussed in Appendix C (d). 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. 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 (c) above: Section 9: Divergence 138 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) 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 div B is a tensorial scalar field, and one can write [ d i v B]'(x') = [div B](x) The operator object (1/dV) ∫dA(x)• acts as a tensorial vector operator so that the result of its action on B is a tensorial scalar. In Section 8 it was shown that d A is a vector density of weight -1 and so is dV. This means that dV' = J dV and d A' = J Rd A so the ratio d A/dV is a tensorial vector. The fact that div B is a tensorial scalar is more obvious from the alternative divergence derivation given in Section 15 (c). The task is now to compute the integral ∫dA(x)•B(x). Appendix B shows that the N-piped faces come in paralle l pairs, so we start by considering pair n. As shown in Section 8 (c), the vector area of the far face of pair n is given by d An(x) = |det(Si j(x'))| en(x) ( Πi≠n dx'i) = g'(x') en(x) ( Πi≠n dx'i) Here |det(S)| = |J| = g'1/2 (Section 5 (k)), and en are the reciprocal base vectors (Section 6) . Quantities dAn, S, en and g' are explicitly shown as functions of space. Appendix B (c) shows that 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 . The contribution to the above divergence inte gral from "far face n" is, approximately, Section 9: Divergence 139 [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) where xfar is taken to be a point at the center of far face n. Recall from Section 7 (s) that B(x) = B'n(x')en where B'n(x') = en(x) • B(x) 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. Inserting the last equation above applied at x' = x'far , B'n(x'far) = en(xfar) • B(xfar) into the previous ≈ equation then gives d An•B(xfar) ≈ g'(x'far) ( Πi≠n dx'i) B'n(x'far) This far face n contribution to the flux integral is now 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) In x'-space, if x'cen is a point at the center of the near face of face pair n, then x'near = x'cen x 'far = x'cen + e'n dx'n where e 'n = axis-aligned basis vector in x'-space , since these two points map into the near and far face n centers in x-space. For any function f, f( x'far) - f(x'near) = (∂'n f(x')) dx'n // no implied sum on n 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'cen) B'n(x'cen)] dx'n In the limit that differentials are very close to 0, replace xcen by x. Then ∫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) Section 9: Divergence 140 where now all N differentials are present in ( Πi dx'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' where d V' = Πi dx'i is the volume of the differential N-piped in Cartesian-view x'-space (Section 8 (a)) . 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) where dV is the volume of the N-piped shown above. In Section 8 (c) item 7 it is shown that dV = (g') 1/2 dV' => (d V'/dV) = 1/ g'(x') so that [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')] // all x'-space coordinates and objects [div B](x) = ∂nBn(x) // all x-space coordinates and objects 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 cl aim 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') and thus the left sides of both equatio ns above could be replaced by [div B]'(x'). (b) Various expressions for div B It is shown above that [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')] 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')] and recall that the B' m are the coefficients of B when expanded on the en, B = B'nen. Section 9: Divergence 141 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 and then [div B](x) = [1/ g'(x') ] ∂'n [g'(x') B'n(x')/ h'n(x') ] For example, spherical coordinate work might use e^1, e^1, e^1 = 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) 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^ [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 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') Comment: Are the equations of the a bove "General" block valid if B is not a tensorial vector? For such a B one might try to make it be tens orial "by definition" as discussed in Section 2 (h). One would then go ahead and define B'n(x') ≡ Rn mBm(x) and claim success. If such a de finition does not result in an inconsistency, then such a B has been moved into the class of tensorial vectors. Example 1 of Section 2 (h) shows have such an inconsistency might arise, a nd 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 Section 9: Divergence 142 (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 the General block says div B = [1/ g'(x') ] ∂'n [g'(x') x'n] = ∂ 'n x'n + [1/ g'(x') ] x'n∂'ng'(x') = 3 + other stuff 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 Sections. (c) Translation from Picture B to Picture M&S 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 everywhere • 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) 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) Section 9: Divergence 143 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 Notice that the scalar function [div B]'(x') → [div B](u) according to the fourth rule above, and that the arguments of all h k(u) are suppressed. As an example, for N=3 the second line above becomes [div B](x) = [1/(h1h2h3)] { ∂1[h2h3 B1(u) ] + cyclic } B = Bn e^n where + cyclic means two other terms with 1,2,3 cyclically permuted. With the replacements B → E , Bn→ En hn → gnn the equation marked above 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 Example 1 for another example. (d) 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: 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 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 Section 9: Divergence 144 which compare to the above [div A](x) = [div A](u) = [1/(h1h2h3)] ∂n [h1h2h3 An(u) / hn] A = An e^n so one should identify the M&F A n with An, the coefficient of e^n. Section 10: Gradient 145 10. The Gradient in curvilinear coordinates This Section is considered in the Picture B context, (a) Expressions for grad f The gradient of f is defined in Cartesian space by [grad f] n ≡ Gn ≡ ∂nf(x) G = grad f = ∂nf(x) n^ Assuming f is a tensorial scalar field under F, then G n = ∂nf(x) are covariant vector field components under F. Since the equation above is then a tensor equation, we know it is covariant in the sense of Section 7 (u). Therefore in x'-space it becomes [grad f]' n = G'n ≡ ∂'nf '(x') According to Section 7 (s) ( based on Section 6 (f)) vector G can be expanded as G = G'1e1 + G'2 e2 +... = ΣnG'n en where en • G = G 'n 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 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 particular useful. To restate, [grad f]( x) = ∂'nf '(x') en // Curvilinear [grad f](x) = ∂nf(x) n^ // Cartesian 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') Section 10: Gradient 146 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 If an expansion on unit vectors is desired, the right equation on the last line can be written, G = [grad f](x) = G'iei = (G'i h'i) e^i = G'i e^i where G'i ≡ h'i G'i so then G'i(x')/h'i = [grad f]'i(x') = ∂'if '(x') = g'ij(x') ∂'jf '(x') G = grad f = G'i e^i 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 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) 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. Since div B was a scalar quantity, this mixture of vectors in x-space with components in x'-space did not arise. For the orthogonal case g' ij = (1/h'i2) δi,j 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) This above equations can be converted from Picture B to Picture M&S using the same rules given in Section 9 (c), which we repeat below Section 10: Gradient 147 • replace x' by u everywhere • 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. 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)) 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^ // M&S 1.05 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 f → φ e^i→ ai h i → gii the equation marked agrees with Moon & Spencer p 2 (1.05). (b) Expressions for grad f • B Sometimes one is interested in the following quantity (back to Picture B) grad f • B Section 10: Gradient 148 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 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) 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') 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 (see Sections 8 and 9) [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) where once again f ( u) = f(x(u)). Comment: According to the second line above, one can write grad f • dx = ∂nf(x) dxn = df = f( x+dx) - f(x) 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 f(x,y), the gradient then points "uphill", and the negative of the gradient then points "downhill". Section 11: Laplacian 149 11. The Laplacian in curvilinear coordinates The Laplacian (also known as the Laplace-Beltrami operator) is defined by lap f = div (grad f) = div G where G ≡ grad f Since f is (by assumption) a tensorial scalar field, grad f is a tensorial vector. Then, as found in Section 9, div(grad f) is a tensorial scalar, meaning [lap f]( x) = [lap f]'( x'). In Cartesian coordinates one writes lap f = ∇ 2f = ∇•∇f = Σn∂n2f but this form gets modified when lap f is expresse d in curvilinear coordinates. Section 9 showed that div G = [1/ g' ] ∂ 'm [g' G' m] where G = G'nen g' = det(g') Section 10 showed that grad f = G = [g'nm (∂'nf ') ] em = G' mem G' m = g'nm (∂'nf ') ∂'nf ' = ∂'n f(x(x')) = ∂ 'nf '(x') Therefore lap f = div (grad f) = div G = [1/ g' ] ∂'m [g' G' m] = [1/ g' ] ∂'m [g' g'nm (∂'nf ')] 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') For an orthogonal coordinate system, g'nm = h'n2 δn,m g'nm = (1/h'n2) δn,m g' = Πi h'i and the first line above simplifies to [lap f]( x) = [1/(Π ih'i)] ∂'m [(Πih'i) (1/h'm2) (∂'mf ') ] Section 11: Laplacian 150 Converting from Picture B to Picture MS gives (see Section 9 (c)) [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) The first line simplifies in the orthogonal case to [lap f]( x) = [1/(Π ihi)] ∂m [ (Πihi) (1/hm2) (∂mf ) ] // orthogonal // M&S 1.09 For N=3 this says, [lap f]( x) = 1/(h1h2h3) { ∂1 [ (h2h3/h1) ∂1f ] + cyclic } With the replacements f → φ h i2 → gii (Πihi) → g the equation marked above agrees with Moon & Spencer p 3 (1.09). Section 12: Curl 151 12. The Curl in curvilinear coordinates The vector curl is defined only in N=3 dimensions (but see section (f) below). Picture B is used. In Cartesian coordinates one writes [curl B]i(x) = [∇ x B(x)]i = εijk∂jBk(x) but when expressed in terms of curvilinear coordinates and components, the form is different. (a) Definition of curl B Consider the x-space differential 3-piped shown on th e right side of the figure in Section 8 (a), 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 th e "far" face does not. As shown in Section 8 (c) item 1, the vector area associated with the face pair n is ( |J| = g'1/2 when g = 1) d An = | det(Si j)| en ( Πi≠n dx'i) = |J| en ( Πi≠n dx'i) = g'1/2 en ( Πi≠n dx'i) n = 1,2,3 where J = det(S i j) is the Jacobian, as in Section 5 (k), and en is a reciprocal base vector, as in Section 6 or Appendix A (called En). Area d An is the out-facing vector area for the far face of pair n, while - d An is the out-facing vector for the near face. 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 which is Section 12: Curl 152 the same as the direction of en. For example, for the bottom face (near face 3) of the 3-piped shown above, this vector points "up", or toward the cente r of the 3-piped. Denote this line integral by ( ∫{B•dx)n 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). 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 Since d An is given above in terms of en, C should be expanded on the ek. As shown in Appendix D (h), C = curl B is in fact a vector field density of weight -1. For an ordinary vector one would write the expansion as C = Σk C'k ek, but the rule for tensor densities says to replace C'k→ J W C'k = J-1 C'k, as explained toward the end of Appendix E (b). Therefore, C = J-1 Σk=13 C'k ek J-1 = (1/ g' ) so that C • dAn = J-1 (Σk C'k ek) • (g'1/2 en ( Πi≠n dx'i) ) = C'n ( Πi≠n dx'i) and then C' n(x') ( Πi≠n dx'i) = ( ∫{B•dx)n // "curl = circulation/area" Our task is to compute this line integral and thereby come up with an expression for C'n(x'), the components of curl B when B is expanded onto the ek in x-space. Comment: This factor of J-1 won't appear in the final results, as will be seen, but we must include it in the analysis in order to claim that C'k are the components of vector C in x'-space. (b) Computation of the line integral This shall be done for the bottom face (n=3) of the x-space differential 3-piped. Since e3 is "up", the circulation is a counterclockwise line integral around the boundary of the bottom face. It is useful to have the above picture near at hand to allow visualizati on of the four contributions to the line integral: Section 12: Curl 153 ( ∫{B•ds)3 ≈ [B(xfront ) - B(xback)] • (e1 dx'1) + [ B(xright ) - B(xleft)] • (e2 dx'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 ei • ej = δij, expand B as follows B = B'jej where B' j(x') = B(x) • ej where the B' j are the covariant components of B in x'-space. This gives ( ∫{B•dx)3 = [B'1(x'front ) - B'1(x'back)] dx'1 + [B'2(x'right ) - B'2(x'left)] dx'2 where x'front = F( xfront ) and similarly for the other three points. In x-space one has d xBF ≡ xback - xfront = e2dx'2 d xRL ≡ xright - xleft = e1dx'1 Applying matrix R gives the corre sponding x'-space equations (recall d x' = R d 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...) Using the fact that f( x'+dx') ≈ f(x') + Σn∂nf(x') dx'n one finds B' 1(x'back) ≈ B'1(x'front ) + (∂ B'1/∂x'2) dx'2 d x' = e'2dx'2 B'2(x'right ) ≈ B'2 (x'left)) + (∂B'2/∂x'1) dx'1 dx' = e'1dx'1 Section 12: Curl 154 so the circulation integral 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) 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) 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 onl y fact needed here is that ε'abc... = εabc... where ε' is the tensor in x'-space, as shown in Appendix D (d). In Cartesian space one can regard εabc... = εabc... as a bookkeeping permutation tens or with the properties given in Sec tion 7 (h). Installing the prime on ε, ( ∫{B•dx)n = ε'nab ∂'aB'b ( Πi≠n dx'i) and this integral is then given entirely in terms of x'-space coordinates and objects. (c) Solving for the curl The equation for the curl obtained at the end of section (a) was C' n(x') ( Πi≠n dx'i) = ( ∫{B•dx)n Insert the section (b) result for ( ∫{B•dx)n to get C'n(x') ( Πi≠n dx'i) = ε'nab ∂'aB'b ( Πi≠n dx'i) The differentials cancel out, so then take dx' i→ 0 and thus shrink the 3-piped around the point of interest x = F-1(x') so that C'n = ε'nab ∂'aB'b C = J-1C'n en B = B'n en curl B = C = [(1/ g' ) ε'nab ∂'aB'b ] en Section 12: Curl 155 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 } Comment 1: In the first line above one can replace [curl B](x) by [∇ x B](x) with the understanding that the LHS is the curl in Cartesian x-space and the RHS is expressing this LH S in terms of x'-space coordinates and objects. The RHS is certainly not equal to ∇' x B' = ε'nab ∂'aB'b(x') n^' . It is to avoid this possible confusion that the curl is written out as the word curl, and the same comment applies to the other differential operators. Comment 2: 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'n en 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 Section 13. Once again, J = g' . (d) Various forms of the curl The first form is that just presented above, with C = curl B, C'n = ε'nab ∂'aB'b C = J-1C'n en B = B'n en curl B = [(1/ g' ) ε'nab ∂'aB'b ] en 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 For practical applications, one usua lly wants both vectors expanded on the e^n unit vectors in this way 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 so that 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 Section 12: Curl 156 To summarize: B'c = Rc dBd g' = g'( x') etc. [curl B](x) = ε'nab [(1/ g' ) ∂ 'aB'b ] en B = B'nen [curl B](x) = ε'nab [(1/ g' ) ∂ 'a(g'bcB'c )] en B = B'nen [curl B](x) = ε'nab [(1/ g' ) h'n ∂'a(g'bc B'c/h'c )] e^n B = B 'n e^n [curl B](x) = εnab ∂aBb(x) n^ // Cartesian B = Bn n^ Converting from Picture B to Picture MS (see Section 9 (c)) 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^ 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. Each of the above results can be written as a determinant using the idea det( Q) ≡ Σi Q1i cof(Q1i) : [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bn en [curl B](x) = (1/ g ) ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 g1cBc g2cBc g3cBc B = Bn en [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 // 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^ Section 12: Curl 157 (e) 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: 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^ Picture M&S: 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 [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 B1 B2 B3 B = Bn en [curl B](x) = (h1h2h3)-1 ⎪⎪⎪⎪ ⎪⎪⎪⎪ e1 e2 e3 ∂1 ∂2 ∂3 h12B1 h22B2 h32B3 B = Bn en [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 With the replacements B → E Bn→ En e^n→ an hi → gii (h1h2h3)-1 → (1/ g ) the equations marked above agree with Moon & Spencer p 2 (1.07) and p 3 (1.07a). (f) The curl in N > 3 dimensions Looking at the basic form of the curl above [curl B]n(x) = εnab ∂aBb(x) // Cartesian Section 12: Curl 158 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) ) ? 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. The rank-2 tensor B b;a – Ba;b = ∂aBb – ∂bBa discussed in Appendix D (h) 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 B]n = εnab [Bb;a – Ba;b ]/2 = εnab Bb;a = εnab [∂aBb – ∂bBa ]/2 = εnab∂aBb . In relativity work, since 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). Writing the N-dimensional contravariant curl co mponents in this manner in Cartesian x-space, [curl B] ij = (Bj;i – Bi;j) 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 had weight -1 ). The general issue of expanding tensors is addressed in Appendix E where this general result is obtained A = Σ ijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... = contravariant components of A in x'-space Applying this to A = [curl B] one gets [curl B] = Σ ij[curl B]'ijei⊗ej = Σij(B'j;i – B'i;j)ei⊗ej 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 where the tangent base vectors en exist as usual in x-space, and B'i = B'i(x') where x' = F(x). Section 13: Vector Laplacian 159 13. The Vector Laplacian in curvilinear coordinates This operator is defined in terms of the vector curl which is only defined for N=3. The context is Picture B. (a) Derivation of the Vector Laplacia n in general curvilinear coordinates The definition of the vector Laplacian of a vector field B(x) is ∇2B ≡ grad(div B) – curl (curl B) . 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 curl B is a vector density of weight -1 (see Appendix D (h)) and curl(curl B) a vector density of weight -2, and so apples ar e being subtracted from oranges to make ∇2B. This tensor conundrum is resolved in Section 15 (g) so we ignore it for now and proceed undaunted. In Cartesian coordinates, one finds that [∇2B]i = ∇2(Bi) ≡ Σn ∂n2Bi 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, M&S use @ in place of ∇2 and we will honor these authors by using that symbol here, so @ B ≡ grad(div B) – curl (curl B) In order to make use of the re sults of earlier sections, define G ≡ grad(f) where f = div B V ≡ curl C where C ≡ curl B so that @ B = G – V Section 10 (a) gives this expression for G, G = grad(f) = ( ∂'kf ') ek in which expression Section 9 (b) allows replacement of f ' as follows, f ' = f '( x') = f( x) = div B = [1/ g' ] ∂ 'i [g' B'i] Section 13: Vector Laplacian 160 so that => G = ∂'k{ (1/ g' ) ∂'i (g' B'i)} ek = ∂'n{ (1/ g' ) ∂'i (g' B'i)} en The second term V is little more complicated. First, from Section 12 (d), C = curl B = ε'nab [(1/ g' ) ∂ 'a{ B'b} ] en = J-1C'n en J = g' V = curl C = ε'ncd [(1/ g' ) ∂ 'c{ J-1C'd} ] en = J-1V'n en (*) where recall from Appendix D (d) that ε'abc... = εabc.. = the usual permutation tensor, but written up and primed so as to be in covariant form. The reason for the factor J-1 in ∂'c{ J-1C'd} was explained in Comment 2 at the end of Section 12 (c) (namely, C is a vector density of wei ght -1). The above lines can be rewritten as C'n = ε'nab ∂'a{ B'b} V'n = ε'ncd ∂'c{ J-1C'd} / / J = g' Then write C' d = g'deC'e = g'de ε'eab (∂'aB'b) and insert this into the V' n equation to get V'n = ε'ncd ∂'c{ (1/g' ) g'de ε'eab (∂'aB'b) } = ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'aB'b) } so V = curl C = J-1V'n en = [ (1/ g' )ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'aB'b) }] en We can then summarize our results: @ 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 = [ (1/ g' ) ε'ncd ε'eab ∂'c { (1/g' ) g'de(∂'a[g'bfB'f]) }] en B = B'nen 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 Section 13: Vector Laplacian 161 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 and to this list we can add the Cartesian form @ B = [∇2Bn ] n^ // Cartesian, n^ = un B = Bn n^ Converting from Picture B to Picture MS gives (see Section 9 (c)) @ 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^ 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. (b) 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 a n d gij = (1/hi)2δi,j things simplify somewhat, @ B = hn e^n [gnk ∂k{ (1/ g ) ∂i (g Bi(u)/hi)} – (1/ g ) εncd εeab ∂c{ (1/ g ) gde (∂a [gbf Bf(u)/hf]) } ] = hn e^n [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]) } ] = h n e^n [ ∂n{ (1/ g ) ∂i (g Bi(u)/hi)} (1/hn)2 Section 13: Vector Laplacian 162 – (1/ g ) εncd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb(u)]) } ] = e^n [ (1/hn) ∂n{ (1/ g ) ∂i (g Bi/hi)} – (hn/g ) εncd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb]) } ] The first term can be written as e^n (1/hn) ∂nT where T = (1/ g ) ∂i (g Bi/hi) To expand the second term, set n = 1 and then write things out explicitly. For the moment, we suppress the leading factor – (h 1/g ) and write ε1cd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb]) } = ε 3ab ∂2{ (1/ g ) h32 (∂a [hb Bb]) } – ε2ab ∂3{ (1/ g ) h22 (∂a [hb Bb]) } c=2 d=3 c=3 d=2 = [ ∂2{ (1/ g ) h32 (∂1 [h2 B2]) } – ∂ 2{ (1/ g ) h32 (∂2 [h1 B1]) } ] a = 1 b = 2 a = 2 b = 1 – [ ∂ 3{ (1/ g ) h22 (∂3 [h1 B1]) } – ∂ 3{ (1/ g ) h22 (∂1 [h3 B3]) } ] a = 3 b = 1 a = 1 b = 3 = ∂2{ (1/ g ) h32 ( ∂1 [h2 B2] – ∂2 [h1 B1] ) } – ∂ 3{ (1/ g ) h22 (∂3 [h1 B1] – ∂1 [h3 B3] ) } Define now Γn in cyclic fashion. 