tensor analysis and curvilinear coordinates, appendices

1 Tensor Analysis and Curvilinear Coordinates: Appendices Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: Oct 21, 2012 These Appendices belong with the document "Tensor Analysis and Curvilinear Coordinates". References and a summary of the Appendices appear there. 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, and u se of a wide Bookmarks pane is recommended. Appendix A: Reciprocal Base Vectors the Hard Way ........................................................................ 4 (a) Definition of E n............................................................................................................................... 4 (b) Simpler notation ........................................................................................................... ................... 5 (c) Generalized Cross Product of N-1 vectors of dimension N ............................................................ 5 (d) Missing Man Formation ...................................................................................................... ............ 7 (e) Apply this Notation to E.................................................................................................................. 7 (f) Compute E m • en............................................................................................................................... 8 (g) Compute E n • Em............................................................................................................................. 9 (h) Summary of relationship between the tangent and reciprocal base vectors .................................... 9 (i) Another Cross Product Notati on and another expression for E ..................................................... 10 Appendix B: The Geometry of Pa rallelepipeds in N dimensions..................................................... 11 (a) Preliminary: Equation of a plane in N dimensions....................................................................... 11 (b) N-pipeds and their F aces in Various Dimensions ......................................................................... 12 The 1-piped ..................................................................................................................................... 12 The 2-piped ..................................................................................................................................... 12 The 3-piped ..................................................................................................................................... 14 The N-piped .................................................................................................................... ................ 15 (c) The question of inward versus outward facing normal vectors..................................................... 16 (d) The Face Area and Volume of N-pipeds in Various Dimensions ................................................ 17 The 2-piped ..................................................................................................................................... 17 The 3-piped ..................................................................................................................................... 18 The 4-piped ..................................................................................................................................... 20 The N-piped .................................................................................................................... ................ 22 (e) Summary of Main Resu lts of this Appendix ................................................................................. 24 Appendix C: Elliptical Polar Coordinates ( N=2, non-orthogonal)................................................ 26 (a) Elliptical polar coordinates............................................................................................... ............. 26 (b) Forward coordinate lines................................................................................................... ............ 27 (c) Inverse coordinate lines................................................................................................... .............. 27 2 (d) Drawing a contravariant vector V in x-space: the meaning of V' n .............................................. 28 (e) Drawing a contravariant vector V' in x'-space: two "Views" ....................................................... 29 (f) Drawing the specific contravariant vector dx in x-space and x'-space .......................................... 32 (g) Study of how dx transforms in th e mapping between x-space and x'-space ................................. 32 (h) Derivation of the Jaco bian Integration Rule................................................................................ ..34 Appendix D: Tensor Densities and the ε tensor ................................................................................ 37 (a) Definition of a tensor density ............................................................................................. ........... 37 (b) A few facts about tensor densities................................................................................................. 38 (c) Theorem about Totally Antisymmetric Tensors: there is really only one: εabc......................... 40 (d) The contravariant ε tensor ............................................................................................................. 41 (e) Some facts about the ε tensor ........................................................................................................ 42 (f) The covariant ε tensor : repeat section (d) as if its weight were not known ................................. 44 (g) Generalized cross products................................................................................................. ........... 45 (h) The tensorial nature of curl B............................................................................................. ........... 45 (i) Tensor E as a weight 0 version of ε : three conventions................................................................ 46 (j) Representation of ε, εε and contracted εε as determinants............................................................ 49 (k) Covariant forms of th e previous section results ............................................................................ 55 Appendix E: Tensor Expansions: direct product, polyadic and operator notation...................... 57 (a) Direct Product Notation.................................................................................................... ............. 57 (b) Tensor Expans ions and Bases ....................................................................................................... 