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] ) and then we have shown that 2nd term (n=1) = – (h 1/g ) ε1cd εdab ∂c{ (1/ g ) hd2 (∂a [hb Bb]) } e^1 = – ( h 1/g ) (∂2 Γ3 – ∂3 Γ2) e^1 = + (h 1/g ) (∂3 Γ2 – ∂2 Γ3) e^1 Therefore the entire first term (n=1) of @ B is given by @ B (first term) = [(1/h 1) ∂1T + (h1/g ) (∂3 Γ2 – ∂2 Γ3) ] e^1 Section 13: Vector Laplacian 163 The other two terms are obtained by cyclic permutation so the final result is then [@ B](x) = [(1/h 1) ∂1T + (h1/g ) (∂3 Γ2 – ∂2 Γ3) ] 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 where T = (1/ g ) ∂i (g Bi/hi) Γ 1 = (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] ) With the replacements B → E Bn→ En e^n→ an hi → gii T → ϒ the result agrees with M&S p 3 (1.11). A more compact summary is this: @ B = [ (1/h n) ∂nT – (hn/g ) εnab∂a Γb ] e^n T = (1/ g ) ∂i (g Bi(u)/hi) Γb = (1/ g ) hb2 εbcd ( ∂c [hd Bd(u)]) (c) The Vector Laplacian in Cartesian coordinates First, one can verify that the last result of section (b) gives the starting point formula for @ B if g = 1 (in u-space). One then has, h i = 1 g = 1 u = F(x) = x ( e^n)i = Sni = δni => e^n = n^, B = Bn e^n = Bn n^ => Bn = Bn so the above 3-line equation block becomes @ B = [ ∂nT – εnab∂a Γb ] n^ T = ∂i (Bi(u)) Γb = εbcd ( ∂c Bd(u)) or Section 13: Vector Laplacian 164 @ B = [ ∂n{∂iBi} – εnab∂a { εbcd ( ∂c Bd)} ] 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)] QED Second, one can verify the claim made earlier that in Cartesian coordinates [@ B]n = ∇2 Bn . To show this, it is necessary to show that (left side from (*) above) ∂n ∂i Bi – εnab∂a εbcd(∂c Bd) = ∂i2Bn ? εnab εbcd ∂a (∂c Bd) = ∂n ∂i Bi – ∂i2Bn ? εbna εbcd ∂a (∂c Bd) = ∂n ∂i Bi – ∂i2Bn ? But since index b now appears only in the ε's, use ( up and down indices same in Cartesian x-space) ε bna εbcd = δncδad – δndδac // Appendix D (j) item 4 so (δncδad – δndδac) ∂a (∂c Bd) = ∂n ∂i Bi – ∂i2Bn ? δncδad∂a (∂c Bd) – δndδac∂a (∂c Bd) = ∂n ∂i Bi – ∂i2Bn ? ∂a (∂n Ba) – ∂a (∂a Bn) = ∂n ∂i Bi – ∂i2Bn ? ∂n (∂a Ba) – ∂a2Bn = ∂n (∂i Bi) – ∂i2Bn ? Since this last equation is true on inspection, QED. Thus it has been shown that, in Cartesian coordinates, [ @ B]n = ∇2 (Bn) = [ grad(div B) – curl (curl B) ]n It is this second (and rather complicated) form [ grad(div B) – curl (curl B) ]n which allowed us to obtain an expression for [ @ B] in general curvilinear coordinates using methods described in this document. Section 14: Summary 165 14. Summary of Differential Operators in curvilinear coordinates The results are given in the Picture M&S context, and are copied from Sections 9-13. The Standard Notation of Section 7 is used throughout. 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 comment at the end of Section 9 (c) 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 Section 3, and the en are the reciprocal base vectors of Section 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 (i). 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. Section 14: Summary 166 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 ___________________________________________________________________________ (a) divergence divergence general: [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: [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: [div B](x) = [1/(h1h2h3)] { ∂1[h2h3 B1(u) ] + cyclic } B = Bn e^n ___________________________________________________________________________ (b) gradient and gradient dot vector gradient general: [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: [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^ // M&S 1.05 Section 14: Summary 167 gradient dotted with a vector: [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) ___________________________________________________________________________ (c) Laplacian Laplacian general: [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: [lap f]( x) = [1/(Π ihi)] ∂m[ (Πihi) (1/hm2) (∂mf ) ] // orthogonal // M&S 1.09 Laplacian orthogonal N=3: [lap f]( x) = 1/(h1h2h3) { ∂1 [ (h2h3/h1) ∂1f ] + cyclic } ___________________________________________________________________________ (d) curl curl general: / / N = 3 o n l y [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 Section 14: Summary 168 [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: [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 ___________________________________________________________________________ (e) vector Laplacian vector Laplacian general: // N=3 only [@ 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^ Section 14: Summary 169 vector Laplacian orthogonal: [@ 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] ) or [ @ B](x) = [ (1/h n) ∂nT – (h n/g ) εnab∂a Γb ] e^n T = (1/ g ) ∂i (g Bi/hi) Γb = (1/ g ) hb2 εbcd ( ∂c [hd Bd]) ___________________________________________________________________________ Example 1: Polar coordinates: a practical curvilinear notation From earlier versions of 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 e1 = r(-sinθ,cosθ) = eθ = r e^θ // = r θ^ e2 = (cosθ,sinθ) = er = e^r // = r^ 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 h1 = hθ = gθθ = r h 2 = hr = grr = 1 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 support 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 Section 14: Summary 170 vθ 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) 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 so v θ = 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^ As an example of a differential operator, consider the divergence for orthogonal coordinates from the above table, [div B](x) = [1/(Πihi)] ∂n [(Πihi) Bn / hn] B = Bn e^n which applied to the present situation reads (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) } Suppose v x and vy are constants. Then the Cartesian expression says [div v](x) = ∂nvn = ∂xvx + ∂yvy = 0 + 0 = 0 The above curvilinear expression gives [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 The notation illustrated here works for any curvilinear coordinates. Section 15: Covariant Method 171 15. Covariant derivation of all curvili near differential operator expressions (a) Review of Sections 9 through 13 Let us review the discussion of Sections 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, 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 Section 9 the object div B was treated as the B flux 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 result (Section 7 (s)), B = B'nen , where the en are tangent base vectors in x-space, while the B'n are components of B in x'-space, we obtained a way to write the Cartesian-space div B in terms of curvilinear x' coordinates and x'-space objects, namely, B'n(x') and the gradient operator ∂'i = ∂/∂x'i. In Section 10 the object grad f was expanded using another Section 7 (s) expansion which is that shown above with the index tilt reversed, 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 Section 11 the Laplacian was written as lap f = div (grad f) and then the results of Sections 9 and 10 for div and grad were used to express lap f in terms of curvilinear coordinates. In traditional notation, one writes ∇2f = ∇ • [ ∇f ] = div (grad f). In Section 12 the object [curl B]n was treated as the circulation line integral around near face n of the same N-piped used in Section 9 for divergence, divide d by the area of that face, again in the limit the N- Section 15: Covariant Method 172 piped shrank to a point. With the same expansion for B shown above, this led to an expression of curl B once again in terms of the curvilinear x' coordinates and the components B'n. In Section 13 the vector Laplacian was treated using the definition @ B ≡ grad(div B) – curl (curl B) and then the results of Sections 9, 10 and 12 on div, grad and curl were recruited to produce an expression for @ B in terms of B'n(x') and x'-space coordinates. (b) 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, E 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 derivatives ] where Q---- means that Q has some number of up and down indices. The following four steps should then be carried out: 1. First, arrange any summed index pairs within [....] so they 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. Second, replace all derivatives w ith covariant derivatives. Thus is done by first writing all derivatives in comma notation, and then by replacing those commas with semicolons. For example: ∂ αBa = Ba,α → Ba;α // B a;α = ∂α Ba – Γn aαBn Objects like ∂ αBa which are not tensors become objects like B a;α which are tensors (see Appendix F (g) through (i) ). Note again that doing this does not affect the Cartes ian value of this object [...], because in Cartesian x-space Γn aα = 0. This is so because, as implied by Appendix F (a) ( where x-space is called ξ-space ), Γc ab ≡ uc • (∂aub) and the basis vectors un are constants in Cartesian space so ( ∂aub) = 0 and then Γc ab = 0. 3. Third, insert a ppropriate powers of g 1/2 as needed to achieve appropriate tensor density weights so that all terms in [...] have the same weight and this we ight equals the weight of Q---- . Tensor densities are discussed in Appendix D, and this weight adjustment idea appears in Appendix D (b) item 3 Example 2. Section 15: Covariant Method 173 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) ). 4. At this point one has Q---- = { tensorized form of the Cartesian expression of interest } 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 (u) the above equation can be converted to x'-space by simply priming all objects within { }. One then has Q' ---- = { tensorized form of the Cartesian expression of interest }' 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 coordinates. (For a vector, recall 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 outlined in Appendix (a) and (b) ). This then is "the covariant method". The method of c ourse applies to tensor analysis applications other than "curvilinear coordinates". Uniqueness. One might obtain two tensorizations of a Cartesian 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 One can then transform both objects to x'-space in the usual manner, {...}' 1a = Ra b{...}1b and {...}' 2a = Ra b{...}2b Therefore, since {...} 1a = {...}2a in x-space, one must have {...}' 1a = {...}'2a in x'-space. Example: In Section (g) 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 . 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 Section 15: Covariant Method 174 (@B)n = {.....} 2n = Bn;j ;j . 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 Verifying this equality is a non-trivial exercise. Both forms are used in Appendix I to compute @B in spherical and cylindrical coordinates. In the following sections, then, we shall in short or der 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 and ∇T in Appendices G, H, K. (c) divergence (Section 9) In Cartesian coordinates, divB = ∂iBi = Bi ,i. The true tensorial tensor which matches this is divB ≡ Bi ;i since Γ = 0 for Cartesian coordinates. Since Bi ;i is a scalar, it is the same in x-space as in x'-space. According to the first J = 1 example of Appendix F (i), Ba ;α = ∂α Ba + Γa αn Bn, so [divB] = [divB]' = B' i ;i = ∂'i B'i + Γ 'i in B'n where B' n are the contravariant components of vector B in x'-space. As shown in Appendix F (d), Γ 'i in = (1/ g' ) ∂ 'n(g' ) and therefore [divB] = ∂'n B'n + (1/ g' ) ∂ 'n(g' )B'n = (1/ g' ) ∂'n (g' B'n). This matches the result obtained in Section 9 by geometric methods. (d) gradient and gradient dot vector (Section 10) If f is a scalar, then G = grad f transforms as an ordinary vector, so it is already a tensor. According to Section 7 (s) or Appendix E (b), the expansion of such a vector may be written G = grad f = G'iei = (∂'if ') ei = (∂'if ') ei Also, grad f • B is a scalar, again already a tensor. Thus [grad f]' • B' = grad f • B = (∂'if')B'i Both these results match those of Section 10. Since f is a scalar, f '(x') = f( x). (e) Laplacian (Section 11) In Cartesian x-space the Laplacian of a scalar function f can be written as Section 15: Covariant Method 175 [lap f] Cart = ∂i∂if = f,i ,i The tensorized version of the Laplacian is th en taken to be the following, using the ,→ ; rule, lap f = f;i ;i Since f ;i ;i is the contraction of a rank-2 te nsor, it is a scalar, and therefore lap f = f ;i ;i = f ' ;i ;i = ∂'i f ' ;i + Γ 'i in f ' ;n where the example B ;a ;α = ∂α B;a + Γa αn B;n from Appendix F (i) is used. Another example shows that for scalar B, B;α = ∂α B . Applying this to f ' ;i and f ' ;n , the above becomes lap f = ∂' i∂'if ' + Γ 'i in ∂'nf ' Again, as shown in Appendix F (d), Γ 'i in = (1/ g' ) ∂ 'n(g' ) and therefore lap f = ∂' n∂'nf ' + (1/ g' ) ∂ 'n(g' ) (∂'nf ') = [1/ g' ] ∂ 'n [g' (∂'nf ')] which matches the result of Section 11. (f) curl (Section 12) As shown in Appendix D (h), C ≡ curl B is a vector density of weight -1 because it contains the ε tensor which has this weight. According to Appendix E (b), th e expansion of such a vector density is given by C = J-1C'iei . Therefore, C = curl B = J-1C'iei = J-1 ε'ijkB'k;jei = (1/ g' ) ε'ijkB'k,jei so [ c u r l B(e)]i = (1/ g' ) ε'ijkB'k,j This matches the result of Section 12. (g) vector Laplacian (Section 13) This differential operator is going to take a bit more work. In Section 13 the vector Laplacian is written @ (instead of ∇2) and in Cartesian coordinates is defined by @B ≡ grad(div B) – curl (curl B) Recall that in Cartesian space (as shown in Section 13 (c)) [grad(div B) – curl (curl B)]n = ∇2(Bn) = (@B)n // Cartesian x-space Section 15: Covariant Method 176 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) In Cartesian space up and down index position does not ma tter and we are just jockeying the indices in search of a tensorized form with contracted indi ces where possible. Writing the above in comma notation gives ( @B)n = (Bj ,j),n – εnab(εbdeBe,d),a and then we are free to replace commas with semicolons since Γ = 0 in Cartesian coordinates (see Appendix F sections (d) and (i) ) : (@B)n = (Bj ;j);n – εnab(εbdeBe;d);a There are several "technical difficulties" visible here. The first term (B j ;j);n is a true vector since it is the contraction of a rank-3 tensor on two tilted indices . In the second term, in order to neutralize the weight of each ε tensor density (each has weight -1), we shall now add a benign factor of g-1/2, using the idea of Appendix D (b) item 3 Example 2 (that is, g-1/2 is a scalar density of weight +1). (The factor is benign since g = 1 in Cartesian coordinates. ) ( @B)n = (Bj ;j);n – g-1/2εnab(g-1/2εbdeBe;d);a Now both objects T b ≡ ( g-1/2εbdeBe;d) and Vn ≡ g-1/2εnab Tb;a are normal vectors, so the above equation says that a normal vector is the di fference between two other normal vectors. Anticipating a few steps ahead when our ship lands in x'-space, we know that ε bdeBe;d = εbdeBe,d due to the e↔ d symmetry of the Γ term in B e;d (see Appendix F (i) Examples with J=1), 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 . Recall that tilt changes like this are only allowed with tr ue tensor indices ( Section 7 (k) ) and that is why the e tilt reversal can only be done after the semicolon in B e;d is installed. The result is then (@B)n = (Bj ;j);n – g-1/2εnab(g-1/2gbcεcdeBe;d);a . Although ε bde is a fine tensor, we really prefer εcde since this is the permutation te nsor, and that is why the gbc factor is added. Section 15: Covariant Method 177 At this point every index is a tensor index, so we ha ve a "true tensor equation" in the sense of Section 7 (u). 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. The point is that we have found a way to write the Cartesian expression of ( @B)n such that both terms are true tensors. Th e equation is therefore now "covariant" and the equation in x'-space can be obtai ned simply by priming all objects: (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d);a 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 since they imply "extra terms". 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) Since Tb;a = Tb,a – Γn baTn (Appendix F (i)) , the above expression is equal to εnab Tb,a because εnab is antisymmetric on a,b while Γn ba is symmetric. Thus, (@B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e;d),a ↑ Now we activate the fact that ε'cdeB'e;d = ε'cdeB'e,d for the exact same symmetry reason to get ( @B)'n = (B'j ;j);n – g'-1/2ε'nab(g'-1/2g'bcε'cdeB'e,d),a ↑ Meanwhile, an Appendix F (i) example that says D;n = D,n s o t h a t ( Bj ;j);n = (B'j ;j),n , since the object 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 . ↑ Since ε'cde is a constant (the permutation tensor, see Appendix D (d) ), 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) . The object (B' j ;j) we know from section (c) above [divB] = B'j ;j = (1/ g' ) ∂ 'j (g' B'j) so that ( @B)'n = ∂'n{(1/ g' ) ∂ 'j(g' B'j)} – g'-1/2ε'nabε'cde ∂'a(g'-1/2g'bc∂'dB'e) In order to compare this result to that of Section 13 , it is necessary to process the indices in the second term as follows, Section 15: Covariant Method 178 g'-1/2ε'nabε'cde ∂'a(g'-1/2g'bc∂'dB'e) g'-1/2ε'nABε'CDE ∂'A(g'-1/2g'BC∂'DB'E) Then take A →c, B→d, C→e, D→a, E→b to get g'-1/2ε'ncdε'eab ∂'c(g'-1/2g'de∂'aB'b) . Then also changing j →i in the first term we have, ( @B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i)} – g'-1/2ε'ncdε'eab ∂'c(g'-1/2g'de∂'aB'b) . This may now be compared with the Section 13 (a) result 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 The results agree, and further processing of this result is given in Section 13. Appendix A: Reciprocal Base Vectors 179 Appendix A: Reciprocal Base Vectors the Hard Way Note : This Appendix is written in the development not ation, not the Standard Notation, though a few equations are translated to the latter form. The rules for translation to Standard Notation are En → en Rij → Ri j Sij → Si j g¯'nm → g'nm g'nm → g'nm . Introduction In Section 6 of the main text the r eciprocal base vectors are defined as En ≡ g'ni ei , and the results given in that section, (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 , are all applicable in the Picture A context with arbitrary metric tensors g' and g, 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 Section 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) Definition of E n The reciprocal base vectors are defined in the following very strange looking and clumsy manner, Appendix A: Reciprocal Base Vectors 180 (Ek)α ≡ det(R) (-1)k-1 εαi1i2i3...ik...iN (e1)i1 (e2)i2 ...... ( ek)ik.......... ( eN)iN 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. (b) 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 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 // (e k)κ is missing, α in κ position (the kth) 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 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 (c) 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 where x and X represent the N th letter of the alphabet. The ε object is again the totally antisymmetric tensor with N indices. In vector notation one writes this symbolically as Appendix A: Reciprocal Base Vectors 181 Q = B x C x D x ... x X / N-1 factors, N-2 crosses 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 } 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 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 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 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 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: Q a ≡ εabc...x BbCcDd.....Xx Q = B x C x D x ... x X / N-1 factors, N-2 crosses Appendix A: Reciprocal Base Vectors 182 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. (d) Missing Man Formation We now make a small variation in the no tation. Start with the above equation, Qa ≡ εabc...x BbCcDd.....Xx , 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 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 and of course it is still true that Q•C = 0, etc. For N=2 the vector no tation is rescued as in (c) above. (e) Apply this Notation to E Compare the above Q α to the section (a) 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 Therefore, the definition of Eκ 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 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 which agrees with the results quoted above. For N =2 ( E's label corresponds to the missing e's label ), Appendix A: Reciprocal Base Vectors 183 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 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 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 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 It is shown next that these N=2 results are special cases of a general fact: Em • en = δm,n . Section 5 (j) showed that em • en = g¯'mn . The other two dot products are now considered. (f) 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 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) = 1 // since RS = 1 We already know that Ek is orthogonal to all the en which form the generalized cross product, therefore Em • en = δm,n which is the "duality relation" discusse d more generally in Section 6 (b). Appendix A: Reciprocal Base Vectors 184 (g) 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 δm,k = Em • ek = Σn An(m) en • ek = Σn An(m) g¯'nk Multiplying both sides 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) Therefore A i(m) = g'mi so, Em = Σn An(m) en = Σn g'mn en which is to say Ek is this linear combination of the ei (this is the definition used in Section 6 (a)) Ek = Σi g'ki ei = g'ki ei // implied sum on i // Std Notation: ek = Σi g'ki ei which may be compared with the previous result Ek ≡ det(R) (-1)k-1 e1 x e2 x ......x eN // ek missing; 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 // recall g' symmetric (h) Summary of relationship between the 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 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 Section 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. 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 Appendix A: Reciprocal Base Vectors 185 (i) 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 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 It is convenient to write this as Q = Πx i=1N-1 A(i) = Πx i A(i) 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) as discussed in section (c) above. This same Πx notation can be applied to the "missing man formation" of (d) above. Suppose Q = A(1) x A(2) x A(3) x ... x A(N) // A(n) is missing One can write this as Q = Πx i=1..N,i ≠n A(i) ≡ Πx i≠n A(i) 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; Ek = det(R) (-1)k-1 Πx i≠k ei 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^ Ek = det(R) (-1)k-1 e3-k x 3^ which is the "trick" notation of section (c) above for the N=2 case. Appendix B: N-piped Geometry 186 Appendix B: The Geometry of Parallelepipeds in N dimensions Introduction 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 Section 3, the normal vectors for the pairs of parallel faces are just the reciprocal base vectors En of Section 6. Section (d) 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. All major results of this Appendix are concisely stat ed in Summary section (e). Since this is a very long section (~15 p), a reader not interested in deta ils would do well to simply read that summary and skip the rest of this Appendix. (a) 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, 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 Therefore, one way to write the equation of a plane in N dimensions is Appendix B: N-piped Geometry 187 r•p^ = p r = (x1, x2, .....xN) 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) N-pipeds and their Faces in Various Dimensions 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. N=1 rvolume1 = α1 e1 0 ≤ α1 ≤ 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 segment). 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 volume of this 1-piped is e 1. The 2-piped Now add another dimension, going to N=2. Introduce a unit vector e2 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 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: N-piped Geometry 188 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 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. It is possible to construct vectors E1 and E2 as linear combinations of e1 and e2 such that the following is true (see Section 6 (b)) Ei• ej = δi,j / / Ek = Σi=12 g'ki ei , see Section 6 (a) 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 Section 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 The first line says (section (a) above) that face 2 lies on a plane which passes through the origin and which has normal vector E^2. 