58 (c) Polyadic Notation .......................................................................................................................... 60 (d) Dyadic Products ............................................................................................................ ................ 61 (e) Transpose notation for dyadics............................................................................................. ......... 62 (f) Large and small dots used with dyadics..................................................................................... ....63 (g) Operators and Matrices for Rank-2 tensors.................................................................................. .64 (h) Expansions of tensors on unit tangent base vectors ...................................................................... 68 (i) Tensor expansions in a mixed basis ......................................................................................... ......76 (j) What is a tensor? .......................................................................................................... .................. 78 Appendix F: The Affine Connection Γc ab and Covariant Derivatives ............................................ 80 (a) Definition and Interpretation of Γ : Γc ab = ec • (∂aeb) = Rc i(∂aRbi) ........................................ 80 (b) Identities of the form ( ∂aRd n) = – Re n Rd m (∂aRem) ...................................................................... 81 (c) Identities of the form ( ∂cgab) = – [gan Γ b cn + gbn Γa cn] ............................................................ 82 (d) Identity: Γd ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab].................................................................... 83 (e) Picture D1 Context ......................................................................................................... ............... 85 (f) Relations between Γ and Γ ' ........................................................................................................... 86 (g) Statement and Proof of the Covariant Derivative Theorem .......................................................... 87 (h) Rule for raising any index on a covarian t derivative of a covariant tensor density....................... 92 (i) Examples of covari ant derivative expressions ............................................................................... 93 (j) The Leibniz rule for the covariant deriva tive of the product of two tensor densities..................... 96 Appendix G: Expansion of ( ∇v) in curvilinear coordinates (v = vector) ...................................... 100 (a) Continuum Mechanics motivation............................................................................................. ..100 (b) Expansion of ∇ v on ei⊗ej by Method 1: Use fact that v b;a is a tensor..................................... 100 (c) Expansion of ∇v on ei⊗ej by Method 2: Use brute force. ........................................................ 102 (d) Expansion on e i⊗ej and e ^i⊗e^j................................................................................................... 104 3 (e) Orthogonal coordinate systems .............................................................................................. .....105 (f) Maple evaluation of ( ∇v) in several coordinate systems ............................................................. 105 Appendix H: Expansion of div(T) in cu rvilinear coordinates (T = rank-2 tensor) ...................... 109 (a) Continuum Mechanics motivation............................................................................................. ..109 (b) Expansion of divT on e n by Method 1: Use fact that Tab ;α is a tensor....................................... 109 (c) Expansion of divT on e n by Method 2: Use brute force ............................................................. 110 (d) Adjustment for T expanded on ( e^i⊗e^j) and divT expanded on e^a............................................ 112 (e) Maple: divT in cylindri cal and spherical coordinates................................................................. 113 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates........................... 115 (a) The first met hod : a review............................................................................................... .......... 115 (b) The first method in spherical coordinates: Maple speaks .......................................................... 117 (c) The first method in spherical coordina tes: putting results in traditional form............................ 120 (d) The second method : Part I.......................................................................................................... 122 (e) The second method : Part II................................................................................................ ......... 123 (f) The second method in spherical coordinates: Maple speaks again............................................. 124 (g) Results for Cylindrical Coordinates from both methods............................................................. 127 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor).......................... 132 (a) Total time derivative as prototype equation ................................................................................ 132 (b) Computation of components ( ∇T)'ijk........................................................................................ 133 (c) Tensor expansions of ∇ T on the u n and en base vectors.............................................................. 134 (d) Tensor expansions of ∇T on the e ^n base vectors......................................................................... 134 (e) Total time derivative equation written in unit-base-vector curvilinear components ................... 136 (f) Shorthand notations and a con tinuum mechanics application ..................................................... 137 (g) Maple computation of the ( ∇T)'ijk components in spherical coordinates.................................. 139 (h) Maple computation of the ( ∇T)'ijk components in cylindrical coordinates ............................... 143 Appendix K: Deformation Tens ors in Continuum Mechanics ...................................................... 145 (a) A Preliminary Deformation Flow Picture.................................................................................... 145 (b) A More Complicated Deformation Flow Picture ........................................................................ 150 (c) Form of a solid constitutive equa tion involving the deformation tensor..................................... 155 (d) Some fluid constitutive equations ............................................................................................... 156 (e) Corotational and other objective time derivatives of the Cauchy stress tensor ........................... 