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 1|. Similarly, E1 • rface1 = E1 • α2e2 = 0 => E^1 • rface1 = 0 E1 • rface1p = E1 • [ α2e2+ e1] = 1 => E^1 • rface1p = 1/E1 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| . Appendix B: N-piped Geometry 189 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. 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. 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 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. A partner is just a copy of the non-partner which is translated by a constant vector. The above results can be summarized in this concise manner: Appendix B: N-piped Geometry 190 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 It is possible to construct vectors E1, E2, E3 as linear combinations of e1, e2, e3 such that the following is true (see Section 6 (b)) Ei• ej = δij / / Ek = Σi g'ki ei , see see Section 6 (a) 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 Section 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 The first line says that face 1 lies on a plane which passes through the origin and which has normal vector E^1. The second 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. 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". 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 Appendix B: N-piped Geometry 191 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. It is possible to construct vectors E1...EN as linear combinations of e1.. eN such that the following is true (see Section 6 (b)) Ei• ej = δij / / Ek = Σi g'ki ei 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 Section 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 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| (c) The question of inward versus ou tward 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 Construct vectors from piped center to face centers ( results here are fairly obvious) (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 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 Appendix B: N-piped Geometry 192 One may conclude that Ei is an outward pointing normal for face ip (far face). Therefore, - Ei is an outward pointing normal for face i, which recall is the face which touches the origin (near face). (d) 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. The 2-piped 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| 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 Appendix A (g) A2 = |e1| E^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 An = |det(S)| En . 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| or | Ek| = |det(R)| | e3-k| k=1,2 // RS = 1 Using the N=2 trick notation from Appendix A (c), Ek = det(R) (-1)k-1 e3-k x 3^ so that | Ek | = | det(R)| | e3-k x 3^| = |det(R)| | e3-k| k = 1,2 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 Appendix A (i): Appendix B: N-piped Geometry 193 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)) Volume . The volume of a 2-piped is the base times the he ight of a parallelogram, familiarly given as by the cross product of the edges, volume(2) = | e1 x e2 | = | εab (e1)a(e2)b | = | det [ e1, e2 ] | = | det(S) | where S is the linearized transformation matrix for N= 2, see Section 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|. The 3-piped Face Area: The faces of a 3-piped are 2-pipeds. For N=2, the 2-piped volume was volume(2) = | εab (e1)a(e2)b | , where e1 and e2 were 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. The first 2 components of each of these 3D vectors ar e 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 being 0. The volume(2) formula expressed in terms of these new 3D vectors is therefore | εab3 (e1)a(e2)b|, where ε is now a 3D ε tensor. The conclusion is that A3 = |εab3 (e1)a(e2)b| 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 A1 = |ε1ab (e2)a(e3)b| A2 = |εa2b (e3)a(e1)b| 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 In Appendix A (e) it was shown that E1 = det(R) e2 x e3 so that e2 x e3 lines up with E1. Regardless of the sign of det(R), we define the vector areas to point in the + E^n directions. Thus, Appendix B: N-piped Geometry 194 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 These equations can be combined into the following single formula An = |e1 x ... x e3| E^n // en missing where the ei are reordered for free due to the absolute value signs. But Appendix A says En = det(R) (-1)n-1 e1 x ... x e3 // en missing so |En| = | det(R) | | e1 x ... x e3 | // en missing Thus, 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)] 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 where the last line uses the shorthand notation of A ppendix A (i). These expressions have the same form as those of the 2-piped. 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 | Here is a drawing showing the last case ( σ = +1), where "base" is A 3 = | e1 x e2 | and "height" is e 3 cosθ , Appendix B: N-piped Geometry 195 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) so that volume(3) = | e3 • e1 x e2 | = | εabc (e1)a(e2)b(e3)c | = | det [ e1, e2, e3] | = | det(S) | These expressions have the same fo rm as those of the 2-piped. The 4-piped Face Area: The faces of a 4-piped are 3-pipeds. For N=3, the 3-piped volume was volume(3) = | ε abc (e1)a(e2)b(e3)c | where e1,e2,e3 were 3D vectors. For the 3-piped which is "f ace 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 A4 = | εabc4 (e1)a(e2)b(e3)c | 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 | Appendix B: N-piped Geometry 196 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 A1 = |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 where, as discussed in Appendix A (c), Q = A x B x C is defined by Q k = εkabc AaBbCc . In Appendix A (e) 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 where the ei are reordered for free due to the absolute value signs. But Appendix A says En = det(R) (-1)n-1 e1 x ... x e4 // en missing so | En| = | det(R) | | e1 x ... x e4 | // en missing 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)] 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)] These expressions have the same form as those of the 2-piped and the 3-piped. 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 | 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 Appendix B: N-piped Geometry 197 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) so that volume(4) = | det(S) | = | det [ e1, e2, e3, e4] | = | εabcd(e1)a(e2)b(e3)c(e4)d | These expressions have the same form as those of the 2-piped and the 3-piped. The N-piped Face Area: The faces of a N-piped are (N-1)-pipeds. If there 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 | where e1,e2,...eN-1 were (N-1)D vectors. 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 all 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 new ND vectors is therefore | ε abc...xN (e1)a(e2)b... (eN-1)x |, where ε is now an ND ε tensor. The conclusion is that AN = | εabc...xN (e1)a(e2)b... (eN-1)x | 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 | 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 where, as discussed in Appendix A (c), Q = A x B x C.... X is defined by Q k = εkabc...x AaBbCc .....Xx Appendix B: N-piped Geometry 198 In Appendix A (e) 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 where the ei are reordered for free due to the absolute value signs. But Appendix A says En = det(R) (-1)n-1 e1 x ... x eN // en missing so | En| = | det(R) | | e1 x ... x eN | // en missing 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)] 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)] These expressions have 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 | = ... 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) where x is the N th letter of the alphabet, so that volume(N) = | det(S) | = | det [ e1, e2, e3, ....eN] | = | εabc...x (e1)a(e2)b(e3)c....(eN)x | These expressions have 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 above. Appendix B: N-piped Geometry 199 (e) 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) 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 demonstrated by the induc tive construction method presented above. 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 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 "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 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 this case, the equations 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 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). 7. The out-facing vector area of face ip of an N-piped can be expressed as Appendix B: N-piped Geometry 200 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 missing 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 following, as discussed in Appendix A (i), ( Ai )α = σ (-1)i-1 εαabc..x (e1)a (e2)b ... (eN)x // where (e i)ί and ί are missing For N=2 the last two expressions for Ai are interpreted as shown in Appendix A (c) Ai = σ (-1)i-1 e3-i x 3^ i = 1,2 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 | where one can regard the tangent base vectors as the columns of the linearized tr ansformation matrix S. Appendix C: Elliptical Polar Coordinates 201 Appendix C: Elliptical Polar Coordinates ( N=2, non-orthogonal) This Appendix is written in the develo pmental notation of Sections 1-6. (a) 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 Writing the first equation above as 1 = x 2/(ρa)2 + y2/(ρb)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 => x 2/a2 + y2/b2 = ρ2 y = bρsinθ y/b = ρsinθ => tan θ = y/x The matrix S is given by S 11 = (∂ x/∂θ) = -aρsinθ S12 = (∂ x/∂ρ) = acosθ S ik ≡ ( ∂xi/∂x'k) S21 = (∂ y/∂θ) = bρcosθ S22 = (∂ y/∂ρ) = bsinθ S = ⎝⎛ ⎠⎞-aρsinθ acosθ bρcosθ bsinθ => det(S) = -ab ρ 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 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 The covariant metric tensor is, Appendix C: Elliptical Polar Coordinates 202 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 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. (b) Forward coordinate lines Here is a Maple plot of some x-space forward coordinate lines (parameters a = 2 and b = 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 which are obtained from the forward transformation equations ρ2 = x2/a2 + y2/b2 tanθ = y/x . (c) Inverse coordinate lines Here is a Maple plot of some x'-space inverse coordinate lines (parameters a = 2 and b = 1) Appendix C: Elliptical Polar Coordinates 203 The coordinate lines in x'-space are plotted using these equations ρ = xi/(acosθ) x i = -10 to +10 21 blue curves( one is a boxy U ) ρ = yi/(bsinθ) y i = -10 to +10 21 red curves ( one is a boxy U) which are obtained from the inverse transformation equations x = aρ cosθ y = bρ sinθ 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. (d) 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 (Section 6 (f)) 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) 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 Section 2 and en are the tangent base vectors of Section 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: Elliptical Polar Coordinates 204 Here is a graphical representation of this vector V in x-space: 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. (e) Drawing a contravariant vector V' in x '-space: two "Views" As with previous examples, the above picture is draw n to the right of an x'-space picture as follows: In Section 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 and en was shown to be a contravariant vector, Appendix C: Elliptical Polar Coordinates 205 e'n = R( x) en. Applying matrix R(x) to the equation V = V'θ eθ + V'ρ eρ one gets the expansion noted above, V' = V'θ e'θ + V'ρ e'ρ which appears first in the list of expansions of V' in Section 6 (f). 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. In the Cartesian 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 The left-side graph shown above 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 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 scalars, as discussed at the end of Section 5 (a). For example, when g ¯' is set to 1, one has | V'| ≠ |V| in the above picture. Another use of this view involves in tegration as will be seen below. In the Curvilinear 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 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 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 Appendix C: Elliptical Polar Coordinates 206 |V'|2 = |V|2 = Vx2 + Vy2 ≠ (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 In writing | e'n| = |en| and | V'|2 = |V|2 above, we use the rule shown in Section 5 (i) which says | A'|2 = |A|2 for any contravariant vector A (|A|2 is a scalar ). Moreover, 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 <= !! so that 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 of the x'-space graph to cause all inte rsection points to have angles which match the metric tensor, which is to say, at each intersection 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 repres entations 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 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. Appendix C: Elliptical Polar Coordinates 207 (f) 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 For V = dx the picture above becomes 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) and once again the "right angle" in the x'-space picture is deceptive. (g) 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: Elliptical Polar Coordinates 208 x'-space x-space Now select the red d x bar on the right and operationally apply the previous picture. 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 lengths are all the same in x-space. The Curvilinear-view lengths of the x' -space bars are all the same, and ar e 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: 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: Elliptical Polar Coordinates 209 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) If a=b, then h θ = ρ and the bar length on the right is then ρdθ as is obvious in polar coordinates. (h) 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 Section 8 ] dA' ≡ dρdθ = the area of a differential patch in Cartesian-view x'-space 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 Section 8 (e ), the ratio of the two patch areas 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 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 (a) which describes an ort hogonal 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 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: Elliptical Polar Coordinates 210 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) or ∫[|J(x')| dθdρ] f(x(x')) = ∫[dxdy] f( x) 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 = dxdy where the meaning of this symbolic equality is the integral statement above, ∫D dxdy f( x) = ∫D' dθdρ |J(x')| f(x(x')) , 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 ≡ dxdy so the above equality of integrals can be written ∫dA f( x) = ∫ dA' |J( x')| f(x(x')) |J( x')| dA' = dA In N dimensions, dA and dA' are differential "volumes ", and the general Jacobian Integration rule takes the form, Appendix C: Elliptical Polar Coordinates 211 ∫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. 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 and therefore ∫dxdy f(x,y) = ∫ dθdρ |J(x')| f(aρ cosθ, bρ sinθ) = ab ∫dθdρ ρ f(aρ cosθ, bρ sinθ) 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 ) which is the familiar result. Appendix D: Tensor Densities 212 Appendix D: Tensor Densities and the ε tensor Picture A is used in this Appendix along with Standard Notation. (a) Definition of a tensor density First, recall from the Section 5 (k) discussion of the Jacobian J, J ≡ det(Si j) = σ sg' / sg = σ(sg'/sg)1/2 = σ(g'/g)1/2 => (g'/g)1/2 = σJ = |J| > 0 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 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. Second, recall our generic sample tensor transformation from Section 7 (j), T ' abc de = Ra a' Rb b' Rc c' Sd' d Se' e Ta'b'c' d'e' which can be rewritten in a more standard wa y using the theorem of Section 7 (q) that Sμ ν = Rνμ, T ' abc de = Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' 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' 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 that 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, |∂x'/∂x| W ≡ [det(∂x'/∂x)]+W = [ det(∂x'i/∂xk) ]+W = J-W Appendix D: Tensor Densities 213 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 . This is the scalar density mentioned in Section 5 (k ) 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 (b) 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. 2. Contracting indices within a tensor density does not alter its weight W. If indices a and d are contracted in the 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. Example 1: 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: Suppose in the outer product the first factor is the scalar density g-W1/2 of weight W1 : g' -W1/2 = J-W1 g-W1/2 // since g' = J2g from Section 5 (k), no R factors since scalar B' c d = J-W2 Rc c'Rdd' Bc' d' // same as in previous example => g' -W1/2B'c d = J-(W1+W2) Ra a' Rc c'Rdd' (g-W1/2 Bc' d') Appendix D: Tensor Densities 214 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' (gW/2 Bc' d') 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 ). 4. Although sometimes authors take a differing stance fo r certain tensors, 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. This means the gab raises an index and gab lowers an index. 5. Raising or lowering an index does not change the weight of a tensor density . Again, using our generic example above, T 'abc de = J-W Ra a' Rb b' Rc c' Rdd' Ree' Ta'b'c' d'e' T 'abc de = g'ex T 'abc dx // raise last index in x'-space T a'b'c' d'e' = ge'e" Ta'b'c' d'e" // lower last index in x-space Therefore T 'abc de = g'ex [ J-W Ra a' Rb b' Rc c' Rdd' Rxe' Ta'b'c' d'e'] = g ' ex [ J-W Ra a' Rb b' Rc c' Rdd' Rxe' (ge'e" Ta'b'c' d'e")] = JW Ra a' Rb b' Rc c' Rdd' (g'ex Rxe' ge'e") Ta'b'c' d'e" = J-W Ra a' Rb b' Rc c' Rdd' (Re e") Ta'b'c' d'e" // Section 7 (o) facts about R 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. 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 . Proof : First form the rank-2 tensor density AiBj which by item 3 has weight W+w. Lower the second index and the mixed rank-2 tensor AiBj by item 5 still has weight W+w. Then contract to get A•B = AiBi and by item 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| . Appendix D: Tensor Densities 215 Proof : |A|2 = A • A which has weight 2W meaning | A'|2 = J-2W |A|2. Therefore | A'| = J-W |A| . 7. As J→1, tensor densities become true tensors. 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 (RRT= 1 and detR = 1) 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. Perhaps we should restrict this comment to underlying transformations F having detS > 0 since passing through detS = 0 is problematical. An example: the cross product considered in sec tion (g) below of N-1 contravariant vectors becomes in this limit an ordinary covariant v ector. If g=1 in x-space, then g' = RRT = 1 in x'-space 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. (c) Theorem about Totally Antisymmetric Tensors: there is really only one: εabc... Theorem : Apart from a scalar factor, there exists only one totally antisymmetric (TA) tensor. 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 . 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.. εabc.. = p e123... Define scalar function f ≡ r 123... / e123... , whatever it might be. Then r abc.. = p(f e123...) εabc.. = p e123... and dividing these two equations one finds, r abc.. = f εabc.. 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. Appendix D: Tensor Densities 216 (d) The contravariant ε tensor Knowing nothing to start, assume that the famous εabc.. totally antisymmetric 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'.. (*) 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 (*) is totally antisymmetric (TA). The argument is given once here and then used later seve ral 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'.. = J -W Ra a' Rb b' ... (–εa'b'c'..) = – ε'abc.. The same result is true for any swap, thus RHS (*) = TA. Since according to section (b) there is only one TA tensor available, apart from a scalar function factor, it follows that ε' abc.. = Kεabc... where K is some scalar function, perhaps just a constant. Equation (*) above then reads K εabc... = J-W Ra a' Rb b' ... εa'b'c'.. (**) Setting in the standard order, one finds that K ε 123... = J-W R1 a' R2 b' ... εa'b'c'.. or K = J -W det(Ri j) = J-W (J)-1 = J-(W+1) so now ε'abc.. = Kεabc... = J-(W+1) εabc... A second assumption is now made: that ε abc.. (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... This assumption is consistent with taki ng W = -1 in the previous equation. 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. Appendix D: Tensor Densities 217 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 (*) above with W = -1 This then is our second example of a tensor density. 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. Sometimes the a ffine connection is called the Levi-Civita connection. (e) Some facts about the ε tensor 1. Consider ( based on section (b) 4 above), ε abc... = gaa' gbb'..... εa'b'c'... This is again in the convention of Weinberg p 99 (4.4.10). 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)] Inserting such a sign renders any ε mixed tensor like ε a bc.. ambiguous when s = - 1, but is acceptable if one promises never to make use of a mixed ε tensor. We shall refer to these two methods as "the Weinberg convention" and the "a dded sign s convention", the former being assumed unless otherwise stated. Install the reference sequence to obtain ε 123.. = g1a' g2b'..... εa'b'c'... = det(gij) = g // Section 5 (k) 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' 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 218 2. Take the same starting point as above ε abc... = gaa' gbb'..... εa'b'c'... The RHS is a totally antisymmetric in indices abc... (see above) and can therefore be written RHS = C εabc... since we showed earlier that there is only one TA tensor apart from scalar C. Therefore ε abc... = C εabc... (*) Insert the reference sequence ε123... = C ε123... = C But in 1 it was just showed that ε 123... = det(g ij) . Therefore C = det(g ij) and then (*) 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... 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... 3. Divide the last two Weinberg convention equations to find that ε'abc... = [det(g' ij)/ det(gij)] εabc... = (g'/g) εabc... From Section 5 (k) 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 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 Section 5 (k). Appendix D: Tensor Densities 219 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. • 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 the section (a) de finition of a tensor density transformation. ) (f) The covariant ε tensor : repeat section (d) as if its weight were not known According to section (b) 5, lowering indices does not change the weight of a tensor density. Section (d) showed that εabc.. 