157 Appendix A: Reciprocal Base Vectors 4 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 E n → en Rij → Ri j Sij → Si j g¯'nm → g'nm g'nm → g'nm . Introduction In Section 6 of the main text the reciprocal base vectors are defined as E n ≡ 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 5 (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, ε 1234 = 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 above 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, ( E k)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) e 2 x e3 a is missing ( E2)α ≡ det(R) εaαc(e1)a(e3)c => E2 = det(R) e 3 x e1 b is missing ( E3)α ≡ det(R) εabα(e1)a(e2)b => E3 = det(R) e 1 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 defi ne 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 6 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 vector s 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 interchange 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 7 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 thi nking 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 t he 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 notation is rescued as in (c) above. (e) Apply this Notation to E Co mpare 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 E κ for N > 2 can be written in this vector notation, E k ≡ det(R) (-1)k-1 e1 x e2 x ......x e N // ek missing; N > 2 The reciprocal base vector E k is thus orthogonal to all the tangent base vectors from which it is constructed (remember ek is missing)! For example, for N=3 the three E vectors are given by E 1 = 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 8 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 in to 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) e 1 x 3^ • e1 = 0 and also E 1•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: E m • en = δm,n . Section 5 (j) showed that e m • en = g¯'mn . The other two dot products are now considered. (f) Compute E m • en One can now compute, for general N, E k • 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 E k is orthogonal to all the en which form the generalized cross product, therefore Em • en = δm,n which is the "duality relation" discussed more generally in Section 6 (b). Appendix A: Reciprocal Base Vectors 9 (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 , E m = 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, E m = Σn An(m) en = Σn g'mn en which is to say E k is this linear combination of the ei (this is the definition used in Section 6 (a)) E k = 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 E k ≡ det(R) (-1)k-1 e1 x e2 x ......x e N // 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 reciproca l 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 Pi cture 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 e n • 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 10 (i) Another Cross Product Notation and another expression for E Go back to th e 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 = Π x i=1N-1 A(i) = Πx i A(i) where in the second form it is understood 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 = Π x 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 e N // ek missing; E k = det(R) (-1)k-1 Πx i≠k ei Once again, for N=2 the Π x symbol implies that ( ek = "missing", e 3-k = the one not missing) Πx i≠k ei = Πx i=1..2,i ≠k ei = e3-k x 3^ E k = 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 11 Appendix B: The Geometry of Parallelepipeds in N dimensions Introduction This Appendi x 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 make 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 24-1 = 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 st ated in Summary section (e). Since this is a very long section (~15 p), a reader not interested in details 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 a rbitrary plane drawn in N space which do es 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 12 r•p^ = p r = (x1, x2, .....xN) where p^ is the unit vector normal to the plane which points "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 v 1 = 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, goin g to N=2. Introduce a unit vector e2 in some arbitrary direction in R2 other than e 1 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 generator 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 a ll 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 13 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 r face2 = 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 e n 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)) E i• ej = δi,j / / Ek = Σi=12 g'ki ei , see Section 6 (a) If one interprets the e n 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: E 2 • 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, E 1 • 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 14 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 ano ther 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 w ill now be shown, lies on a plane which is parallel to that of the generator piped, but which does not pass through the origin. In N=3 dimensions, the generator 2-piped and its part ner 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 partner piped (add 4 line segments), to get 4 additional side faces. Two of these faces necessarily touch the origin, and the other tw o 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 r face3 = 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 15 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 e 1, 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 S ection 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 ano ther 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 p ass through the origin. The generator (N-1)-piped and its partner 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 ha lf do not. Each face is an (N-1)-piped. It is convenient to refer to the partner face of a pair 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: r volumeN = 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 16 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 E 1...EN as linear combinations of e1.. eN such that the following is true (see Section 6 (b)) E i• ej = δij / / Ek = Σi g'ki ei If one interprets the e n vectors as tangent base vectors, then the N vectors En are the corresponding reciprocal base vectors which are discussed in S ection 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 t hat, 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 E i • {(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 17 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 e mbark now on another long marc h to inductively arrive 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 Ar ea. The area of a 2-piped face (a line