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 interesting to see what happe ns when the same method used in section (d) 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'.. ] (*) Section (e) 2 noted that ε a'b'c'... is a totally antisymmetric tensor, and therefore as in section (d) one concludes that the RHS of (*) is also totally antisymmetric and can be written as RHS(*) = K εabc.. , so (*) then says K ε abc.. = J-W [ Raa' Rbb'.... εa'b'c'.. ] (**) Use section (e) 2 to set εa'b'c'.. = det(gij) εa'b'c'.. inside the bracket, K ε abc.. = J-W [ Raa' Rbb'.... det(g ij) εa'b'c'..] , and then install the reference sequence on both sides K ε 123.. = J-W [ R1a' R2b'.... det(g ij) εa'b'c'..] But section (e) 1 says ε 123.. = det(g ij), so cancel det(g ij) on both sides to get K = J -W [ R1a' R2b'.... εa'b'c'..] = J-W det(Rij) = J-W det(Sj i) = J-W J = J-(W-1) So here the result is K = J -(W-1) whereas in section (d) the result was K = J-(W+1) . In the current case, since (*) and (**) have the same RHS, setting the LHS's equal says ε'abc.. = K εabc.. = J-(W-1) εabc.. But section (e) 3 said that ε' abc.. = J2 εabc.. and therefore one gets W = -1. Appendix D: Tensor Densities 220 Thr conclusion is that εabc... transforms with weight -1, the same as εabc..., so (*) becomes ε' abc.. = J [ Raa' Rbb'.... εa'b'c'.. ] (g) Generalized cross products In Appendix A (c) the following cross product of N-1 v ectors is considered (converted now to standard notation) Q a ≡ εabc...x BbCcDd.....Xx or Q = B x C x D .... x X If the vectors B,C,D..X are all contravariant vectors, th en applying the rule of s ection (b) 3, one concludes that, since ε is a tensor density of weight -1 and since a ll 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 Similarly, one may consider Q a ≡ εabc...x BbCcDd.....Xx . 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 (h) The tensorial nature of curl B It has just been shown that C = A x B is a vector density of weight -1, this being a special case of the generalized cross product discussion above. As noted in section (a) 7, if R happens to be a (global) rotation, then C is in fact a tensorial vector. 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 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 where the second term is the same as the first term, since – ε nab ∂bBa = εnba ∂bBa = εnab ∂aBb . Recall now from Section 7 (v) that the cova riant derivative of a vector is given by Appendix D: Tensor Densities 221 Bb;a = ∂aBb – Γc ab Bc where the affine connection Γc ab is symmetric under a ↔b. Therefore B b;a – Ba;b = [∂aBb – Γc ab Bc] - [∂bBa – Γc ba Bc] = ∂aBb – ∂bBa Therefore C n can be expressed as C n = εnab [Bb;a – Ba;b ]/2 so by the same ε anti-symmetry noted above the final result is Cn = εnab Bb;a . The major feature of B b;a -- as discussed in Section 7 (v) -- is that it is a rank-2 tensor if B is a tensorial vector. The weight addition rule of section (b) 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 = curl B is a vector density of weight -1. If the underlying transformation F is a rotation, C becomes an ordinary vector as per section (a) 7. (i) Tensor E as a weight 0 version of ε : three conventions 1. Equations in the "Weinberg Convention" In this section it is assumed that g ij and gij raise and lower indices of the ε tensor just as they do for any other tensor (Weinberg convention). In sec tions (d) and (e) it was established that ε abc... = g εabc... ε123.. = +1 ε123... = g ε'abc... = g'ε'abc... ε'123.. = +1 ε'123... = g' ε'abc... = J2 εabc... = (g'/g) εabc... ε is a rank-N tensor of weight W = -1 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 Section 5 (k)) 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 It was shown in Section 5 (k) that g' = J2g 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 of section (b) 3, one Appendix D: Tensor Densities 222 concludes at that Eabc... transforms as a tensor of weight (+1) + (-1) = 0, and so Eabc... is an ordinary tensor. This is the motivation of the above definitions . It was shown at the end of Section 7 (u) that a tensor density equation with matchi ng weights is "covariant", so one is not surprised to see the second line above being the same as the first line but everything is primed (s = s'). Raising indices on both sides gives Eabc... ≡ |g|-1/2 εabc... => E123... = |g|-1/2 E'abc... ≡ |g'|-1/2 ε'abc... => E'123... = |g'|-1/2 To compare Eabc... and Eabc... , Eabc... = |g|-1/2 εabc... = |g|-1/2 g εabc... = s |g|-1/2 |g| εabc... = s |g|1/2 εabc... Eabc... = |g|-1/2 εabc... so that, Eabc... = s|g| Eabc... = g Eabc... Summarizing, 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... 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 (e). Here is the previous secti on redone in the "added sign s convention": In section (e) it was established that ( g = det(g ij)) ε abc... = sgεabc... ε123.. = +1 ε123... = sg = |g| // g → sg ε'abc... = sg'ε'abc... ε'123.. = +1 ε'123... = sg' = |g'| // g' → sg' 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 Section 5 (k) ) Appendix D: Tensor Densities 223 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 But 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 de finitions 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 To compare Eabc... and Eabc... , Eabc... = |g|-1/2 εabc... = |g|-1/2 |g| εabc... = |g|-1/2 |g| εabc... Eabc... = |g|-1/2 εabc... so that E abc... = |g| Eabc... 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... 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 (k) 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 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... Appendix D: Tensor Densities 224 Eabc... EABC... = |g|-1 εabc... εABC... E'abc... E'ABC... = |g'|-1 ε'abc... ε'ABC... Notice that in all three conventions, 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. (j) Representation of ε , εε and contracted εε as determinants 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) 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 permutati on 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 2 ↔3 on the right above, each pe rmutation 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 can be written as K εabc...x : ΣP p P2(Ma1Mb2 Mc3.....MxN) = K εabc...x Setting abc..x to 123..N, one gets. ΣP p P2(M11M22 M33.....MNN) = K 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 because p = ε abc...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) ε abc...x Consider now the following matrix where ab c..x is some permutation of 123...x, Appendix D: Tensor Densities 225 where M(abc..) = Ma1 Ma2 Ma3 ... M aN M b1 Mb2 Mb3 ... M bN M c1 Mc2 Mc3 ... M cN ... M x1 Mx2 Mx3 ... MxN 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) and therefore Σ P p P2(Ma1Mb2 Mc3.....MxN) = det(M(abc..) ) = det(M) εabc...x 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, correspon ds to the product of the diagonals of that matrix. 2. Application of the theorem to M = δ: a representation of ε Apply the above theorem to matrix M = 1 ≡ δ, the identity matrix, having M ij = δi,j. Clearly det( δ) = 1 and one then has Σ P p P2(δa,1δb,2 δc,3.....δx,N) = det[δ(abc..) ] = εabc...x Thus is obtained the famous representation of εabc..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 where, for future use, each row v ector has been given a name like Ra where ( Ra)i = δa,i . The conclusion then is that Appendix D: Tensor Densities 226 which is the same as εabc...x = ΣP p P2(δa,1δb,2 δc,3.....δx,N) . 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' 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') } 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' A typical element of this matrix is given by Rc• Rb' = (Rc)i(Rb')i = δc,i δb',i = δc,b' so that matrix can be written as δ a,a' δa,b' δa,c' .... δa,x' Appendix D: Tensor Densities 227 δb,a' δb,b' δb,c' .... δb,x' δc,a' δc,b' δc,c' .... δc,x' ≡ δ(abc..x; a'b'c'..x') .... δx,a' δx,b' δx,c' .... δx,x' where we have made up a name for this matrix as shown. The conclusion then is that which is the same as ε abc...x εa'b'c'...x' = ΣP p P2(δa,a'δb,b'δc,c'.....δx,x') As usual, the argument of P 2 is the product of the diagonal elements of the matrix. 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: Σa εabc...x εab'c'...x' 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. Then Σ a εabc...x εab'c'...x' = εAbc...x εAb'c'...x' = ΣP p P2(δA,Aδb,b'δc,c'.....δx,x') = ΣP p P2(δb,b'δc,c'.....δx,x') (*) 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' Appendix D: Tensor Densities 228 The permutation sum shown on the right side of (*) 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, Σa εabc...x εab'c'...x' = [cof δ(abc..x; a'b'c'..x')]aa 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) 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' 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 hit va lue, 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 ! Σ P p P2(δA,AδB,B'δc,c'δd,d'.....δx,x') = 2!ΣP p P2(δc,c' δd,d'.....δx,x') (*) 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, Σa,b εabc εabc' = 2!ΣP p P2(δc,c') = 2 δc,c' If c = c' = 2, then this says Σ a,b εab2 εab2 = ε132 ε132 + ε312 ε312 = 1 + 1 = 2 Appendix D: Tensor Densities 229 The permutation sum shown on the right side of (*) is the determinant of δ(abc..x; a'b'c'..x') but with the first 2 rows and columns crossed out. Therefore, Σa,b εabcd...x εabc'd'...x' = 2! { [cof δ(abc..x; a'b'c'..x')]aa}bb The conclusion then is that (implied summation on a,b on the LHS) 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 The conclusion then is that (implied summation on a,b,c on the LHS) 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' an example being ε abc2 εabc2 = 3! δ22 = 3! = ε1342 ε1342 + ε1432 ε1432 + 4 more terms = 1+1+4 = 6 The final point is that at which all indices are summed, with result εabcd...x εabcd...x = N! and example of which is ε abcεabc = ε1232 + ε2132 + 4 more terms = 1 + 1 + 4 = 6 Appendix D: Tensor Densities 230 6. Summary of Results • • • • εabcd...x εabcd...x' = (N-1)! δx,x' εabcd...x εabcd...x = N! (k) Covariant forms of the previous section results The results above were all developed in Ca rtesian x-space where up and down indices on the ε's did not matter. The rules for converting any result a bove to covariant form are as follows: Weinberg convention: • write the left side as either |g|-1 ε***** ε***** or as E***** E***** .The objects with indices as shown by asterisks are true tensors (weight 0). • write the right side replacing every δa,b by δa b → ga b as shown in Section 7 (m). Then the right side will also be a true tensor. Example : The εε product for N=2 with no indices summe d was written above as (g = 1) εabεa'b' = ⎪⎪ ⎪⎪ δaa' δab' δba' δbb' = δa,a' δb,b' – δa,b' δb,a' Appendix D: Tensor Densities 231 The covariant form is as follows, where now g is some arbitrary metric tensor for x-space, EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪ga a' ga b' gb a' gb b' = ga a' gb b' – ga b' gb a' 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' because true tensor equations are "covariant"(Section 7 (u)). 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 rais ing 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' Ea bEa'b' = |g|-1 εa bεa'b' = ⎪⎪ ⎪⎪ga a' ga b' gba' gbb' = ga a' gbb' – ga b' gba' EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪gaa' gab' gba' gbb' = gaa' gbb' – gab' gba' and of course in x'-space the equations are the same but everything is primed. 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 first equation above becomes ( εa'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') The second equation is undefined (when s=-1), and the third equation is EabEa'b' = |g|-1 εabεa'b' = ⎪⎪ ⎪⎪gaa' gab' gba' gbb' = gaa' gbb' – gab' gba' 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 E: Tensor Expansions 232 Appendix E: Tensor Expansions: direct product, polyadic and operator notation This entire section uses the general Picture A context where x-space need not be Cartesian, The Standard Notation is used throughout. (a) Direct Product Notation The key tool required for the expression of tensor expansions is the notion of a direct product of n tensorial vectors defined in this simple way, ( A⊗B⊗C ...)abc... ≡ AaBbCc..... ( A⊗B⊗C ...)a bc... ≡ AaBbCc..... etc The tensor A⊗B⊗C ... is nothing more than the outer product of vectors A,B,C as discussed in Section 7 (a) for contravariant vectors, but later extended to any mixture of vector types. As noted in Section 7 (j), one can define a direct product of two rank-2 tensors in this way, (M⊗ N)ab,AB ≡ MaANbB // rank = n = 2; number of tensors = I = 2 (M⊗ N)ab ,AB ≡ Ma ANb B etc and then the same idea can be applied to form a direct product of tensors of any rank, for example (M⊗ N)ab,AB,αβ = MaAαNbBβ etc // rank = n = 3; number of tensors = I = 2 On the left side the number of groups of indices eq uals the tensor rank n, and the number of indices within each group matches the number I of tensors being direct-product-multiplied. In what follows, only the direct product of vectors 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' ... where of course the indices abc can be tilted in any way desired according to Section 7 (k). Appendix E: Tensor Expansions 233 (b) Tensor Expansions and Bases Let bi be an arbitrary complete set of basis vectors in x-space. As shown in Section 6 (b) there exists a unique set of dual ("reciprocal") basis vectors bi (also in x-space) such that bi• bj = δi j. Consider then the following expansion of a rank-3 tensor A A = Σijk αijk (bi⊗bj⊗bk) where ( bi⊗bj⊗bk ...)abc = (bi)a (bj)b (bk)c . The coefficients αijk can be obtained by dotting both sides with ( bi'⊗bj'⊗bk') and using ( bi'⊗bj'⊗bk') • (bi⊗bj⊗bk) = bi'• bi bj'• bj bk'• bk = δi' iδj' jδk' k . The result is then (unpriming indices) α ijk = A • (bi⊗bj⊗bk) = Aabc (bi⊗bj⊗bk)abc = Aabc (bi)a (bj)b (bk)c . (*) 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. In this manner, a tensor A of any rank can be expa nded on an arbitrary complete set of basis vectors, and the coefficients of that expansion can be obt ained by the inversion shown above for rank 3. Two special bases are of interest. The ui are the axis-aligned basis vectors in x-space as discussed in see Section 7 (s). For these basis vectors, one has ( ui)a = δia and ( ui)a = δi a . If one considers this expansion, A = Σijk αijk (ui⊗uj⊗uk) where ( ui⊗uj⊗uk ...)abc = (ui)a (uj)b (uk)c = δia δjb δkc then the coefficients are found to be αijk = A • (ui⊗uj⊗uk) = Aabc δi a δj b δk c = Aijk so the coefficients are exactly the x-space contravariant components of the tensor A. Thus A = Σijk Aijk (ui⊗uj⊗uk) On the other hand, if ei are the tangent base vectors in x-space (see Sections 3), the dual vectors are the ei and from Section 7 (s) one has ( 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 Appendix E: Tensor Expansions 234 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 and thus the coefficients in this case are exactly th e x'-space contravariant components of tensor A, as shown in Section 7 (j). Thus, A = Σijk A'ijk (ei⊗ej⊗ek) . Expansions like the above are the generalizations to tensors of any rank of these vector expansions stated in Section 7 (s), A = Σiαi bi αi = bi • A // arbitrary basis A = ΣiAi ui // axis aligned unit vectors A = ΣiA'i ei // tangent base vectors 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 = Σijk... Aijk... (ui⊗uj⊗uk...) Aijk... = contravariant components of A in x-space A = Σijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... = contravariant components of A in x'-space Orthonormal basis . If the basis vectors bi happen to be ort honormal as defined by bi• bj = δij then bi = bi because the dual basis is unique. As indicated in (*) above, this implies that coefficient αijk is unchanged if any or all indices are lowered, as if these αijk were components of a tensor in some Cartesian space. That Cartesian space is in fact the x'-space that would arise if transformation F were custom-selected such that the bi were the tangent base vectors ei for that F, for then g' ij = ei • ej = δi,j so that x'-space would in fact be Cartesia n. But for a pre-determined F, the αijk are just some coefficients and are not components of a tensor relati ve to F, and it just happens that αijk = αijk etc. An example of orthonormal basis vectors arises if bi = e^i ≡ ei/h'i and x'-space has a diagonal metric tensor g' ab = h'a2δa,b. One then has e^i • e^j = δi,j since e^i • e^j = ei • ej / (h'i h'j) = g'ij/ (h'i h'j) = h'i2δij/ (h'i h'j) = δi,j . Appendix E: Tensor Expansions 235 Then since the dual basis is unique, one has e^i = e^i and then αijk(any up/down) = Aabc (e^i)a (e^j)b (e^k)c = A'ijk (h'ih'jh'k) where the last expression comes from the third expa nsion shown above. Expansi ons on the unit versions of the tangent base vectors e^i are discussed more in section (h) below. Tensor density . If A is a tensor density of weight W, the ge neral rule is to make this replacement: A'ijk... → J WA'ijk... so the third general expansion above would be written A = J W Σijk... A'ijk... (ei⊗ej⊗ek...) A'ijk... = contravariant components of A in x'-space As justification for this rule, start with a regular tensor transformation for A, A'ijk... = Ri i' Rj j' Rk k'..... Ai'j'k'... The rule gives J W A'ijk... = Ri i' Rj j' Rk k'..... Ai'j'k'... or A' ijk... = J-W Ri i' Rj j' Rk k'..... Ai'j'k'... which is the correct form for the transformation of a tensor density of weight W (Appendix D). Expansion of tensor-like objects. If Aijk is some "tensor like" object having three indices (such as ∂iTjk) one can still do the three expansions shown above but th e results would have to be restated this way: A = Σijk... αijk... (bi⊗bj⊗bk...) αijk... = Aabc... (bi)a (bj)b (bk)c... A = Σ ijk... Aijk... (ui⊗uj⊗uk...) Aijk... = components of A in x-space A = Σ ijk... Aijk... (ei⊗ej⊗ek...) Aijk... = Ri a Rj b Rk c Aabc Since A is not a tensor, in this case Aijk... are not the contravariant components of tensor A in x'-space relative to the transformation x' = F(x). (c) Polyadic Notation Some fields of study historically use "polyadic notation" as follows, ( ABC ...) ≡ A⊗B⊗C ... Appendix E: Tensor Expansions 236 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 ... Sometimes this basis vector notation is compressed even more, to wit, i j k ... ≡ (bibjbk...) ≡ bi⊗bj⊗bk ... 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 do not "commute". For example i j = ( bibj) = bi⊗bj => ( i j )nm = (bibj)nm = (bi⊗bj)nm = (bi)n (bj)m ( j i )nm = (bjbi)nm = (bj⊗bi)nm = (bj)n (bi)m ≠ (i j )nm and therefore one cannot write i j = j i. The general expansion stated above now appears as A = Σijk... αijk... (bi⊗bj⊗bk...) = Σijk... αijk... (bibjbk...) = Σijk... αijk... (i j k ... ) where α ijk... = A • (bi⊗bj⊗bk...) = A • (bibjbk...) = A • (id jd kd ... ) = Aabc... (bi)a (bj)b (bk)c... 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. (d) 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 In this notation, the expansion given above for a rank-2 tensor becomes A = Σij αij (bibj) αij = Aab(bi)a (bj)b = Aab (bibj)ab . Appendix E: Tensor Expansions 237 Notice 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 Cartesian. In section (g) below 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. (e) Transpose notation for dyadics Superscript T as usual indicates the transpose of a vector or matrix. For a vector V, certainly Vi = (VT)i, meaning the object in the ith row of V is the same as the object in the ith column of VT . Therefore one can express the dyadic product in this more down-to-earth manner, ( ab)ij ≡ aibj = ai (bT)j = ( abT)ij or ab = abT 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 and one sees that in fact ( ab)ij = (abT)ij = aibT j = aibj . Meanwhile, the object aTb is just a number, aTb = a • b = (a1 a2) ⎝⎛ ⎠⎞ b1 b2 = a1b1 + a2b2 = a scalar (if a and b are vectors) . 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 ... In the special case that the bi are the unit vectors ui , and assuming N = 2 dimensions, one has A = Σ nm Anm unum = Σnm Anm unumT = A11 u1 u1T + A12 u1 u2T + A21 u2 u1T + A22 u2 u2T = a matrix with A12 in the upper right corner Appendix E: Tensor Expansions 238 where un is a column unit vector and unT is the corresponding row unit vector (see comments in Section 3 (c) about "unit" vectors). For example, u1u2T= ⎝⎛ ⎠⎞ 1 0 ( 0 1) = ⎝⎛ ⎠⎞ 0 1 0 0 . 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 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. The direct product or polyadic notation seems clearest for rank n > 2. (f) Large and small dots used with dyadics 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 = d iAijcj . 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 . 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 The same small dot is used to indicate the product of two dyadics, which is to say, matrix multiplication A•B ≡ AB 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 . It is this larger dot • which was the subject of Section 5 (i); (3) The next section provides an explanation of the small dot as part of an operator interpretation for dyadics. Appendix E: Tensor Expansions 239 (g) Operators and Matrices for Rank-2 tensors Operator concept. As discussed in Section 5 (i), x-space and x'-space of Picture A are both N-dimensional real Hilbert Spaces with scalar product indicated by the large dot •, and one can regard V as a vector in either space. Expressed as a "vector" in x-space one can write, as done a bove with generic basis bi , V = Σi [V(b)]i bi // [V(b)]i are the coefficients of this expansion . Moreover, one can regard a rank-2 tensor A as an "operator" in this Hilbert space, A = Σ ij [A(b)]ij bibjT Application of (bT)n on the left and bm on the right, and then a double use of (bT)nbi = bn • bi = δn i gives [A(b)]nm = (bT)n A bm 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. The idea of A as operator has an abstract meaning distinct from the matrix A ij. In the above equation the symbol A is this abstract operator and (bT)n A bm has a meaning distinct from our interpretation of it in terms of the matrix combination of three objects. In the matrix interpretation, one writes (bT)n A bm = [(bT)n]i Aij [bm]j = [bT]i Aij [bm]j and only then does A become a "matrix". This matrix happens to be the contravariant Aij matrix because we happened to select the un basis to write the components like [bm]j = uj • bm. Bra-ket Notation. For the author of this document, the bra-ket notation commonly used in quantum mechanics (Paul Dirac 1939) provides a clean way to look at a rank-2 tensor A as an operator. It is true that in quantum mechanics one usually deals with infinite dimensional Hilbert spaces and complex numbers, but the formalism applies just as well to real Hilbert spaces with finite dimensions. In bra-ket notation one writes bi → |bi>, biT→ <bi| , so that the above equations become |V> = Σ i [V(b)]i |bi> <b j|bi> = δji orthogonality of the basis [V(b)]i = <bi| V > 1 = Σi | bi><bi| completeness of the basis < U | V > = U • V = scalar product A = Σ ij [A(b)]ij | bi> <bj| [A(b)]ij = <bi | A | bj > . In this 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 as the "matrix element of the operator A in the bi basis ". In general, |b i> and |bi> are different vectors. In this notation, based on what was presented earlier, one can write, Appendix E: Tensor Expansions 240 Anm = <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 In the first of these three lines, one can raise and lower indices with g ab and gab on both sides of the equation. On the second line this can be done with g'ab and g'ab. It was shown in Section 6 (b) that bn = wnmbk and conversely bn = wnmbk where w nm is the metric tensor g' one would get for some underlying transformation F b which causes bn to be its tangent base vectors. Thus, on the third line above we can raise and lower indices on each side with wab and wab where w nm = bn • bm . Notice in the last three equations that th e 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. [A(b)]nm are the components of a rank-2 tensor in only two cases -- those shown in the first pair of equations above. In the first case Anm are components of a tensor in x-space, and in the second case the A'nm are components of a tensor in x'-space. In a consistent notation one might write Anm = [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> . 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> . One could then define the following tensor-like object, B n i ≡ [bn]i . 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 Bni = [bn]i = (BT)i n , where we use the notion of the transpose of a tilted matr ix described in Section 7 (i) item 8. One then has [A (b)]n m = Bn i Ai j(BT)j m . Since all the matrices are tilted the same way and summed indices are contractions, this is one of the "legal" Standard Notation matrix forms and we then write, A (b) = BABT or more precisely [A(b) = BABT ]SN,dt Appendix E: Tensor Expansions 241 where SN,dt means Standard Notation, down-tilt, as described in Section 7 (i) item 7. 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 bm . 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. It was shown in Section 7 (i) that in standard notation R is real orthogonal, so in fact one has for the bm = em basis, A(e) = BABT = R A RT = R A R-1 = R A S . Specifically in this case, [A (e)]n m = Rn iAi jSj m = Rn iRmjAi j = A'n m . More on bra-ket notation and its relation to the small dyadic dot. Consider the following facts, <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> where |(Ac) > = a new Hilbert space vector which results when A is applied to |c>, A|c> <(A Td) | = a new transpose Hilbert space vector wh ich results when A is applied to <d|, <d|A . So one has this general idea that <d | A | c > = <d |Ac > = <A Td | c> A | c > = |(Ac)> <d | A = <(A Td) | . 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 > Appendix E: Tensor Expansions 242 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 (again, one might have g ≠1) Tnm = (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 . In the generic bn basis one has [( AB)(b)]nm = <bn |(AB)| bm > . 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 the small-dot notation U • B emphasizes the idea of an operator acting on a vector, equivalent to U| B> . Here then are a few quotes from Morse and Feshbach ( an = un) to illustrate some of the notation described above. These author s are working in Cartesian space (g=1) where up and down indices don't matter. ( The first item here is A • c = ΣijAijcj ui from the start of section (f). ) Notice the impressive name "idemfactor" for the identity operator 1 = Σi | ai><ai| = Σi aiaiT = Σiaiai. Appendix E: Tensor Expansions 243 (h) Expansions of tensors on unit tangent base vectors We start with the general Picture A (and late r specialize to orthogonal coordinates), In section (b) above it was estab lished 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'... . Since en = h'n e^n, this same expansion for 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...) where the unit-vector expansion coefficients are given by [A (e^)]ijk... = h'ih'jh'k......A' ijk... = 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'... Coefficient notation. In a curvilinear coordinates applicati on of these expansions, the expansion coefficients are usually written in the following manner, [A (e^)]ijk... = Ax'ix'jx'k ...... 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 = θ and x'3 = φ one might write [A(e^)]2213 = Aθθrφ . Since [A (e^)]ijk... is not a tensor (with respect to F), there is no particular reason to put the indices "up" and for that reason they are usually written down, as in A θθrφ . Appendix E: Tensor Expansions 244 Matrices M and N. It is convenient now to define Ma b ≡ h'a Ra b so that then [A(e^)]ijk... = Mi i'Mj j'Mk k'...... A i'j'k'... . Defining Na b to be the inverse of Ma b, one has N a b ≡ h'b-1Sa b = h'b-1Rba so A ijk... = Ni i'Nj j'Nk k'...... [A(e^)]i'j'k'... . To verify that this N is the correct inverse or M, M a kNk c = (h'a Ra k)( h'c-1Rck) = (h'a/ h'c) Ra k Rck = (h'a/ h'c)δa c = δa c making use of the orthogonalit y rule of Section 7 (r), R a k Rck = δa c . M and N can be written in terms of the tangent base vectors as follows: Mn i ≡ h'n Rn i = h'n(en)i Ni n = h'n-1Rni = h'n-1(en)i = ( e^n)i which says that N i n = { e^1 , e^2, .... } -- the columns of Ni n are the unit tangent base vectors. Rank-1 tensors. For a vector, the above coefficient relation is written [A(e^)]i = Mi j Aj or A(e^) = M A and A = N A(e^) where A has these two familiar expansions, A = Anun = [A(e^)]n e^n [A(e^)]n = Ax'n for example [A(e^)]1 = Ar . Rank-2 tensors. Here the coefficient relation is [A(e^)]ij = Mi i'Mj j'A i'j' = Mi i' A i'j' Mj j' which can be written [A (e^)]nm = Mn i A ij Mm j = Mn i A ij (MT)jm Mn i = h'n(en)i . Appendix E: Tensor Expansions 245 Defining bn = h'nen , then [ bn]i = h'n(en)i = Mn i = Bn i of section (g) . Meanwhile, from Section 6 (b) we know that wnk = bn • bk = h'nh'k(en • ek) , and then w nk = (w-1)nk . In any event, whatever w nk is, it is the object which can lower the n or m indices on both sides of the above equation. Lowering just the m index and then reversing the j tilt gives [A (e^)]n m = Mn i A ij Mmj = Mn i A ij (MT)jm = Mn i A i j (MT)j m and we replicate the section (g) result with B = M : A (e^) = M A MT / / [ A(e^) = M A MT ]SN,dt . 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 th en to define x"-space as the space whose basis vectors are the e^n unit vectors, which are generally not orthogonal. The relation Ma b ≡ h'a Ra b given above can be written in down-tilt form as M = HR where Hi j ≡ diag(h'1, h'2.....). It then follow that N = M -1 = R-1H-1 = SH-1 Finally, note that x = xiui = x'iei = x'i(h'ie^i) = x"ie^i => x"i = h'i x'i or x" = H x' which shows that the transformation from x'-space to x" -space is linear with matrix H. We can now show all three spaces in the same picture as follows: This picture shows that the transformation directly from x-space to x"-space is FM(x) = H F(x) . Appendix E: Tensor Expansions 246 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 equations 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 A ijk... = 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 A" = M A M T // rank-2 tensor, matrix notation 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), and H is the diagonal matrix of scale 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. At this point the 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 pict ure so that x-space is Cartesian with g = 1, and x'- space is the space of a set of orthogonal curvilinear coordinates x'. In this scenario, F(x) is 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. Orthogonal curvilinear coordinates application We now switch to picture B (g=1) and assume that the x' i are orthogonal coordinates, and our three-frame picture above then becomes Appendix E: Tensor Expansions 247 In this situation, x-space is Cartesian with g ab = gab = δab and g'ab = h'a2δab and g'ab = h'a-2δab. Since the metric tensors are diagonal, one could write H = g'ab , but we continue to use H. In what follows, the plan is simply to exercise both the developmental and standard notations with regard to the M and N matrices. To this end, we first collect the following facts from Section 7 (o), R ab = Ra b' gb'b = Ra b Rab = g'aa'Ra' b' gb'b = g'aa'Ra'b = h'a2 Rab = h'a2 Ra b Rab= g'aa'Ra' b = h'a2 Ra b or Rab = Ra b R ab = Rab = h'a2 Ra b Ra b = h'a-2 Rab . Also from Section 7 (11), g' ab = Ra a'Rb b'ga'b' = Ra a'Rb a' = Ra cRb c g'ab = Sa' a Sb' b ga'b' = Raa' Rbb'ga'b' = Raa' Rba' = RacRbc or g'ab = Ra cRb c and g'ab = RacRbc . In this scenario, regardless of what R and S are, M is a "rotation" (verified below), where we include in this term possible axis reflections. What we really mean is that in developmental notation M is a real- orthogonal matrix, MM T = 1. Since N=M-1, N is then also a rotation. To prove that M is a rotation in developmen tal notation, the standard notation equation Ma b ≡ h'a Ra b can be reverse-translated to M ab = h'aRab . Then (now g' = RRT from Section 5 ( l) ) (MMT)ac = MabMcb = h'aRab h'cRcb = h'ah'c RabRT bc = h'ah'c(RRT)ac = h' ah'cg'ac = h'ah'c [ h'a–2 δa,c] = δa,c => MMT = 1 . Proving the same thing directly in standard notation requires showing that M a b Mc b = δa,c ( see the end of Section 7 (i) ) Ma b Mc b = h'a Ra b h'c Rc b = h'a h'c Ra b Rc b = h'a h'c Ra b (h'c-2 Rcb) = (h'a/h'c) (Ra b Rcb) Appendix E: Tensor Expansions 248 = (h' a/h'c)δa c = δa c = δa,c where use was again made of the orthogonality rule of Section 7 (r), R a b Rcb = δa c. Relation beween M and N. Looking at Ma b Mc b = δa,c and knowing that M a b (M-1)b c = δa c = δa,c one concludes that (M-1)b c = Mc b . But (M-1)b c = Nb c so N b c = Mc b // reminder: this does not say that N = MT in standard notation which can be verified from the above expressions for N and M. Interpretation of N and M. Since en = S un ( Section 3 (a) with e'n = un) and since en = h'ne^n , it follows that e^n = 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 = Na bδb n = Na b(un)b or e^n = N un and ( e^n)a = Na n . // => un = M e^n 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". Above it was shown that Na n = Mn a , therefore Mn a = Na n = (e^n)a The rotation matrix Na n = (e^n)a has the orthogonal basis vectors e^n as its columns, while the rotation matrix Mn a = (e^n)a has the orthogonal basis vectors e^n as its rows. From this point of view, it seems pretty reasonable that MN = 1. It might be noted that, in our situation with Cartesian x-space and orthogonal coordinates, e^n = e^n : e^n = en/|en| | en|2 = en• en = g'nn = h'n-2 so e^n = en h'n = h'n g'nn en = h'nh'n-2 en = h'n-1 en = e^n . Appendix E: Tensor Expansions 249 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. One implication of the above picture relates to tensor equati ons being covariant, as discussed in Section 7 (u). If one has a tensor field equation in x-space, Q ad c(x) = Hab(x)Tb c(x) Bd(x) , 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") . 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 observa tion 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 Fram e 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 example, which mi ght apply to fluid dynamics, has this covariance at each point x in the fluid, and it happens that the rotation is a different rotation at different points x, but it is always a rotation. 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 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 s o h r = 1 and h θ = r . The N and M matrices may be computed as follows: Ma b ≡ h'a Ra b = ⎝⎛ ⎠⎞ 1 0 0 r ⎝⎛ ⎠⎞cosθ sinθ -sinθ /r cosθ/r = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ = Rz(-θ) Appendix E: Tensor Expansions 250 Na b ≡ Sa b h'b-1 = ⎝⎛ ⎠⎞cosθ -rsinθ sinθ rcosθ ⎝⎛ ⎠⎞ 1 0 0 1/r = ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ = Rz(θ) Therefore, the relation between a rank-2 tensor's e^n-expanded form components and the Cartesian form components is given by the expression stated above for rank-2 tensors, A(e^) = M A MT or ⎝⎛ ⎠⎞ Arr Arθ Aθr Aθθ = ⎝⎛ ⎠⎞ cosθ sinθ -sinθ cosθ ⎝⎛ ⎠⎞ A11 A12 A21 A22 ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ . // Lai p 316 Problem 5.71 Airy functions . In isotropic elastic stress analysis for states of plane stress and plane strain, the 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, the above equation becomes ⎝⎛ ⎠⎞ Trr Trθ Tθr Tθθ = ⎝⎛ ⎠⎞cosθ sinθ -sinθ cosθ ⎝⎛ ⎠⎞∂22φ – ∂1∂2φ – ∂1∂2φ ∂12φ ⎝⎛ ⎠⎞cosθ -sinθ sinθ cosθ (*) where ∂ 1 = cosθ ∂r - (sinθ/r)∂θ ∂1 = ∂/∂x1 ∂ 2 = sinθ ∂r + (cosθ/r)∂θ ∂2 = ∂/∂x2 Using Maple's dchange function, one can have Maple compute ⎝⎛ ⎠⎞ Trr Trθ Tθr Tθθ from (*) to be // Lai p 264 (5.27.3) 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. 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), Tij = λ tr(E)δij + 2μEij or the same thing Tij = λ tr(E)δij + 2μEij Appendix E: Tensor Expansions 251 where λ and μ are Lamé's constants. With respect to transformation F M, this is a "true tensor equation" (tr(E) = E kk is scalar under rotations), so according to Sec tion 7 (u) it is "covariant" and in x"-space may be written T"ij = λ tr(E")δij + 2μE"ij or [T(e^)]ij = λ [T(e^)]kk δij + 2μ[E(e^)]ij . For example, using the notati on convention described above, Trr = λ [ Trr+ Tθθ] + 2μ Err Trθ = 2μ Erθ . Notice that δ" ij = δij under transformation F M , since δ"ij = Mi aMj bδab = Mi aMj a = δij, whereas under transformation F one has δ 'ij = Ri aRj bδab = Ri aRj a = g'ij. By way of contrast, the Ca rtesian-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 E rθ ≠ (∂ruθ + ∂θur)/2. In fact, this relation is E = [( ∇u)T + (∇u)] / 2 and ( ∇u) for polar coordinates is computed in Appendix G and one ends up with E rθ = (∂ruθ + (1/r) ∂θur - uθ/r) / 2 . (i) Tensor expansions in a mixed basis Recall the rank-n tensor expa nsion from section (b) above, A = Σijk... αijk... (bi⊗bj⊗bk...) αijk = A • (bi⊗bj⊗bk) where αijk... are the coefficients of the expansion of A on the direct product basis shown. A might be a tensor, or it might be a tensor-like object. To make explicit the fact that the coefficients depend on the choice of basis, one might write (one b for each i ndex, number of b's is the rank of the tensor) , α ijk... = [ A(b,b,b...)]ijk... . 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 e xpansion of a rank-3 tensor, A = Σijk [ A(b,b,b)]ijk (bi⊗bj⊗bk) [ A(b,b,b)]ijk = A • (bi⊗bj⊗bk) . 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) . 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 : Appendix E: Tensor Expansions 252 [ 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 . Alternatively, one could relate this co efficient to the coefficients expanded on ui⊗uj⊗uk, [ A (b,e,u)]ijk = A • (bi⊗ej⊗uk) = Aabc (bi)a (ei)a (ui)a as was shown near the start of section (b). In the case of a rank-2 tensor, one has the option of using the other notations discussed above, A = Σ ij [ 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| matrix bra-ket where [ A (b,u)]ij = A • (bi⊗uj) = Aab (bi)a (uj)b = (bi)T A (uj j) = < bi| A | uj > . 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) where the hats are replicated into the superscript tensor label. Example of a mixed-basis expansion of a tensor The identity tensor can be expanded this way, since ui • uj = (ui)Tuj = δi j = < ui| uj> , 1 = Σj uj ⊗ uj = Σj uj(uj)T = Σj ujuj = Σj | uj>< uj| . direct product matrix dyadic bra-ket The ui are related to the ei according to uj = Ri j ei . Appendix E: Tensor Expansions 253 Proof : (uj)a = Ri j (ei)a => δja = Ri j Ria which is an orthogonalit y rule of Section 7 (r). It follows that 1 = Σ ij Ri j ei ⊗ uj = Σij Ri j ei(uj)T = Σij Ri j eiuj = Σij Ri j | ei><uj| direct product matrix dyadic bra-ket where Ri j = (ei)T 1 uj = ei • uj = eiuj = < ei | uj> // as in Section 7 (s) matrix dot dyadic bra-ket This then is a mixed-basis expansion of the identity tensor. Another form would be 1 = Σ ij Rij ei ⊗ uj = Σij [1(e,u)]ij ei ⊗ uj following the notation discussed above, leading to this rather obscure way of writing Rij , Rij = [1(e,u)]ij . (j) What is a tensor? We are now in a better position to examine so me possible answers to this question. (1) A tensor is an operator like A which lives inside a direct product Hilbert Space. We can associate with this tensor A a la rge variety of up-indexed objects such as [A (b,e,u)]ijk in the example above. If one has at hand K different bases of interest, then for a tensor of rank n there would be Kn possible up-indexed objects. In the case of rank 2, these K2 different indexed objects are matrices. Given the un and en bases used throughout this document, there are two special up-indexed objects of rank 3: [A(u,u,u)]ijk and [A(e,e,e)]ijk which we abbreviate as [A(u)]ijk and [A(e)]ijk or as [A]ijk and [A']ijk . The indexed object [A]ijk is a set of N3 contravariant components of a rank-3 tensor in x- space, and [A']ijk is a set of N3 contravariant components of the same rank-3 tensor in x'-space, where these two spaces are linked by a transformation x' = F(x) which has a linearized form d x' = R d x at a point x. The two sets of contravariant components are related by [A ']ijk = Ri i'Rj j'Rk k' [A]i'j'k' . These sets of components "transform as a rank-3 te nsor with respect to the underlying transformation x' = F(x). As outlined in Section 7, the all-up indexed te nsor of rank n is just one of a family of 2n tensors where the indices take all possibly up and down po sitions, and each such tensor has a corresponding transformation rule, such as [A ']i jk = Ri i'Rjj'Rk k' [A]i' j'k' for A = Σijk [ A']i jk (ei⊗ej⊗ek) . For this definition of "tensor" as an abstract opera tor A, there are many possible set pairs of components (indexed objects where the values of all indices are set in all possibl e ways) which do NOT transform as a Appendix E: Tensor Expansions 254 rank-3 tensor with respect to F as just described. The component sets that do transform as tensors are those for which the components are coefficients of an expansion of A on a direct product basis where the individual basis vectors are selected from ei or ei, or are selected from ui or ui. If some other generic basis vector bi appears, then the component set does not transform as a tensor under F. Definition (1) is basically the definition of "tensor" used in this document. (2) Another definition of tensor might be: a tensor is any of the in dexed objects mentioned in (1) above. The set of components like [A (b,e,u)]ijk is called a "tensor" because it is a possible coefficient set that can be obtained by expanding the "tensor operator" A on suitable basis vectors. Just as a matrix is sometimes written without its indices, so this tensor object might be represented just as A(b,e,u). The components of this particular indexed object do not transform as either end of the transformation rule stated above, so this tensor is a tens or, but does not transform as a tensor. (3) A third possible definition: a tensor is any i ndexed object each of whose indices ranges from 1 to N where N is the dimension of one's space of interest. By this definition, any NxN matrix A ij would be a tensor of rank 2. It would be very unlikely that a random matrix like this would be part of the transformation rule [A']ij = Ri i'Rj j' [A]i'j' so this matrix A ij is then a tensor, but it probably doesn't transform as a tensor. In fields of physics involving relativity, the first definition is normally used, and an indexed object is called a tensor only if it transforms the way a tensor should transform with respect to a transformation of interest. For example, the affine connection Γ c ab is never called a tensor. In most other areas of physics definition (3) seems more common, where any matrix is a rank-2 tensor, also known as a second-order tensor. The whole subject of second order tensors is then identified with linear algebra where the operators are matrices. It may turn ou t that a particular matrix is in fact a tensor by definition (1) with respect to rotations. This is the case for basic matrices involved in equations whic h must be covariant. In continuum mechanics, which generally uses definition (3), there are many tensors which do not transform as tensors under rotations or other transformations. In th at field, when a tensor in fact transforms as a tensor, it is called an objective tensor (sometimes an indifferent tensor). Equations which are covariant in the sense of Section 7 (u) are called "frame indifferent". See Appendix K for examples. Appendix F: Affine Connection 255 Appendix F: The Affine Connection Γc ab and Covariant Derivatives (a) Definition and Interpretation of Γ : Γc ab = ec • (∂aeb) = Rc i(∂aRbi) Context can be confusing in a discussion of the affine connection Γ, so we start with 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 writing, an application in which the Γ object appears frequently. Recall from Section 1 that the metric tensor G is a diagonal matrix whose elements are independently +1 or -1. And recall from Section 5 (b) that in x-space, g ab = RaiRbjGij. If G = 1, then the xi coordinates of x-space are the "curvilinear coordinates" and the ξi are the "Cartesian coordinates". In Picture C1 the tangent base vectors en exist in ξ-space and the components of en are given by (en)i = Rni, as in Example 1 of Section 3 (polar coordi nates). Since matrix R is the linearization of transformation F at ξ and since x = F(ξ), one can regard R as a function either of ξ or x. We choose x as the variable and write [ en(x)]i = Rni(x). For example, Example 2 of Section 3 (spherical coordinates) showed that eφ(x) = rsinθ φ^(r,θ,φ). In any event, en(x) varies with x, and one wonders just how en varies with x. For a small variation d x in the x-space coordinates, one has, d( en)i = ∂j(en)i dxj ∂ j ≡ ∂/∂xi or (d en)i = (∂jen)i dxj . Since en and (∂jen) are both vectors in ξ -space, and since the ek are known to form a complete basis in ξ- space, it must be possible to expand ( ∂jen) on the ek with some appropriate coefficients, call them Γk jn : (∂ jen) = Γk jn ek . Dotting this equation into ek gives Γk jn = ek • (∂jen) = Rk i(∂jRni) where we recall from Section 7 (s) that ( ek)i = Rk i and ( en)i = Rni. These coefficients Γk jn(x) comprise a tensor-like field called the affine connection , so we shall regard the above line as the definition of Γk jn in the context of Picture C1 above. With more standard index names, the above becomes Γc ab = ec • (∂aeb) = Rc i(∂aRbi) . Appendix F: Affine Connection 256 From Section 7 (q) we know that, for Picture C1, R c i = (∂xc/∂ξi) Rbi = (∂ξi/∂xb) (∂aRbi) = (∂2ξi/∂xa∂xb) = (∂bRai) Therefore one can write Γ c ab = Rc i(∂aRbi) = (∂ xc/∂ξi) (∂2ξi/∂xa∂xb) = ∂xc ∂ξi ∂2xi ∂ξa∂ξb , which form appears in Weinberg p 100 (4.5.1). Notice again that ( ∂bRcn) = (∂cRbn) and that Γc bc is symmetric on the lower two indices. In th e next section an alternate form for Γ is derived, so both forms will be stated here: Γc ab = Rc i(∂aRbi) // ∂a = ∂/∂xa Γc ab = – Rbi (∂aRc i) (b) Identities of the form ( ∂aRd n) = – Re n Rd m (∂aRem) The identities 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 The second two lines are just restatements of the first line, but all are derived below. Corollary: The first identity above allows an alternate fo rm for the affine connection in terms of R: Γc ab = Rc i(∂aRbi) // as in section (a) above Γc ab = – Rbi (∂aRc i) // alternate form Our context is: . Appendix F: Affine Connection 257 Proof: These identities are a simple consequence of the f act that RS = 1 which in standard notation is written δc b = Rc αRbα (one of the orthogonality rules). So, 0 = ∂a(δd e) = ∂a(Rd mRem) = Rd m (∂aRem) + Rem (∂aRd m) QED 1 (*) Apply Σe Re n to both sides of (*) to get ( or, just use the Inversion Rule of Section 7 (r) ) 0 = Re 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) Q E D 2 Alternatively, apply Σ d Rdn to both sides of (*) 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) Q E D 3 (c) 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 (∂ cgab) = + [g an Γn cb + gbn Γn ca] 2 Proof of 1: ( ∂cgab) = – [gan Γ b cn + gbn Γa cn] The LHS is given by LHS = ( ∂cgab) = ∂c(Ra iRb i)Gii = Ra i(∂cRb i)Gii + Rb i(∂cRa i)Gii For the RHS, the Γ objects can be replaced by their al ternate definitions from section (a) Γc ab = – Rbi (∂aRc i) // from section (a) Γb cn = – Rni (∂cRb i) // b →n then c→b then a→c Γa cn = – Rni (∂cRa i) Appendix F: Affine Connection 258 The RHS of the claimed identity 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)} = Ra 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 = R a i (∂cRb i) Gii + Rb i (∂cRa i) Gii = LHS QED Proof of 2: ( ∂cgab) = + [g an Γn cb + gbn Γn ca] The LHS is given by LHS = ( ∂ cgab) = ∂c(RaiRbi)Gii = Rai (∂cRbi)Gii + Rbi (∂cRai)Gii For the RHS, the Γ objects can be replaced by their primary definitions from section (a) Γ c ab = Rc i(∂aRbi) Γn cb = Rn i(∂cRbi) // c →n then a→c then