segment) is just the length of the edge which is the face, A 1 = |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, A 1 = |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 A n = |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)| | E 1| = | e2| and |det(S)| | E2| = | e1| or |E k| = |det(R)| | e3-k| k=1,2 // RS = 1 Using the N=2 trick notation from Appendix A (c), E k = det(R) (-1)k-1 e3-k x 3^ so that | E k | = | det(R)| | e3-k x 3^| = |det(R)| | e3-k| k = 1,2 since e 3-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 . Appendix B: N-piped Geometry 18 One can restate A n = |det(S)| En using the cross product notation presented in Appendix A (i): 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) = | e 1 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 Ar ea: 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 e 1 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 li es 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 A 3 = |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 A 1 = |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 A1 = |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 Appendix B: N-piped Geometry 19 In Appendix A (e) it was shown that E1 = det(R) e2 x e3 so that e 2 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, 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 A n = |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 E n = det(R) (-1)n-1 e1 x ... x e3 // en missing so |E n| = | 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 A n = |det(S)| E n = 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 Appendix 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) = | A 1 • e1 | = | A2 • e2 | = | A3 • e3 | or volume(3) = | e 2 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 20 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) = | e 3 • e1 x e2 | = | εabc (e1)a(e2)b(e3)c | = | det [ e1, e2, e3] | = | det(S) | These expressions have the same form as those of the 2-piped. The 4-piped Face Ar ea: 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 21 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 E 1 = det(R) e2 x e3 x e4 so that e 2 x e3 x e4 lines up with E 1. 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 E n = det(R) (-1)n-1 e1 x ... x e4 // en missing so |E n| = | 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 A n = |det(S)| E n = 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) = | A 1 • e1 | = | A2 • e2 | = | A3 • e3 | = | A4 • e4 | or volume(4) = | e 2 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 22 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 [ e 1, 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 Are a: 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 whic h 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 e i 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) form ula expressed in terms of these new ND vectors is therefore | εabc...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 23 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 E n = det(R) (-1)n-1 e1 x ... x eN // en missing so |E n| = | det(R) | | e1 x ... x eN | // en missing Thus, A n = |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 A n = |det(S)| E n = 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) = | e 2 x e3 x e4....eN • e1 | = ... Using ε notation one can write the first case as e2 x e3 x e4...eN • e1 = (e 1)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 [ e 1, 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 24 (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 points "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 2 N vertices as demonstrated by the inductive construction method presented above. 3. The locus of points making up the (closed) interior of an N-piped spanned by e 1...eN is given by rvolumeN = Σn=1Nαnen 0 ≤ αn ≤ 1 The tails of all the vectors e 1...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 e n are the tangent base vectors associat ed 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 dime nsional 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 25 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 followi ng, 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 [ e 1, e2, e3, ....eN] | = | εabc...x (e1)a(e2)b(e3)c....(eN)x | where one can regard the tangent base vectors as the co lumns of the linearized transformation matrix S. Appendix C: Elliptical Polar Coordinates 26 Appendix C: Elliptical Polar Coordinates ( N=2, non-orthogonal) This Appendi x is written in the develo pmental notation of Sections 1-6. (a) Elliptical polar coordinates The 2D "elliptic" coordinate sy stem has coordinate lines which are orthogonal ellipses and hyperbolas. When rotated about its two symmetry axes, this system 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 ρ a, 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 27 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 M aple 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 inver se coordinate lines (parameters a = 2 and b = 1) Appendix C: Elliptical Polar Coordinates 28 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 pl ots 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 = V 1(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 29 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 e dges 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 drawn 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 e n was shown to be a contravariant vector, Appendix C: Elliptical Polar Coordinates 30 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 on e 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'-space 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 app lications 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 integration 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 31 |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 Vi ew 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 inters ection 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 successfu lly 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 represen tations 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 32 (f) Drawing the specific contravariant vector dx in x-space and x '-space Since d x is t he primordial contravariant vector, everythi ng 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 dx ' 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 re presentative 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 le ft point generally to the northeast. Appendix C: Elliptical Polar Coordinates 33 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. Tho se 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 34 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 = d θ => 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 no w 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. Th