i→k Γn ca = Rn i(∂cRai) The RHS of the claimed identity may then be written RHS = g an Γn cb + gbn Γn ca = { R akRnkGkk}{Rn i(∂cRbi)} +{RbkRnkGkk}{Rn i(∂cRai)} = R ak (Rnk Rn i)(∂cRbi)Gkk + Rbk(Rnk Rn i)(∂cRai)Gkk = R ak δk i(∂cRbi)Gkk + Rbk∂k i(∂cRai)Gkk = R ai (∂cRbi)Gii + Rbi (∂cRai)Gii = LHS QED (d) 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] . Appendix F: Affine Connection 259 Recall in our definition above, Γd ab = Rd k(∂aRbk), that Γ was given in terms of R matrices. The following corollary ( derived at the end of this section ) concerns contraction of the upper Γ index with a lower one: Γa an = (1/2) gad ∂ngad = (1/2)(1/g) ∂ng = (1/ g ) ∂n(g ) . Proof: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] We know that g ab = RaeRbe Gee // sum on e gdc = Rd iRc i Gii // sum on i 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 permutations of the first line : ∂ cgab = [Rbe (∂cRae) + Rae(∂cRbe) ]Gee ∂agbc = [Rce (∂aRbe) + Rbe(∂aRce) ]Gee ∂bgca = [Rae (∂bRce) + Rce(∂bRae) ]Gee . The last four lines can be inserted into the Ri ght Hand Side of our desired identity to obtain (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) ] 1 2 3 4 5 6 Due to the symmetry noted above in section (a), ( ∂iRje) = (∂jRie), 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) = Rd i δe i Gii Gee (∂aRbe) = R d e Gee Gee (∂aRbe) = Rd e(∂aRbe) = Γd ab QED Appendix F: Affine Connection 260 Proof of Corollary: The abovementioned 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 ) The first and third terms of the second expression cancel due to symmetry gad∂agnd – gad∂dgan = gad∂agnd – gda∂agdn = gad∂agnd – gad∂agnd = 0 and the fact that (1/g) ∂ ng = gab(∂ngab) is proved as follows: (1) g ab = (g-1)ab = cof(gab)T/det(gab) = cof(g ab)/g => cof(g ab) = g gab (2) g = det(g ab) = Σab gabcof(gab) => ∂g/∂gab = cof(g ab) = g gab (3) ∂ ng = ∂ g/∂xn = (∂ g/∂gab)( ∂gab/∂xn) = g gab (∂ngab) => (1/g) ∂ng = gab(∂ngab) . The final form shown is just calculus : g -1/2 ∂n(g1/2) = g-1/2 (1/2) g-1/2 ∂n(g) = (1/2) (1/g) ( ∂ng) . (e) Picture D1 Context Our main context of interest is Picture A, . For the proof given in the next section below, it is us eful to think of Picture A as the top part of this Picture D1, The relationship between the R's and S's are these Appendix F: Affine Connection 261 R = R' R-1 = R' S => R' = R R S = R R'-1 = R S' The tangent and reciprocal base vectors in ξ-space associated with transformations F and F ' are these: (en)i = Rni ( en)i = Rn i x-space (e'n)i = R'ni ( e'n)i = R'n i x'-space Now there are two affine connections, Γc ab ≡ (∂xc/∂ξn) (∂2ξn/∂xa∂xb) = Rc n ∂a (∂ξn/∂xb) = Rc n(∂aRbn) ∂a = ∂/∂xa = [ ec]i • (∂a[eb]i) = ec • (∂aeb) Γ 'c ab ≡ (∂x'c/∂ξn) (∂2ξn/∂x'a∂x'b) = R'c n ∂'a (∂ξn/∂x'b) = R'c n(∂'aR'bn) ∂'a = ∂/∂x'a = [ e'c]i • (∂'a[e'b]i) = e'c • (∂'ae'b) (f) 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) Γ 'c ab = Rc d Raα Rbβ Γd αβ – Rbβ(∂'aRc β) Γ 'c ab = Rc d Raα Rbβ Γd αβ – Raα Rbβ (∂αRc β) // Weinberg p 102 (4.5.8) If the second term were not presen t, the relation would state that Γ d αβ transforms as a mixed rank-3 tensor in the usual manner (Section 7 (j)). Since the second term is present, Γd αβ is not a tensor. Proof of the first relation : We now make use of Picture D1 shown above. Start with the Γ ' definition given above, Γ 'c ab ≡ R'c n(∂'aR'bn) = (R R)c n ∂'a(RR)bn = Rc dRd n ∂'a(RbβRβn) // R' = R R = R c dRd nRbβ(∂'aRβn) + Rc d(Rd nRβn)( ∂'aRbβ) = R c dRd nRbβ([Raα∂α]Rβn) + Rc d(δd β)( ∂'aRbβ) // ∂'a = Raα∂α in first term only = R c dRaαRbβRd n(∂αRβn) + Rc β (∂'aRbβ) = R c dRaαRbβ Γd αβ + Rc α (∂'aRbα) // Γd αβ ≡ Rd n(∂αRβn) QED Appendix F: Affine Connection 262 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) => R c α (∂'aRbα) = – Rbα(∂'aRc α) = – Rbβ(∂'aRc β) = – R bβ Raα(∂αRc β) and this gives the other two relations stated above. (g) 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 indices "down"). In th e next section it will then be shown how to adjust the theorem if one or more of the tensor density indices is "up". Covariant Derivative Theorem: The covariant derivative (B abc..x; α as defined below) of a covariant tensor density of ra nk n and weight W (B abc..x ) transforms as a covariant tensor density of rank n+1 and weight W. 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, contr action, and so on. The first term 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) 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 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 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 (u) 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 section (d), (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. Appendix F: Affine Connection 263 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' or B'ABC..X; α = J-WRαα'{RAa'RBb'..... RXx'} * B a'b'c'..x'; α' or, in gory detail, ∂' 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'{RAa'RBb'..... RXx'} * / / R H S { ∂α'Ba'bc'..x' – Γn a'α'Bnb'c'..x' – Γn b'α'Ba'nc'..x' – .. – Γn x'α'Ba'b'c'.. n del a-term b-term x-term + ( W / 2 g ) ( ∂α'g) Ba'b'c'..x' } Proof: 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 We first claim that this 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 ) del RHS = J-W Rαα'{RAa'RBb'..... RXx'} {∂α' Ba'bc'..x' } Unpriming all the Latin indices and setting α' = β shows that these two terms indeed match. Appendix F: Affine Connection 264 2. Show that the a-related terms balance. The a'-term on the LHS is this – Γ 'n AαB'nBC..X The connection between Γ ' and Γ given in section (f) says Γ 'c ab = Rc d Raκ Rbσ Γd κσ – Raκ Rbσ (∂κRc σ) or Γ 'n Aα = Rn d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ) . Inserting the above for Γ 'n Aα and the transformation rule for B', the LHS a'-term becomes – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { J-W Rnn'RBbRCc....Rxx Bn'bc..x } and to this we must add the c ontribution called del'-a above. Meanwhile, the RHS a-term is this J-W Rαα'{RAa'RBb'..... RXx'}{– Γn a'α'Bnb'c'..x' } // remove Latin primes, then n →n' = – { J -W Rαα'{RAaRBb..... RXx}{Γn' aα'Bn'bc..x } so we have to show that – { 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 } ? where now the del'-a term has been added in to the LHS, and in so doing index a → n'. We can see that the factors J-W RBbRCc....Rxx Bn'bc..x are the same on both sides so they can be removed to give a simpler relation which we must show is valid: – { R n d RAκ Rασ Γd κσ – RAκ Rασ (∂κRn σ)} { Rnn' } + R αβ (∂βRAn') = – { R αα'{RAa }{Γn' aα'} In the first term one sees Rn d Rnn' = δdn' which pins d to n' in that term only, so the above becomes – R Aκ Rασ Γn' κσ + Rnn'RAκ Rασ (∂κRn σ) + Rαβ (∂βRAn') = – R ασRAκ Γn' κσ . The first term on the left cancels the term on the right, and do β→σ in the third term to get RAκ Rασ {Rnn'(∂κRn σ)} + R ασ (∂σRAn') = 0 . Appendix F: Affine Connection 265 Then cancel the common R ασ factor and use the symmetry ( ∂κRn σ) = (∂σRn κ) to get (∂σRAn') = – Rnn'RAκ (∂σRn κ) Now in this order do σ→a, A→d, n→e, n'→n, κ→m to get (∂aRdn) = –Ren Rdm (∂aRe m) . But this is seen to be the third identity of section (b)! 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)} { Rnn'RBbRCc....Rxx Bn'bc..x } + R αβ (∂βRAn')RBb..... RXx Bn'bc..x = – { R αα'{RAaRBb..... RXx}{Γn' aα'Bn'bc..x } In reversing the sequence, one of course adds back in 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. In the previous equation, which was shown true, do A,a ↔B,b (indices) and B n'bc..x → Ban'c..x to get the following known-valid equation : – { Rn d RBκ Rασ Γd κσ – RBκ Rασ (∂κRn σ)} { Rnn'RAaRCc....Rxx Ban'c..x } + R αβ (∂βRBn')RAa..... RXx Ban'c..x = – { R αα'{RBbRAa..... RXx}{Γn' bα' Ban'c..x } . The first line is in fact the b'-term (– Γ 'n BαB'AnC..X ), the second line is del'-b, and the RHS is the b-term. This shows that the three b-related terms match LHS = RHS. 4. Similarly, the c,d.....x terms match . Just repeat item 3 above for each extra index. 5. It remains to show that the thr ee terms so far neglected match as well . These terms are LHS: R αβ(∂β J-W) RAaRBb..... RXx Babc..x // the del'-J term + (W/2g') (∂ 'αg') B'ABC..X // the LHS W term RHS: J-W Rαα'{RAa'RBb'..... RXx'}(W/2g) ( ∂α'g) Ba'b'c'..x' // the RHS W term That is, one must show that Appendix F: Affine Connection 266 Rαβ(∂β J-W) RAaRBb..... RXx Babc..x + (W/2g') (∂ 'αg') B'ABC..X = J-W Rαα'{RAa'RBb'..... RXx'}(W/2g) ( ∂α'g) Ba'b'c'..x' . Expanding B' ABC..X in the second term gives R αβ(∂β J-W) RAaRBb..... RXx Babc..x + (W/2g') (∂ 'αg') J-W{RAa'RBb'..... RXx'} * Ba'b'c'..x' = J -W Rαα'{RAa'RBb'..... RXx'}(W/2g) ( ∂α'g) Ba'b'c'..x' . Now unprime the primed Latin indices and 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 . Next, remove all common factors (and associated implied sums) to get R αα' (∂α' J-W) + (W/2g') (∂ 'αg') J-W = J-W Rαα' (W/2g) (∂α'g) . Since ∂' α = Rαα'∂α' this becomes (∂' α J-W) + (W/2g') (∂ 'αg') J-W = J-W (W/2g) (∂'αg) . Move the second term to the RH S and do the derivative to get (-W) J-W-1(∂'α J) = J-W (W/2)[ (∂'αg)/g – (∂'αg')/g' ] so it remains then to show that J -1(∂'α J) = (1/2)[ (∂ 'αg')/g' - (∂'αg)/g ] From Section 5 (k) one has J2 = 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 ] = [ g( ∂'α g') – g' (∂'α g)]/g2. (*) But (finally) evaluation of the LHS of (*) gives [ g(∂'α g') – g' (∂'α g)]/g2 Appendix F: Affine Connection 267 which shows that equation (*) is valid. Once again, by reversing the above sequence of steps, one shows that the remaining three terms are in balance. QED (h) Rule for raising any index on a covariant deriv ative of a covariant tensor density. The Rule is stated at the end of this section before the examples. Consider the general form given in section (g) for a covariant derivative 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 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 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 . The second term here can be combined with the b-term to give del-extra + 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 . The first identity of section (c) reads (∂ cgab) = – [gai Γ b ci + gbi Γa ci] or [– (∂ cgab) – gai Γ b ci] = gbi Γa ci // now do c →α, b→n, a→β or [– (∂ αgβn) – gβi Γ n αi] = gni Γβ αi or [– (∂αgβn) – gβb Γ n αb] = gni Γβ αi . Appendix F: Affine Connection 268 Therefore, del-extra + b-term = g ni Γβ αi Banc..x = Γβ αi Bai c..x = Γβ αn Ban c..x so it has been shown that Baβ 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 Here then is a comparison B abc..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 Rule for raising some non-last index q: (1) In all terms, raise the corresponding B index. (2) in the q correction term, make the replacement – Γ n qα → + Γq nα ( = Γq αn ) Corollary : In order to construct the covariant derivative of any rank-n tensor density B------ (indices in any positions), write out the above form and use a covariant correction term for each covariant index (such as –Γn aαBnb c..x for the a index above) and use a contravariant correction term for each contravariant index (such as + Γb αn Ban c..x for the b index above). 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. (i) Examples of covariant derivative expressions It will be assumed that all B objects in the exampl es are true tensors unless otherwise specified. 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 The above examples also apply if B is a scalar density of any weight W. Since ∂ αB is then a vector density of weight W (the a ddition rule), one will have ( ∂'αB') = J-WRαα'(∂α'B) and no "correction terms" are required. Appendix F: Affine Connection 269 Example J=1: (covariant derivative of a vector B) Ba;α = ∂α Ba – Γn aαBn covariant rank-2 tensor // 2nd term is symmetric on a ↔α B a ;α = ∂α Ba + Γa αn Bn mixed rank-2 tensor To obtain the other two possibilities, it is necessary to apply the metric tensor g αβ and gαβ∂β = ∂α , B a;α = ∂α Ba – gαβΓn aβBn mixed rank-2 tensor B a;α = ∂α Ba + gαβΓa βn Bn contravariant rank-2 tensor Example J=2: (covariant derivative of a rank-2 tensor) B ab;α ≡ ∂α Bab – Γn aαBnb – Γn bαBan covariant rank-3 tensor Ba b;α ≡ ∂α Ba b + Γa αn Bn b – Γn bαBa n etc. B ab ;α ≡ ∂α Bab – Γn aαBnb + Γb αn Ban B ab ;α ≡ ∂α Bab + Γa αn Bnb + Γb αn Ban . 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 B a bc;α ≡ ∂α Ba bc + Γa αn Bn bc – Γn bαBa nc – Γn cαBa bn ... B abc ;α ≡ ∂α Babc + Γa nαBnbc + Γb nαBanc + Γc nαBabn Special J=2 application to the metric tensor: gab;α ≡ ∂α gab – Γn aαgnb – Γn bαgan = 0 by section (c) identity 2 g a b;α ≡ ∂α ga b + Γa αn gn b – Γn bαga n = 0 + Γa αb – Γa bα = 0 g ab ;α ≡ ∂α gab – Γn aαgnb + Γb αn gan = 0 – Γb aα + Γb αa = 0 g ab ;α ≡ ∂α gab + Γa αn gnb + Γb αn gan = 0 by section (c) identity 1 Appendix F: Affine Connection 270 The middle lines use the fact that gi j = δi j. 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 gab;α = ga b;α = gab ;α = gab ;α = 0 . // Weinberg p 105 (4.6.16,17,18) 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 and Γ = 0. Then in any x'-space g' ab;α = 0 as well since g' ab;α = Raa' Rbb' Rαα' ga'b';α' . Example J=2: (double covariant derivatives) Consider again the J=1 examples from above B a;α = ∂α Ba – Γn aαBn Ba ;α = ∂α Ba + Γa αn Bn . 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 B;a ;α = ∂α B;a + Γa αn B;n where we have simply inserted a semicolon in each term. Consider again the J=2 examples from above, B ab;α = ∂α Bab – Γn aαBnb – Γn bαBan Ba b;α = ∂α Ba b + Γa αn Bn b – Γn bαBa n This applies to any rank-2 tensors B ab or Ba b. According to sections (i) and (h) above, B a;b and Ba ;b are bona-fide rank-2 tensors, and therefore B a;b;α = ∂α Ba;b – Γn aαBn;b – Γn bαBa;n Ba ;b;α = ∂α Ba ;b + Γa αn Bn ;b – Γn bαBa ;n 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 . The next examples are for tensor densities : (The trivial J=0 cases were already considered above.) Appendix F: Affine Connection 271 Example J = 1 (vector density of weight W) // recall Γκ κα = (1/2g) ∂αg 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 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 Ba b;α = ∂α Ba b + Γa αn Bn b – Γn bαBa n + W Γκ κα Ba b Bab ;α = ∂α Bab – Γn aαBnb + Γb αn Ban + W Γκ κα Bab Bab ;α = ∂α Bab + Γa αn Bnb + Γb αn Ban + W Γκ κα Bab . As noted in Appendix D (b) item 3 Example 2, adding a factor g W/2 to a tensor density of weight W neutralizes the weight, and the result is a regular tens or. 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 (gW/2B);α = ∂α(gW/2B) contravariant vector Example J=1: (covariant derivative of a vector density B of weight W) (g W/2Ba);α = ∂α (gW/2Ba) – gW/2 Γn aα Bn covariant rank-2 tensor (gW/2Ba);α = ∂α (gW/2Ba) + gW/2 Γa αn Bn mixed rank-2 tensor Example J=2: (covariant derivative of a tensor density B of weight W) (gW/2Bab);α = ∂α (gW/2Bab) – Γn aα(gW/2Bnb) – Γn bα(gW/2Ban) covariant rank-3 tensor (gW/2Ba b);α = ∂α (gW/2Ba b) + Γa αn (gW/2Bn b) – Γn bα(gW/2Ba n) etc. (gW/2Bab);α = ∂α (gW/2Bab) – Γn aα(gW/2Bnb) + Γb αn (gW/2Ban) (gW/2Bab);α = ∂α (gW/2Bab) + Γa αn (gW/2Bnb) + Γb αn (gW/2Ban) (j) The Leibniz rule for the covariant derivativ e 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 product rule is this: (A----B----) ;α ≡ A----;α B---- + A---- B----;α // Weinberg p 105 (4.6.14) (*) Recall from the Covariant Derivative Theorem of section (g) above that A---- and A---- ;α have the same weight, call it W A. Similarly, B---- and B---- ;α have the same weight W B. According to the outer product rule of Appendix D (b) item 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. Appendix F: Affine Connection 272 Proof: Start with 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---- The "index correction terms" are those Γ terms discussed in the previous sections. One can then write out the two terms on the RHS of (*) 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---- ] Meanwhile, the LHS of (*) can be written as (A----B----) ;α = (A----B----) ,α + ( all index correction terms) + (W A+ WB) Γκ κα (A----B----) 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 which is to say ∂α(A----B----) = (∂ αA----)B---- + A---- (∂α B----) . 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 invol ving A indices) + (terms involving B indices) . 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) . 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 } Just so, in the general case one has Appendix F: Affine Connection 273 (A----B----) ;α|ICT = ( all index correction terms) = { index correction terms for A---- } B---- + A---- { index correction terms for B---- } and this then shows that the index correction te rms on the two sides of (*) do in fact match. QED 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 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 Example with a scalar function A and a vector B : (AB b);α = A;αBb + ABb;α = A,αBb + ABb;α // for a scalar A, A ;α = A,α Example with a scalar constant A and a vector B: (AB b);α =A(Bb;α) / / s i n c e A ;α = A,α = 0 so a scalar constant can always be extracted from (AB b);n to give A(B b;n). A tensor constant like εabc cannot be extracted in this manner since εabc ;α ≠0. In fact ε abc ;α = Γa nαεnbc + Γb nαεanc + Γc nαεabn ε123 ;α = Γ1 nαεn23 + Γ2 nαε1n3 + Γ3 nαε123 = Γ1 1α + Γ2 2α + Γ3 3α . A more general example: (A abcBde);α ≡ Aabc ;α Bde + Aabc Bde;α An example with the metric tensor: (g abB----b----);α = gab ;α B----b---- + gab B----b----;α . But g ab ;α = 0 as shown at the end of section (h). Therefore, (B---- a----);α = gab B----b----;α Appendix F: Affine Connection 274 which says that raising an index "commutes" with covariant differen tiation -- one can raise an index ignoring the fact that : α is sitting there. But we already know this must be true because we know that the object B---- b----;α is a true tensor, and gab can raise any index on a true tensor. Appendix G: Expansion of ( ∇v) 275 Appendix G: Expansion of ( ∇v) in curvilinear coordinates (v = vector) This appendix assumes the usual curvili near coordinates context, Picture B (a) Continuum Mechanics motivation 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 where the indices are the reverse of the normal dyadic definition of Appendix E, ( BA)ij ≡ BiAj . 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 ( eg, Lai (3.4.3) and (3.4.8) ), 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). 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). The object ( ∇v), called the velocity gradient, is of great in terest in fluid mechanics. Correspondingly, the object (∇u) is of great interest in the theory of elastic solids, where u is the displacement field. The matrix ∇A is not a differential operator since the deri vative does not act on the vector standing to the right of ∇A, but one is still often interested in expressing ∇A in curvilinear coordinates. This is done in the following sections, where we replace the generic vector A by generic vector v and this v has nothing to do with the velocity v mentioned above. (b) Expansion of ∇v on ei⊗ej by Method 1: Use fact that v b;a is a tensor. Appendix G: Expansion of ( ∇v) 276 The covariant derivative v b;a is discussed in Appendix F (g,h,i). We shall define a true tensor object ( ∇v) as vb;a so that (∇v)ba ≡ vb;a = [vb,a – Γ c ab vc] // v b,a ≡ ∂avb (∇v)'ba ≡ v'b;a = [v'b,a – Γ 'c ab v'c] // v' b,a ≡ ∂'av'b In Cartesian space Γ = 0 so one has (∇v)ba = vb,a = ∂avb so (∇ v)ba aligns with our object of interest in Cartesian space. As shown in Appendix E (b) 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] where the ui are the Cartesian basis vectors in x-space, while ei are the reciprocal base vectors. According to Appendix F (e), 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, so Γ 'c ab = Rc i(∂'aRbi) Γ 'c ij = Rc k(∂'jRik) . Inserting this into our expansions above 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] 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 Appendix E (g): [ ( ∇v)(e)]ij = <ei|∇v| ej> = ( ei)T (∇v) ej = ei • (∇v) ej = ei • (∇v) • ej bra-ket matrix dot of vectors dyadic An alternate form is obtained using the identities of Appendix F (b) (adjusted from Picture C1 to Picture D1 shown in that Appendix ), (∂' aRdn) = – Ren Rdm (∂'aRe m) (∂'jRik) = – Rek Rim (∂'jRe m) Then R c k(∂'jRik) = – Rc k Rek Rim (∂'jRe m) = – δc e Rim (∂'jRe m) = –Rim(∂'jRc m) Appendix G: Expansion of ( ∇v) 277 so that ∇ v = Σij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = v'i;j = [(∂'jv'i) + Rim(∂'jRc m) v'c] . We shall use this second form for [( ∇ v)(e)]ij below. Comment 1 : A variation of the above development would be to start this way ∇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] which quickly leads to a result similar to the above, where ∇ v = Σij [(∇v)(e)]i j ei⊗ej [(∇v)(e)]i j = v'i ;j = [(∂'jv'i) – Rcm (∂'jRi m) v'c] . Comment 2 : The idea that [( ∇v)(e)]i j = [∂'jv'i + Γ ' i jc v'c] can be reached by this alternate path: dv i = (∂jvi)dxj (chain rule) => d v = (∇ v)dx Expand: d x = dx'i ei and v = v'n en => d v = dv'n en + v'n den , but d en = (∂ 'jen)dx'j = Γ 'i jn ei dx'j // from definition of Γ in Appendix F (a), but Picture A so d v = dv'n en + v'n Γ 'i jn dx'j ei = dv'i ei + v'n Γ 'i jn dx'j ei = (dv'i + v'a Γ 'i ja dx'j )ei . But dv' i = (∂ v'i/∂x'j)dx'j = (∂'jv'i) dx'j so dv = (∂ 'jv'i) dx'j + v'a Γ 'i ja dx'j )ei = [∂'jv'i + v'a Γ 'i ja ] dx'j ei . Then d v = (∇ v)dx => [∂'jv'i + v'a Γ 'i ja ] dx'j ei = (∇ v) dx'j ej or [∂' jv'i + v'a Γ 'i ja ] ei = (∇v) ej => ei• (∇v) ej = ei• [∂'jv'i + v'a Γ 'i ja ] ei = (∂'jv'i + v'a Γ 'i ja ) => [( ∇ v)(e)]i j = ei• (∇v) ej = [∂'jv'i + Γ 'i jc v'c ] (c) 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 Appendix G: Expansion of ( ∇v) 278 notation ei(ej)T which works for rank-2 tensor expansions. Recall that ei(ej)T is an NxN matrix as discussed in Appendix E (e). In this brute force method, start with the Cartesian space expansion of Appendix E, (∇ v) = Σcd(∇v)dc uducT = Σcd(∂cvd) uducT // note that ud = ud in Cartesian space 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)] The next step is to express the matrix uducT as a linear combination of ee efT. In Section 3 (b) it was shown that the tangent base vectors transform in this way e'n = Σi Ri n ei where ( e'n)i = δni (*) For present purposes, we write this as ud = Σe Re d ee where ( ud)e = δde Therefore ud = Σe Red ee Then we can change this colu mn vector to a row vector, ucT = Σf Rfc efT and so uducT = Σef Red Rfc ee efT = Red Rfc ee efT Inserting these two results gives (∇v) = Σcd(∇v)dc uducT = Σcd(∂cvd) uducT = { Ri c [(∂'iRj d) v'j + Rj d (∂'iv'j)] } { Red Rfc ee efT } = { R i c [(∂'iRj d) v'j + Rj d (∂'iv'j)] } { Red Rfc ee efT } = { ( R fc Ri c)[Red (∂'iRj d) v'j + (Red Rj d)(∂'iv'j)] } ee efT = δ fi [Red (∂'iRj d) v'j + δej (∂'iv'j)] ee efT = [ R ed (∂'fRj d) v'j + (∂ 'fv'e)] ee efT Appendix G: Expansion of ( ∇v) 279 = Σef [(∇v)(e)]ef ee efT where [(∇ v)(e)]ef = (∂'fv'e) + Red (∂'fRj d) v'j or [(∇v)(e)]ij = (∂'jv'i) + Rim (∂'jRj m) v'j and this agrees with the second form obtained by Method 1. (d) Expansion on e i⊗ej and e ^i⊗e^j Reversing the covariant tilts in the above expansion 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] Then since ei = h'ie^i one gets yet another expansion (m ore generally see Appendix E (h) ), ∇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] . 