ere is no need to define or consider any Curvilinear- view area in x'-space because the Cartesian-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 di fferent, 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 tw o 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 shape 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 35 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, they 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 orange one on the ri ght with some dxdy patch that approximates it in area, like the green patch. The statement is about an integration. Thus one has lim Σ i 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 "Jacobian Integration Rule" of the section title. The integral on the left is well defined in 2D cal culus, 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 "volum es", and the general Jacobian Integration rule takes the form, Appendix C: Elliptical Polar Coordinates 36 ∫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 37 Appendix D: Tensor Densities and the ε tensor Picture A is used in this Ap pendix along with Standard Notation. (a) Definition of a tensor density First, recall fr om 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 38 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 prett y obvious that a sum of two index-similar tensor densities 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 density is formed from two smaller 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 39 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 for 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")] = J W 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 we ight 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 40 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 linear ized 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 contravari ant 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 T ensors: 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 41 (d) The contravariant ε ten sor 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 assump tion, 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 several 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 taking 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 ε123...N = ε'123...N = +1, but we shall follow Weinberg. Appendix D: Tensor Densities 42 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), εabc... = 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 "added sign s conve ntion", 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 43 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. These 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 44 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) defi nition of a tensor dens ity 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 ch ange 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 interesti ng to see what happens 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 45 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, then applying the rule of section (b) 3, one concludes that, since ε is a tensor density of weight -1 and since all the RHS vectors have weight 0, the object Q a is a covariant vector density of weight = -1 , and thus has this transformation rule Q' a = J RabQb 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 Cn = 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 covariant derivative of a vector is given by Appendix D: Tensor Densities 46 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 s ection (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 sections (d) and (e) it was established that εabc... = 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 (sg = |g|, s= sign(g) = sign(g') as in Section 5 (k)) E abc... ≡ |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 47 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 E abc... ≡ |g|-1/2 εabc... => E123... = |g|-1/2 E'abc... ≡ |g'|-1/2 ε'abc... => E'123... = |g'|-1/2 To compare E abc... and Eabc... , E abc... = |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, E abc... = s|g| Eabc... = g Eabc... Summarizing, E 123... = |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 section 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 48 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... , E abc... = |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... E abc... 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 on ly |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 E abc... ≡ 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 E abc... ≡ 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 49 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 being 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 2 nd indices of the N matrix elements, and the sum is over all N! such permutations. There are many ways to arrive at a given permuta tion of 123...N by doing pairwise swaps, but for all these ways, the number of swaps S will be either even or odd. The parity p of a permutation is defined then as (-1)S and this p appears in the above sum. If one were to swap 2 ↔3 on the right above, each permutation would have S → S+1 since an extra swap is needed to undo 2 ↔3. Thus, all parities p → -p and in fact the whole object negates. But the swap 2↔3 is the same as b ↔c since M b3 Mc2 = Mc2 Mb3. Applying this argument to any pair of indices, one concludes that Q abc..x is totally antisymmetric and therefore 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 given 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 50 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, corresponds 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 R a where ( Ra)i = δa,i . The conclusion then is that Appendix D: Tensor Densities 51 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') = det { ⎝⎜⎛ ⎠⎟⎞ Ra Rb ... Rx ( Ra' Rb' ... Rx') } which is the determinant of this matrix R a• 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 R c• 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 52 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 these 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 53 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 set {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 54 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 permutations, 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 55 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 above 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 δ a,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 summed was written above as (g = 1) εabεa'b' = ⎪⎪ ⎪⎪ δaa' δab' δba' δbb' = δa,a' δb,b' – δa,b' δb,a' Appendix D: Tensor Densities 56 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) = sε abc.. (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 i nvolving 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 57 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 e xpansions 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 58 (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 ( b i'⊗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 obtained 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 ( u i⊗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 = Σ ijk 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 ( e i⊗ej⊗ek ...)abc = (ei)a (ej)b (ek)c = Ria Rjb Rkc Appendix E: Tensor Expansions 59 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 ge neralizations to tensors of any rank of these vector expansions stated in Section 7 (s), A = Σ iαi bi αi = bi • A // arbitrary basis A = ΣiAi ui // axis aligned unit vectors A = Σ iA'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 orthonormal 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 Cartesian. 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 60 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 expansion shown above. Expansions 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 stud y historically use "polyadic notation" as follows, ( ABC...) ≡ A⊗B⊗C ... Appendix E: Tensor Expansions 61 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, ( b ibjbk...) ≡ bi⊗bj⊗bk ... Sometimes this basis vector notation is compressed even more, to wit, i j k ... ≡ (b ibjbk...) ≡ bi⊗bj⊗bk ... although this notation seems to be mostly used when the b i are the unit vectors ui. In all these notations, one must be aware th at 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 give n above for a rank-2 tensor becomes A = Σij αij (bibj) αij = Aab(bi)a (bj)b = Aab (bibj)ab . Appendix E: Tensor Expansions 62 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 tran spose 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 = ab T 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 b i are the unit vectors u i , 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 63 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 Someti mes 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 ≡ c TA ; (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 64 (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 above with generic basis b i , V = Σ i [V(b)]i bi // [V(b)]i are the coefficients of this expansion . Moreover, one can regard a ra nk-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 (or 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 65 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 because the operator has ch anged, 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, wh ile 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 matrix 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 summe d 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 66 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 > = d T 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 operator 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, <(A Td) | 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 67 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 discussi on, 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 = [ ( u n)T]a Aa (BT)b [um]b = δn a Aa Bb δm b = AnBm . In the generic b n basis one has [( AB)(b)]nm = <bn |(AB)| bm > . It is to emphasize this operator view of a dyadic that Morse and Feshbach 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 authors 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 68 (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 established that one can expand a tensor A on the tangent base vectors en as A = Σijk... A' ijk... (ei⊗ej⊗ek...) A' ijk... = contravariant components of A in x'-space A' ijk... = Ri i'Rj j'Rk k'...... A i'j'k'... . 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 applica tion 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 69 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 70 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 = x iui = 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 F M(x) = H F (x) . Appendix E: Tensor Expansions 71 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, dx " = 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 F M(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 72 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, MMT = 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 73 = (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 e n = 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 intuitively 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 74 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 equations 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 te nsors with respect to the underlying x" = F M(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 observation axes are related by un = M(x ) e^n(x) where M(x ) is a rotation. If the first equation describes something at location x in the realm of Newtonian mechanics, we expect the equation to have the same form in both Frame S and 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 75 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 le ft 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-sp ace 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 76 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 notation 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 Cartesian-coordinates equation E ij = (∂iuj + ∂jui)/2, which relates strain tensor E ij to the vector displacement u of a continuum particle, is not a "true tensor equation", so 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 ran k-n tensor expansion 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 b i 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 77 [ 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 coefficient 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 78 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 exam ine some 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 larg e 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 u n 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 2 n 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 operato r A, there are many possible set pairs of components (indexed objects where the values of all indices are set in all possible ways) which do NOT transform as a Appendix E: Tensor Expansions 79 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 b i 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 indexed 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 tr ansform as either end of the transformation rule stated above, so this tensor is a tensor, 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 matric es involved in equations which 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 i ndifferent 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 80 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 re lativity 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 coordinates). 