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] ∇ 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 = g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] ∇v = Σij [(∇v)(e^)]ij e^i⊗e^j [(∇v)(e^)]ij = h'i h'j g'iag'jb [(∂'bv'a) + Ram(∂'bRc m) v'c] 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)] . Another choice is to use the v'n components of v obtained when expanding v on the e^n , Appendix G: Expansion of ( ∇v) 280 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 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)] As discussed in Section 14 Example 1, the components v'n are convenient since they all have the same dimensions. Moreover, when a specific curvilinear system is selected, one can dispense 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^ . (e) Orthogonal coordinate systems In this case g' ab = h'a2δa,b and g'ab = h'a-2δa,b and things simplify. The a bove four expansions 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] and the last object noted above becomes (m →d and c→b) 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)] 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. (f) Maple evaluation of ( ∇v) in several coordinate systems This object P ij 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, Appendix G: Expansion of ( ∇v) 281 Then one needs the scale factors h n' from the metric tensor g ¯ = STS ( dev notation Section 5 ( l) ) Appendix G: Expansion of ( ∇v) 282 The terms T1,T2 and T3 shown above are then entered, Pij ≡ [(∇v)(e^)]ij = h'i-1 h'j-1 [ (∂'jh'i) v'i + h'i(∂'jv'i) + h'i2Ri d(∂'jRb d) (h'bv'b)] T2 T3 T1 The terms are then added and simplified and out pops the result, Below are some sample results (including the above): Appendix G: Expansion of ( ∇v) 283 (∇v) = Σij Pij e^i e^jT Pij = [(∇v)(e^)]ij • Pij in polar coordinates (where 1,2 = r, θ) : // agrees with Lai (2.23.23) • P ij in cylindrical coordinates (where 1,2,3 = r, θ,z) : // agrees with Lai (2.34.5) The polar coordinates results are seen to be upper left 2x2 piece of the cylindrical results. • Pij in spherical coordinates (where 1,2,3 = r, θ,φ) : // agrees with Lai (2.35.25) 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 ( divT ) 284 Appendix H: Expansion of div(T) in curvilinear coordinates (T = rank-2 tensor) The object of attention in this Appendix, divT , is expressed this way in Cartesian coordinates, (divT)i = ∂jTij , 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 (b) below, divT will be redefined as a true tensorial vector which equals ∂jTij in Cartesian coordinates. (a) 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) 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. Comment . 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 (u) under these kinds of transformations. As in the case of ∇v, one is interested in expressing divT in general curvilinear coordinates. (b) Expansion of divT on e n by Method 1: Use fact that Tab ;α is a tensor. As shown in the examples of Appendix F (i), the following object transforms as a rank-3 tensor, T ab ;α ≡ ∂αTab + Γa αn Tnb + Γb αn Tan = ∂α Tab // x-space where g=1, Γ = 0 T 'ab ;α ≡ ∂'αT 'ab + Γ 'a αn T 'nb + Γ 'b αn T 'an // x'-space Contracting b with α yields the following tensorial vector equations, (divT) a = Tab ;b = ∂bTab // x-space where Γ = 0 (divT)'a = T 'ab ;b ≡ ∂'bT 'ab + Γ 'a bn T 'nb + Γ 'b bn T 'an // x'-space Note that the Cartesian space statement (divT)a = ∂bTab is obtained, as noted in the opening comments above. According to Section 7 (s) or Appendix E (b), the vector divT can be expanded as Appendix H: Expansion of ( divT ) 285 divT = Σa(divT)'a ea (divT)'a = [(divT)(e)]a // divT in the en basis From Appendix F (a), adjusted from Picture C to Picture A context (primes on ∂ ), Γ 'c ab = – Rbi (∂'aRc i) Γ 'a bn = – Rni (∂'bRa i) // b →n then a→b then c→a Γ 'b bn = – Rni (∂'bRb i) => (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 The conclusion is that divT = Σa[(divT)(e)]a ea [(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. (c) Expansion of divT on e n by Method 2: Use brute force Start with the known expansion of divT in Cartesian x-space divT = Σi [divT]i ui = Σi (∂jTij) ui Compute (∂jTij) = (Ra j∂'a)( Rb'i Rc'j T 'b'c') = Ra 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' = ( Ra 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 Then use the same ui expansion as in Appendix G (c), ui = Σe Rn i en Appendix H: Expansion of ( divT ) 286 Combining one gets, 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 = { δ n 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 where [(divT) (e)]n = (∂'a T 'na) + Ra j (∂'a Rc'j) T 'nc' + Rn i (∂'a Rb'i) T 'b'a From Appendix F (b) item 3 one has (Picture C1 → Picture A so primes on ∂'s), (∂' aRdn) = – R en Rdm (∂'aRe m) (∂'aRc'j) = – Rej Rc'm (∂'aRe m) // d→c', n→j (∂'aRb'i) = – Rei Rb'm (∂'aRe m) so then [(divT)(e)]n = (∂'a T 'na) + Ra j (∂'a Rc'j) T 'nc' + Rn i (∂'a Rb'i) T 'b'a = ( ∂' 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 = ( ∂' a T 'na) – Rc'm (∂'aRa m) T 'nc' – Rb'm (∂'aRn m) T 'b'a 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' . Replace summation indices m →i, a→b, = ( ∂'b T 'nb) – Rb'i (∂'bRn i) T 'b'b – Rc'i (∂'bRb i) T 'nc' . 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 Appendix H: Expansion of ( divT ) 287 This is seen to match the result of Method 1 of the pr evious section. The brute force method is in fact not too bad and requires no explicit use 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 shown above, still with all down-tilt R ma trices, but the index structure is different. This other form appears when one does the "brute force" method starting with ( ∂jTij) instead of with (∂jTij). Of course in Cartesian space these two objects must be the same. (d) Adjustment for T expanded on ( e^i⊗e^j) and divT expanded on e^a In the above, it has been assumed that T ij is a rank-2 tensor so that, in the notation of Appendix E, T = ΣijTij(ui⊗uj) = ΣijT 'ij(ei⊗ej) 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) One then has [T (e^)]ij = h'ih'jT 'ij => T 'ij = h'i-1 h'j-1 [T(e^)]ij 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 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] = 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 } = h' a * { ∂ 'b (h'a-1 h'b-1) [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) – Rni (∂'bRb i) h'a-1 h'n-1 [T(e^)]an } 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) Appendix H: Expansion of ( divT ) 288 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 } 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 will specialize to orthogonal systems so the last form above 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 Even for an orthogonal curvilinear coordinate system , the form of this tensor divergence is amazingly complicated. Here it is expressed in terms of the down-tilt R matrix and the scale factors and as usual repeated indices are summed. (e) 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, Appendix H: Expansion of ( divT ) 289 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) • For polar coordinates P1 and P2 are given by the first two lines above with the last terms set to 0. These polar results then agree with Lai p58 (2.33.32,33). • spherical coordinates (where 1,2,3 = r,θ ,φ) : The expressions above agree with Lai p 65 (2.35.33,34,35). Appendix I: Expansion of ( @B) 290 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates This appendix assumes the usual curvilinear coordinates context, Picture B, . 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 In continuum mechanics, it appears for example in the Navier/Cauchy equation which describes the small vector displacement u field in an isotropic elastic solid, ρo∂t2u = ρ0B + (λ+μ)∇e + μ ∇2u e = div u = dilatation // Lai p 216 (5.6.9) 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) where ρ is the mass density and p is pressure. (a) The first method : a review In Section 13 (c) it was 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 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) Appendix I: Expansion of ( @B) 291 and this is frequently done (see examples cited above). 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) . Since [@B]n agrees with ∇2(Bn) in Cartesian coordinates, and since we know how to write div, grad and curl in curvilinear coordinates (Sections 9,10,12 or Se ction 15), the right hand side of the above equation provides our "first method" of writing @B in curvilinear coordinates. In Section 15 (g) it was shown that the proper "tensorization" of th e above equation is given by (@B)n = (Bj ;j);n – g-1/2εnab(g-1/2εbdeBe;d);a @B = (@B)n un 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 where (B'j ;j) = div B. The object @B is a normal vector (weight 0), assuming B is a normal vector. Doing various simplifying steps, we then arrived at the following expression 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] ) } 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 expans ion of a vector, italic font has been used for the components. As shown in various places, one has B' n = B'n/h'n en = h'n e^n B = B'nen = B'ne^n h ' n = scale factor and this then gives ( @B)'n = ∂'n{(1/ g' ) ∂ 'i(g' B'i/h'i) } @B = (@B)'n en – (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 – 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 for computation in Maple, Appendix I: Expansion of ( @B) 292 (@B)'n = (1/h'n) ∂'n{(1/ g' ) ∂'i(g' B'i/h'i)} @B = (@B)'n e^n – ( h ' n/g' ) ε'ncd ε'dab ∂'c { (1/g' ) h'd2(∂'a[h'bB'b]) } . 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 uglier (Appendix D (j) item 3) and then one would want to use the εε product as is. (b) The first method in 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^ . Maple begins in the same manner as shown earlier in Appendix G (f), the idea bei ng that this code could be used for any coordinate system, Appendix I: Expansion of ( @B) 293 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 expr ession will be used below in parsing the vector Laplacian results : The subs command used here (and more intensely below) removes the arguments of the function f for purely cosmetic reasons (see A Maple User's Guide n earby for details, operands section). The arguments are added in the first place to prevent Maple from thinking the unspecified function f is a constant. Next comes a low-budget implementation of the permutation tensor eps(a,b,c) = εabc, Appendix I: Expansion of ( @B) 294 The two terms of the above ( @B)'n expression shown at the end of s ection (a) are then duly entered, In the following code, we use ( in line with the notation shown at the star t of this section) q1_ = ( @B)'1 = (@B)r q2_ = ( @B)'2 = (@B)θ q3_ = ( @B)'3 = (@B)φ Appendix I: Expansion of ( @B) 295 See comment above about the Maple subs commands (purely cosmetic). (c) The first method in 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: ∇ 2f = (1/r2)∂r(r2∂rf) + (1/r2sinθ)∂θ(sinθ∂θf) + (1/r2sin2θ)∂φ2f = 1 + 2 3+4 5 1 2 3 4 5 1 2 5 3 4 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 ] 3 4 5 1 2 Therefore Appendix I: Expansion of ( @B) 296 (@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 ] 5 3 4 1 2 (@B)φ = ∇2(Bφ) - (1/r2) [csc2θBφ - 2cscθ∂φBr - 2cotθ cscθ∂φBθ ] 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 ] 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 Once again, written out in gory detail, these three expressions are (in the same order r, θ,φ) : and each expression contains nine terms. 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. Appendix I: Expansion of ( @B) 297 (d) The second method : Part I In the first method, described in section (a) 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 . At the end of Section 15 (b) it was noted that th ere is an alternative tensorization , namely (@B)n = B'n;j ;j . These two tensors must be the same since the tensorization of a Cartes ian form equation is unique, but it is not so easy to show. Be that as it may, our "second method" is to use this B'n;j ;j tensorization to compute once again the components of the vector Laplacian. Appendix F (i) has among its examples the followi ng covariant derivative of a rank 2 tensor, B ab ;α ≡ ∂α Bab + Γa αn Bnb + Γb αn Ban and since B a;b is a rank-2 tensor one can write (indi ces are substituted in the second line) B a;b ;α ≡ ∂α Ba;b + Γa αk Bk;b + Γb αk Ba;k Bn;j ;j ≡ ∂j Bn;j + Γn jk Bk;j + Γj jk Bn;k // Γ'j jk = (1/ g ) ∂k(g ) as in App F (d) . Another example in that Appendix shows that (the lo wer three lines are index substitutions of the first) Ba;α = ∂α Ba + gαbΓa bs Bs Bn;j = ∂j Bn + gjbΓn bs Bs Bk;j = ∂j Bk + gjbΓk bs Bs Bn;k = ∂k Bn + gkbΓn bs Bs Therefore, Bn;j ;j = ∂j[∂j Bn + gjbΓn bs Bs] + Γn jk[∂j Bk + gjbΓk bs Bs] + Γj jk[∂k Bn + gkbΓn bs Bs] Combining the second last term with the first gives Bn;j ;j = [∂j ∂j Bn + Γj jk(∂kBn )] + ∂j(gjbΓn bs Bs) + Γn jk[∂j Bk + gjbΓk bs Bs] + Γj jk[gkbΓn bs Bs] 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] Appendix I: Expansion of ( @B) 298 The first two terms 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) in the sense that we earlier wrote ( Section 15 (e) ) , ∂'j∂'jf ' + (1/ g' ) ∂ 'k(g' ) ∂'kf = (1/ g' ) ∂ 'k [g' (∂'kf ) = lap (f) Therefore 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) + Extra Terms So we begin to see why the scalar Laplacian of a B component appears in ( @B)n. (e) The second method : Part II Unfortunately, we don't want to see lap (B'n), we want to see lap( B'n) ! Consider then, lap (B'n) = lap ( B'n/hn) = [∂'j ∂'j (B''n/hn) + (1/ g' ) ∂'k(g' ) (∂'k (B''n/hn) )] and one computes the pieces as follows: ∂ 'j ∂'j (B'n/hn) = ∂ '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) ∂'k (B'n/hn) = [(∂'kB'n) h-1 n + B'n(∂'k h-1 n) ] so that 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) + 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 Appendix I: Expansion of ( @B) 299 The conclusion so far is that ( @B)n = Bn;j ;j = lap(B'n) + Extra Terms = h-1 n lap(B'n) + Other Terms + Extra Terms As before, our interest is with the expansion @B = (@B)'n e^n where then (@B)'n = lap( B'n) + h'n [Other Terms + Extra Terms] 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. The Terms shown are these, just quoting from above, 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] 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) 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 and "lowering" the differential operators appropria tely, 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 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 (f) The second method in spherical coordinates: Maple speaks again In method 1, there was no need in the Maple calculation for the affine connection object , Γ 'd ab = (1/2) g'dc [ ∂'ag'bc + ∂'bg'ca – ∂'cg'ab] 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)] . 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 very sparse , and this explains why we are not going to be simply 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 : Appendix I: Expansion of ( @B) 300 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 To maintain generality, however, we let Maple compute Γd ab = G(d,a,b) from the first equation above, so Γ d ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] The Extra Terms are then entered, as shown above : (continued on next page) Appendix I: Expansion of ( @B) 301 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 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) OT1 OT2 OT3 The final results are then generated : Appendix I: Expansion of ( @B) 302 These are seen to match the unnumbered terms in the q1_,q2_,q3_ expressions of section (c) above, those terms which get added to the scalar Laplacian contribution. Quoting from method 1 above, ( @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 ] (g) 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^ . The scalar Laplacian of an unspecified function f : ∇ 2f = (1/r)∂r(rf) + (1/r2)∂φ2f + ∂z2f = 1 + 2 3 4 1 2 3 4 The components of the vector Laplacian are found by the "first method" to be Appendix I: Expansion of ( @B) 303 1 2 4 3 Therefore, (@B)r = ∇2(Br) - Br/r2 - 2∂φBφ/r2 . 3 4 1 2 Therefore, ( @B)φ = ∇2(Bφ) - Bφ/r2 + 2∂φBr/r2 . 4 3 1 2 Therefore, ( @B)z = ∇2(Bz) as befits a component which is Cartesian. In this manner, we end up with 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 Appendix I: Expansion of ( @B) 304 Once again, written out in gory detail, these three expressions are (in the same order r, φ,z) : The "second method" produces these results for the terms which are added to the scalar Laplacian, and these are seen to agree with the unnumbe red terms in q1_, q2_ and q3_ shown above. In cylindrical coordinates the Γ object is even sparser than in spherical coordinates. One has Appendix I: Expansion of ( @B) 305 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 Γr = ⎣⎢⎡ ⎦⎥⎤ 0 0 0 0 -r 0 0 0 0 Γφ = ⎣⎢⎡ ⎦⎥⎤ 0 1/r 0 1/r 0 0 0 0 0 Γz = ⎣⎢⎡ ⎦⎥⎤ 0 0 0 0 0 0 0 0 0 so only 3 of 27 components are non-vanish ing. Recall from Appendix F (a) that Γ just describes how the tangent base vectors move as x' moves, ( ∂ 'jen(x')) = Γ ' k jn(x') ek(x'), and cylindrical coordinates is just polar coordinates with z tacked on so the tangent base vectors don't move much, e3 = z^ not at all. 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 We take this as a useful prototype equation to work with for two reasons. First, it contains our object of interest, which is the gradient of a rank-2 tensor. Second, this equation plays a role in the continuum mechanics of non-Newtonian fluids as discussed below in section (f). One can define, in Cartesian coordinates, a ( ∇T) object : (∇T) ij k ≡ ∂kTij so that d tTij = ∂tTij + (∇ T)ij k vk . 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 (b) 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 k vk 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 . Appendix I: Expansion of ( @B) 306 In our usual formalism, x'-space is the space of some generic curvilinear coordinates x'n (not necessarily orthogonal) and then the above equation tells us the form taken by the time derivative equation in curvilinear coordinates, and it remains only to compute the objects ( ∇T')ij k. 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 dtT'ij = ∂tT'ij + (∇ T)'ijk v'k // curvilinear coordinates and then we can deal with the pure covariant tensor components ( ∇T)' ijk . Appendix J: Expansion of ( ∇T) 307 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor) This appendix assumes the usual curvili near coordinates context, Picture B (a) Total time derivative as prototype equation 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 We take this as a useful prototype equation to work with for two reasons. First, it contains our object of interest, which is the gradient of a rank-2 tensor. Second, this equation plays a role in the continuum mechanics of non-Newtonian fluids as discussed below in section (f). One can define, in Cartesian coordinates, a ( ∇T) object : (∇T) ij k ≡ ∂kTij so that d tTij = ∂tTij + (∇ T)ij k vk . 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 (b) 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 k vk 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 . In our usual formalism, x'-space is the space of some generic curvilinear coordinates x'n (not necessarily orthogonal) and then the above equation tells us the form taken by the time derivative equation in curvilinear coordinates, and it remains only to compute the objects ( ∇T')ij k. Appendix J: Expansion of ( ∇T) 308 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 and then we can deal with the pure covariant tensor components ( ∇T)'ijk . (b) Computation of components ( ∇T)'ijk The covariant derivative is discussed in Appendix F (g,h,i). We shall define a true tensor object ( ∇T)abc as Tab;α so that ( see App F, J=2 example with B ab;α, take B→T and α→c ) (∇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 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 Cart esian x-space (first line above) the basis vectors are the fixed un which don't change, so Γc ab ≡ 0. In curvilinear x'-space, Γ'c ab ≠ 0. Since T ab;c transforms as a true rank-3 tensor, its defining equation is "covariant" (Section7 (u)) so that in x'-space the equation 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, 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 above 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] // Appendix F (d), g' = metric tensor and then we know how to write the time de rivative equation in any curvilinear coordinates d tT'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 In Appendix I (b) we show Maple code to compute the covariant metric tensor g' ij from the curvilinear coordinates defining equa tions (such as x = rsin θcosφ for sphericals). Then Appendix I (f) shows the extra code for computing g'ij and Γ 'd ab . The reader can then add a few ex tra lines to have Maple compute the Appendix J: Expansion of ( ∇T) 309 desired (∇T)'abc using the equation (*) above. Later in this appendix we shall use Maple to compute certain related quantities which we can then verify against a known source. (c) 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 Appendix E (b) one can expand the rank-3 tensor Tab;c in various ways. The first line below shows expansions 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 As usual, up and down index positions are the same for the first line in Cartesian space, but are significant on the second line. 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 curvilinear x'-space. So, we now ha ve three different notations for the expansion components: Tij;k = [(∇T)(u)]ijk = (∇T)ijk T'ij;k = [(∇T)(e)]ijk = (∇ T)'ijk . Just to fill things out, here are the corre sponding expansions for rank-2 tensor T, T = Σij Tij ui⊗uj = Σij [T(u)]ij ui⊗uj = Σij [T(u)]ij ui⊗uj T = Σij T'ij ei⊗ej = Σij [T(e)]ij ei⊗ej = Σij [T(e)]ij ei⊗ej Tij = [T(u)]ij T'ij = [T(e)]ij and then for a rank-1 tensor v, v = Σi vi ui = Σi [v(u)]i ui = Σi [v(u)]i ui v = Σi v'i ei = Σi [v(e)]i ei = Σi [v(e)]i ei vi = [v(u)]i v'i = [v(e)]i (d) 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 Appendix J: Expansion of ( ∇T) 310 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 in Appendix E (h). We start with the expansion given above, ∇T = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek 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 . Since the e^n don't transform as vectors, and since the [(∇T)(e^)]ijk are not tensorial components, we let the indices on [(∇T)(e^)]ijk arbitrarily be "down". Putting them up gives a false impression of a covariant matching index tilt. 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 where we follow our convention that the components of unit-base-vector tensors are written in script. (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 with the definition T'ij ≡ [T(e^)]ij = h'i h'j [T(e)]ij = h'i h'j T'ij . Finally, for a vector v, v = Σi [v(e^)]i e^i where [v(e^)]i = h'i [v(e)]i with the definition v'i ≡ [v(e^)]i = h'i [v(e)]i = h'i v'i . In terms of the scripted unit-base-vector components, our three expansions are: Appendix J: Expansion of ( ∇T) 311 ∇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 . 