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( e n)i = ∂j(en)i dxj ∂ j ≡ ∂/∂xi or (d e n)i = (∂jen)i dxj . Since e n and (∂jen) are both vectors in ξ-space, and since the e k 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 e k 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 81 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 the 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 th e 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 82 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 alternate 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 83 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 Γ d ab may be expressed entirely in terms of the metric tensor, Γ d ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] . Appendix F: Affine Connection 84 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 Right Hand Side of our desired identity to obtain (RHS)d ab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] = ( 1 / 2 ) (R d 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 85 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 useful 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 86 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 present, 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 87 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 the 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 te nsor density of rank n+1 and weight W. The implication is that all the indices including α of Babc..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'-space is defined as follows: (Weinberg p 104 4.6.12) B abc..x; α ≡ ∂α Babc..x – Γn aαBnbc..x – Γn bαBanc..x – .... – Γn xαBabc..n // x-space del a-term b-term x-term + ( W / 2 g ) ( ∂ αg) Babc..x 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 (as 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 88 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 89 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 90 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 secti on (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 the 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-re lated 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 three 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 91 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 RHS 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 92 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 93 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 exam ples 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 94 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 the 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 95 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 Weinberg 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 derivativ es 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 96 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 ar e 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 97 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 involving 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 98 (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 terms 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 99 which says that raising an index "commutes" with covariant differentiation -- 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) 100 Appendix G: Expansion of ( ∇v) in cu rvilinear coordinates (v = vector) This appendix assumes the usual curvilinear coordinates context, Picture B (a) Continuum Mechanics motivation Although t he 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 sees, 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 => DA /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 interest in fluid mechanics. Correspondingly, the object (∇u) is of great interest in the theory of elastic solids, where u is the displacement field. The matrix ∇A is not a differential operator since the derivative 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) 101 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 = Σ ij 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 = Σ ij v'i;j ei⊗ej v' i;j = [(∂'jv'i) – Rc k(∂'jRik) v'c] or ∇v = Σ ij [(∇v)(e)]ij ei⊗ej [(∇v)(e)]ij = [(∂'jv'i) – Rc k(∂'jRik) v'c] Note that the expression shown contains 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) 102 so that ∇v = Σ ij [(∇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 = Σ ij [(∇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 => e i• (∇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 ju st 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) 103 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) = Σ cd(∇v)dc uducT = Σcd(∂cvd) uducT // note that u d = 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 u ducT 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 u d = Σe Re d ee where ( ud)e = δde Therefore u d = Σe Red ee Then we can change this colu mn vector to a row vector, ucT = Σf Rfc efT and so u ducT = 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) 104 = 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 e i = 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 = Σ ij [(∇v)(e)]ij ei⊗ej [( ∇v)(e)]ij = [(∂'jv'i) + Rim(∂'jRc m) v'c] ∇v = Σ ij [(∇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) 105 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 cas e g'ab = h'a2δa,b and g'ab = h'a-2δa,b and things simplify. The above 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 = Σ ij [(∇v)(e)]i j ei⊗ej [( ∇v)(e)]i j = [(∂'jv'i) – Rcm (∂'jRi m) v'c] . ∇v = Σ ij [(∇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) 106 Then one needs the scale factors h n' from the metric tensor g ¯ = STS ( dev notation Section 5 ( l) ) Appendix G: Expansion of ( ∇v) 107 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) 108 (∇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 ) 109 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 te nsorial 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 known 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 = ρ a 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 fo llowing 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 ) 110 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 = Σ a(∂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 u i expansion as in Appendix G (c), u i = Σe Rn i en Appendix H: Expansion of ( divT ) 111 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 ) 112 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 matrices, 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 ) 113 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 above are valid for general curvilinear coordinates, orthogonal 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 o bject [(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 ) 114 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) 115 Appendix I : The Vector Laplacian in Spherical and Cylindrical Coordinates This appendix assu mes the usual curvilin ear 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 usually 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 fo r 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 describes 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) 116 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) 117 (@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 stated . 