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 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 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, which we shall refer to below, avoids summations in curvilinear coordinates and thus has no need for our scripted components. (e) Total time derivative equation written in unit-base-vector curvilinear components Recall from section (a) our prot otype time derivativ e equation of interest in x'-space d tT'ij = ∂tT'ij + (∇ T)'ij k v'k . How does one write this equation in terms of unit-base-vector tensor components? From section (d) we had (no implied sums here) T'ij = h'i h'j T'ij v'k = h'k v'k so the above d t equation can be processed as follows: dtT'ij = ∂tT'ij + (∇ T)'ij k v'k 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 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 d t T'ij = ∂t T'ij + Q'ijk v'k Q' ijk ≡ h'i h'j h'k-1 (∇T)'ij k (*) Appendix J: Expansion of ( ∇T) 312 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 prop erty (see application below), it is a property of space at point x. Recall that no assumption has been made that the curvilinear coordinates are orthogonal. Section (b) 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 : (*) shows how one writes the total time derivative equation in arbitrary curvilinear coordinates. If we now assume the coordinates are orthogonal, then (∇T)' ij k = gkk'(∇T)'ijk' = hk2δk,k' (∇T)'ijk' = hk2(∇T)'ijk and then Q' ijk ≡ 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 and so for orthogonal coordinates (*) may be written dt T'ij = ∂t T'ij + (∇T)'ijk v'k (∇T)'ijk ≡ h'i h'j h'k (∇T)'ijk . // orthogonal We shall assume from now on that the curvilinear coordinates are orthogonal. As a reminder, here is what the above equation sa ys if i = j = 1, using the notation scheme just described above : d t T'11 = ∂t T'11 + (∇T)'11k v'k dt Trr = ∂t Trr + (∇T)rrr vr + (∇ T)rrθ vθ + (∇ T)rrφ vφ (f) Shorthand notations and a continuum mechanics application Consider our original Cartesia n time derivative equation, dtTij = ∂tTij + (∇ T)ij k vk . One could regard (∇T) ij k as the components of a vector (∇ T)ij labeled by fixed values i and j, so that [(∇ T)ij]k = (∇T)ij k . Then one can say d tTij = ∂tTij + (∇ T)ij• v . The next step is to suppress the ij labels, since the equation above is true for any i and j, Appendix J: Expansion of ( ∇T) 313 dtT = ∂tT + (∇T) • v . This is only a shorthand notation , no precision justification is required. We can do the same thing to the equation written in curvilinear coordinates d t T'ij = ∂t T'ij + (∇T)'ijk v'k [(∇T)'ij]k = (∇T)'ijk dt T'ij = ∂t T'ij + (∇T)'ij • v'k d t T' = ∂t T' + (∇T)' • v' and then we have d tT = ∂tT + (∇T) • v dt T' = ∂t T' + (∇T)' • v' which gives the nice impression that the equation is written in a coordinate-independent manner. In the book of Lai, the • is omitted and the shorthand notations are written dtT = ∂tT + (∇T) v dt T' = ∂t T' + (∇T)' v' ( * ) 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 or d tTij = ∂tTij + (∇T)ijk vk dt T'ij = ∂t T'ij + (∇T)'ijk v'k An application of our total time derivative equation app ears in the first line of Lai p 470 , DA1/Dt = ∂A1/∂t + (∇ A1) v 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. In the nota tion described above, th is would be written dtA'1 = ∂tA'1 + (∇A1)' v' which is just an example of equati on (*) above. The detailed meaning is dt (A'1)ij = ∂t (A'1)ij + (∇A1)'ijk v'k Appendix J: Expansion of ( ∇T) 314 and for i = j = 1 this says (spherical coordinates) , d t[(A1)rr] = ∂t[(A1)rr] + (∇A1)rrr vr + (∇ A1)rrθ vθ + (∇ A1)rrφ vφ . The tensor A 1 is the first "Rivlin-Ericksen tensor" associ ated with the flow of non-Newtonian fluids (rheology). The A i tensors, briefly mentioned in Appendix K (d), 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) where (∇v) is the gradient-of-vector object treated in 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 equati on 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 (see table in next section). So, our next task is to compute the ( ∇T)'ijk which appears in our generic equation above, dt T' = ∂t T' + (∇T)' v' dt T'ij = ∂t T'ij + (∇T)'ijk v'k (g) Maple computation of the (∇T)'ijk components in spherical coordinates From section (e) we have, for arbitrary curvilinear coordinates, ( ∇T)'ijk ≡ h'i h'j h'k (∇T)'ijk and from section (b) , (∇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] . g' = metric tensor for x'-space But we are now assuming only orthogonal coordinates, so g'ab = δa,bha2 g'ab = δa,bha-2 => ( ∇T)'ijk = g'ii'g'jj'g'kk'(∇T)'i'j'k' = (h'i h'j h'k)-2 (∇T)'ijk and then ( ∇T)'ijk = (h'i h'j h'k)-1 (∇T)'ijk . Appendix J: Expansion of ( ∇T) 315 We want the result expressed in terms of the T'ij and not the T' ij, so recall 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) Then (∇T)'abc = ∂'cT'ab – Γ'n acT'nb – Γ'n bcT'an (∇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) = 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) where we break the ∂'k term in two pieces for Maple technical reasons. So the Maple program will compute ( ∇T)'ijk = (h'i h'j h'k)-1 (∇T)'ijk 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 The Maple code is similar to that reported in Ap pendix I (b) and (f). The code computes the affine connection from the metric tensor, Then the terms shown above are entered ( T'ij = Te[i,j], h' i = hp[i], Γ 'd ab = G(d,a,b), etc.) Appendix J: Expansion of ( ∇T) 316 The terms are then added, multiplied by (h' i h'j h'k)-1, and then displayed, and here are the resulting values for ( ∇T)'ijk (for example, ( ∇T)'112 = (∇T)rrθ ) Appendix J: Expansion of ( ∇T) 317 The strange " symbols in the above table should be ignored, just a Maple artifact. These results agree with the spherical coordinates 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 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 and we quote from Appendix H Appendix J: Expansion of ( ∇T) 318 . The Lai method of computing ( ∇T)'ijk is different from ours. Lai uses an "affine connection" which is geared to the unit tangent base vectors, which in our notation would be written ∂'je^i = Γ'(Lai) ijke^k , // Lai p 502 (8A.12) whereas "the true" affine connection measures th e change of the full tangent base vectors, (∂'jen) = Γ'k jn ek . // Appendix F (a) It is not hard to show that the connectio n between these two affine connections is, Γ' (Lai) ijk = h'i-1 [h'kΓ'k ji– ∂'j(h'i) δi,k] = h'k-1[– h'i Γ'i jk + δi,k(∂'jh'i) ] where the second form can be obtained from the first using this identity from Appendix F (c) (∂ cgab) = – [gan Γ b cn + gbn Γa cn] . The object Γ 'd ab in spherical coordinates has 9 of its 27 co mponents non-zero, as shown at the start of Appendix I (f), but Γ 'd ab = Γ 'd ba means there are only 6 distinct non-vanishing components. Correspondingly, the Γ'(Lai) ijk object has 6 of its 27 components non-zero (Lai p 503 (8A.14)). Lai uses M ijk = (∇T)'ijk and in Lai notation the expansion is ∇T = Σijk Mijk e^ie^je^j (p 501) which provides an example of the polyadic notation described in our Appendix E (c) ( e^ie^je^j= e^i⊗e^j⊗e^k). (h) Maple computation of the (∇T)'ijk components in 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 ) Appendix J: Expansion of ( ∇T) 319 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 at the end of Appendix I (g), only 2 of which are distinct. Correspondingly Γ'(Lai) ijk has only 2 of 27 non zero as shown in Lai p 502 (8A.13). It should be emphasized that this same simp le Maple code can be used to compute the ( ∇T)'ijk for any system of orthogonal curvilinear coordinates in an y number of dimensions N. And the code implied at the end of section (b) and in the middle of section (e) with Q' ijk computes ( ∇T)'abc and (∇T)'ijk for non-orthogonal as well as orthogonal coordinates. Appendix K: Deformation Tensors 320 Appendix K: Deformation Tensors in Continuum Mechanics Tensor-like objects appear everywhere in continuum m echanics. As was noted in Section 2 (k), and later in Appendix E (i), continuum mechanics texts gene rally refer to all objects having indices as being "tensors", whether or not these objects actually transf orm as tensors with respect to some underlying transformation. The most commonly appearing tens ors have two indices and are just 3x3 matrices associated with 3D space. In this category, there are several kinds of stress tensors, and many kinds of strain and 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 app lied 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 section 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 pr ovided. Section (c) 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 discusse d in Section 7 (u). Similarly, sections (d) and (e) consider covariant constitutive equations for fluids (a) A Preliminary Deformation Flow Picture Recall the general nature of the Pictures appearing through this document, such as The arrow represents an underlying generally non-linear transforma tion between x-space and x'-space given by x' = F(x), while R( x) (S = R-1) is the linearized-at-point- x version of the transformation which defines the notion of a vector as in d x' = R d x (see Section 2). Since F will have another meaning below, we shall change the transformation name so that x' = F(x). Now consider 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 below. The linearized R matrix objects asso ciated with the two transformations are shown to the right and are given the names R t0 = F on the bottom and R t= Ft on the top. The transformation x' = Ft(x,τ) is a spatial coordinate transf ormation only, the time coordinate τ is a parameter. In the picture below, time increases in the upward direction, so τ > t > t0. Appendix K: Deformation Tensors 321 Fig 1 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 "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 when 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, Fig 2 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 (3.18.13) Appendix K: Deformation Tensors 322 where the matrix F ij is called "the deformation gradient". It is also known as "the deformation gradient tensor" even though it is not a "tensorial tensor" with respect to any identifiable transformation. In Section 5 (o) the above equation was identified with d x' = R( x)dx with R(x) here being F(x,t). The picture above is then the second figure shown in Section 2 (left-ri ght switched). 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 , one finds that F ij = (∂ xi/∂Xj) ≡ ∂jXi or F = (∇ x) // Lai p 105 (3.18.4) where the gradient ∇ is with respect to X, so it is really ∇ = ∇(X). Thus the name "deformation gradient". [ 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 a 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 1 above. It is entirely analogous to the lower flow, but names are changed. We get from the lower flow to th e upper flow by making these replacements : t0 → t d X → dx Fto(X,t) → Ft(x,τ) S0 → S Fto(X,t0) = 1 → Ft(x,t) = 1 t → τ d x → dx' Fto(X,t) → Ft(x,τ) S → S' F to(X,t0) = 1 → Ft(x,t) = 1 The equations corresponding to those shown above are d x' = Ft(x,τ) dx dx' i = (Ft)ijdxj // Lai p 457 (8.7.2) 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) where the gradient ∇ is with respect to x, so it is really ∇ = ∇(x). 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 ". Appendix K: Deformation Tensors 323 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 . 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(X,t) - Ft0-dt0 (X,t) ] / dt0 , but this goes against the grain of the idea that X is a material coordinate at 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 makes the differentiation process clearer , and will be used below for that purpose. The frames of reference and the cameraman Each of the three frames of reference S 0, S and S' in Fig 1 has its own set of orthonormal basis vectors which we are completely free to set in any manner. 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 ma in concern will be with the dumbbells like d X, dx and d x', 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, d x = dx1e^1(S) + dx2 e^2(S) + dx3 e^3(S) d X = dX1e^1(S0) + dX2 e^2(S0) + dX3 e^3(S0) Once the axes are chosen, the value of the F ij are determined, for example, F 12 ≈ (dx1)/(dX2) . Appendix K: Deformation Tensors 324 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 just inserted here "for interest". The expansions use the 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) , where we write the expansion in both direct product and matrix form as in Appendix E. This is a "mixed basis expansion" as discussed in Appendix E (i). Consider the application of this expansion to d X : { Σij Fij [e^i(S)] [e^j(S0)]T } 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)] = Σij Fij ΣkdXk [e^i(S)] δj,k = [ Σij Fij dXj] e^i(S) = [ d x i ] e^i(S) // using the fact that F d X = dx = d x . 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. • 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: Ft = 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) The verification is similar to the above, Appendix K: Deformation Tensors 325 { Σij (Ft)ij [e^i(S)] [e^j(S)]T } 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' . 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. (b) A More Complicated Deformation Flow Picture Consider now this flow picture, Fig 3 There is much to be said about this drawing. The left side is the same as in Fig 1 shown above. The picture is simplified in that it shows only the linearized R-type transformations like F and not the full transformations like F, and these R-type matrices are now la beled right on the transformation arrows. Appendix K: Deformation Tensors 326 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 Here x0 is a randomly selected center 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 the picture 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 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 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 F t*Q(t) = Q( τ)Ft => F t* = Q(τ)FtQ(t)T or F t*(x*,τ*) = Q(τ) Ft(x,τ)Q(t)T t* = t 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! which is the matrix form for the transformation of a rank-2 tensor as shown, for example, in Section 5 (f). But we have just seen that F* = Q(t) F so the required Q(t)T on the right is missing. We conclude Appendix K: Deformation Tensors 327 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), wh ich is very different frame 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 (u) 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 te nsor = the Piola deformation tensor C = FTF = the right Cauchy-Green deformation tensor = the Finger deformation tensor F = RU = VR R = rotation U,V = symmetric positive definite Tensors B and C are defined simply 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 114 (3.21.1,2,4) 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 = FF T = (VR)(VR)T = VRRTVT = VVT = VV = V2 // Lai p 125 (3.25.1) C = F TF = (RU)T(RU) = UTRTRU = UTU = U2 // Lai p 115 (3.22.1,2) 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 . Appendix K: Deformation Tensors 328 C* = F*TF* = (QF)T(QF) = FTQTQF = FTF = C => C* = C 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* Consider, F* = QF = Q(RU) = (QR) (U) = R*U* 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 . Next write F* = QF = Q(VR) = (QVQ T)(QR) = V* R* 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* . So here is a summary for our tensors of interest. Only two of the five deformation tensors actually transform as tensors. The refere nces 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 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 . Appendix K: Deformation Tensors 329 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 W t*(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 . A tensor W t which transforms as a rank-2 tensor (is objectiv e) with respect to rotation Q(t) must satisfy the rule 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 Next comes C t ≡ FtTFt : Ct* = 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) What about the left and right relative stretch tensors V t and Ut? F t= RtUt = VtRt F t* = Rt*Ut* = Vt*Rt* Consider, F t* = Q(τ) Ft QT(t) = Q(τ) RtUtQT(t) = [Q(τ ) RtQT(t)] [Q(t)U tQT(t)] = R t*Ut* . Since [Q( τ) R tQT(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 Appendix K: Deformation Tensors 330 Rt* = Q(τ) RtQT(t) and U t* = Q(t)U tQT(t) // U t is a rank-2 tensor Finally, write F t* = Q(τ) FtQT(t) = Q(τ)VtRtQT(t) = [Q(τ )VtQT(τ)] [Q(τ) RtQT(t)] = V t* Rt* By the same argument used severa l times above, we conclude that R t* = Q(τ)RtQT(t) and V t* = Q(τ)VtQT(τ) // V t is not a rank-2 tensor, τ ≠ t The rule for transforming R t is the same as found a few lines above. 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 ) / / L a i ( 8 . 1 3 . 8 ) Notice that among the "normal" tensors, B and V are objective, whereas among the "relative tensors" it is C t and Ut that are objective. All the other tensors are "non-objective". (c) Form of a solid constitutive equa tion 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 section (b) 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, so is any power of B, B* 2 = [QBQT][QBQT] = Q B2QT etc. 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). Appendix K: Deformation Tensors 331 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 B2. 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 (u), 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 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 only covari ant 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. (d) Some fluid constitutive equations It was noted in section (b) that the relative deformation tensors are appropria te when one is interested in time derivatives of the tensors. It was also noted that 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 , 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) These A n(x,t) tensors appear in various models of "non-Newtonian" fluid behavior, the general study of which is called rheology, based on the Greek word for a cu rrent flow (a rheostat controls electric current), Appendix K: Deformation Tensors 332 // 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. 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) 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) Here v is the fluid velocity vector and d t = d/dt = D/Dt. Again, ( ∇v) is the subject of Appendix G. (e) 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 Clarification of the above equation One can think of the above equation T* = QTQT as involving operators in Hilbert Space, as outlined in Appendix E (g). In the upper part of Fig 3 above we show four different frames of reference called S, S*, S' and S'* each of which has its ow n 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, here 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 Appendix K: Deformation Tensors 333 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 Section 2 (i) 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 where the argument is appropriate to the space of interest. 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) = Σ ab Qab(t) ua ⊗ ub . 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 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 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) Q T + [ (dQ/dt) T QT + Q T (dQ/dt)T ] , so there are two extra unwanted terms. Just as B = FF T 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. Appendix K: Deformation Tensors 334 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). J t(τ) ≡ RtT(τ) T(τ ) Rt(τ) // J t(x,τ) ≡ RtT(x,τ) T(x,τ) Rt(x,τ) where R t(τ) is the rotation which appears above in section (b), where we had (showing τ arguments), R t*(τ) = Q(τ) Rt(τ)QT(t) . The tensor R t(τ) is non-objective due to appearance of Q( τ) instead of Q(t) on the left (see comments on Wt above). 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 (see prev section) Rt*(τ) = Q(τ) Rt(τ)QT(t) // how R t(τ) transforms, where F t(τ) = Rt(τ)Ut(τ) Jt(τ) ≡ RtT(τ) T(τ ) Rt(τ) // definition of J t(τ) in frame S Jt*(τ) ≡ Rt*T(τ) T*(τ ) R*t(τ) // corresponding J t* in frame S* 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) . (*) 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. Appendix K: Deformation Tensors 335 Consider now the limit of (*) as τ → t. One finds, Jt(τ) ≡ RtT(τ) T(τ ) Rt(τ) Jt(t) ≡ RtT(t) T(t) R t(t) = 1 T(t) 1 = T(t) and J*t(τ) ≡ R*tT(τ) T*(τ ) R*t(τ) J*t(t) ≡ R*tT(t) T*(t) R* t(t) = 1 T*(t) 1 = T*(t) . In this limit, the corotation R t-1(τ) has come to a halt, and (*) becomes a statement that T is objective. More interestingly, we can apply ∂ τn = ∂n/∂τn to both sides of (*) to get d τn Jt*(τ) = Q(t)[ d τn Jt(τ) ] QT(t) . Taking the limit τ→t then gives [dtn Jt*](t) = Q(t) [d tn Jt](t) QT(t) which says that d tnJt are all objective tensors. And in particular, for n = 1, (d tJt)* = Q(t) (d tJt) QT(t) , 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) 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) 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) We suppress the t subscript on J L just to avoid having to write (J L)t(τ). In the table at the end of section (b) one sees that F t transforms the same way R t does, so one can repeat the above analysis 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, J L(τ) ≡ FtT(τ) T(τ ) Ft(τ) // = T∪ n Appendix K: Deformation Tensors 336 and these are the "Oldroyd lower convected derivatives " (Lai p 485, called 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 484 (8.19.21) The Oldroyd Upper convected derivatives Finally, consider again the non-objective way that F t transforms (table end of section (b)) F t*(τ) = Q(τ)Ft(τ)QT(t) => (F t-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 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, J U(τ) ≡ Ft-1(τ) T(τ ) Ft-1,T(τ) // = T^n 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) 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 Appendix K: Deformation Tensors 337 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 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 Section 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 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 created 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 . Appendix K: Deformation Tensors 338 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 that transfor ms as in Section 5 (f) 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 . With this long-winded introduction, we now consider the corotational stress tenser J t(τ) from above, J t(τ) ≡ RtT(τ) T(τ ) Rt(τ) Since R t(τ) is a rotation, R tT(τ) = Rt-1(τ), so we have, suppressing τ, Jt = Rt-1T Rt . 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 given above a J U ≡ Ft-1 T (Ft-1)T 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 we have J L ≡ FtT T Ft and this cannot be written in the form J L = R-1 M (R-1)T so this does not fit into our interpretive Appendix K: Deformation Tensors 339 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, JU ≡ Ft-1 T Ft-1T // upper JL ≡ FtT T Ft // lower 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 per Section 5 (h) for vectors. We can interpret the above two equations in this manner J ≡ F t-1 T Ft-1T // upper J¯ ≡ FtT T¯ Ft // lower which we compare with the first equations in Section 5 (f) where we change generic matrix name M to T, T' = R T R T // contravariant rank-2 tensor transforms this way T¯' = ST T¯ S // covariant rank-2 tensor transforms this way . 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 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 In standard notation the two equations J ≡ F t-1 T Ft-1T // upper J¯ ≡ FtT T¯ Ft // lower become Jij = (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 and this explains the meaning of the words "upper" and "lower" in respect to the Oldroyd objects. See footnote on Lai page 485. References 340 References 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, which seems more comprehensible than LRK. P. Lucht. The most recent version of the doc ument you are reading and related documents are downloadable at http://user.xmission.com/~rimrock . If not there, search on or the document title. 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). This is the place to go to find explicit expressions for differential operators in specific curv ilinear coordinate systems (and very much more). References 341 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 (u) above). 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)