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 foll owing 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 A ppendix G (f), the idea being that this code could be used for any coordinate system, Appendix I: Expansion of ( @B) 118 The last object is g' , where hp[n] = h' n . The next chunk of code computes the scalar Laplacian of an unspecified function f, and this expression will be used below in parsing the vector Laplacian results : The subs command used here (and more intensely be low) 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 th inking 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) 119 The two terms of the above (@ B)'n expression shown at the end of section (a) are then duly entered, In the following code, we use ( in line with the notation shown at the start of this section) q1_ = (@ B)'1 = (@B)r q2_ = (@ B)'2 = (@B)θ q3_ = (@ B)'3 = (@B)φ Appendix I: Expansion of ( @B) 120 See comment above about the Maple subs commands (purely cosmetic). (c) The first method in spherical coordinates: putting results i n 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 acti ng 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) 121 (@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 Lapl acian acting on the component. This question is answered in the following section. Appendix I: Expansion of ( @B) 122 (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 tensori zation of a Cartesian form equation is unique, but it is not so easy to show. Be that as it ma y, 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 (indices 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 lower three lines are index substitutions of the first) B a;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) 123 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) 124 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 algebrai c 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) 125 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 -sinθ cosθ 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) 126 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) 127 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) 128 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 ma nner, 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) 129 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 te rms which are added to the scalar Laplacian, and these are seen to agree with the unnumbere d 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) 130 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-vanishing. 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 then 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, d tT'ij = ∂tT'ij + (∇ T)'ij k v'k . Appendix I: Expansion of ( @B) 131 In our usual formalism, x'-space is the space of some generic curvilinear coordinates x'n (not necessarily orthogonal) and then the above e quation 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) 132 Appendix J: Expansion of ( ∇T) in curvilinear coordinates (T = rank-2 tensor) This appendix assumes the usual curvilinear coordinates context, Picture B (a) Total time derivative as prototype equation The total time derivative of a contravariant rank-2 ten sor 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 then 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, d tT'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 e quation 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) 133 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 a bove) the basis vectors are the fixed u n 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 de fining 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 "ten sorization": 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 equations (such as x = rsinθ cosφ 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 extra lines to have Maple compute the Appendix J: Expansion of ( ∇T) 134 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 e xpansions on x-space basis vectors, while the second line shows expansion on the reciprocal and tangent base vectors (which also exist in x-space) : ∇T = Σijk Tij;k ui⊗uj⊗uk = Σijk [(∇T)(u)]ijk ui⊗uj⊗uk = Σijk [(∇T)(u)]ijk ui⊗uj⊗uk ∇T = Σijk T'ij;k ei⊗ej⊗ek = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek = Σijk [(∇T)(e)]ijk ei⊗ej⊗ek 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 have three different notations for the expansion components: T ij;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 = Σ i 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 tensors 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) 135 major complications (see below) since such compon ents are non-covariant (neither contravariant nor covariant nor mixed). On the other hand, since th e 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) 136 ∇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 = Σ i v'i e^i v'i = h'i v'i . The scripted tensor components have indices whic h are integers, i = 1,2...N. Once we actually select a particular curvilinear coordinate system, one can replace the indices with curvilinear coordinate names and then, since those names indicate that one is talking about x'-space 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 always necessary when summations like Σijk are involved, unless one is willing to write out all the terms in the sum, wh ich 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 derivative equation of interest in x'-space dtT'ij = ∂tT'ij + (∇ T)'ij k v'k . How does one write this equation in terms of unit-b ase-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) 137 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 cu rvilinear 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 says 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 Cartesian 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, sin ce the equation above is true for any i and j, Appendix J: Expansion of ( ∇T) 138 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 wr itten 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 appears 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 expressed 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, this 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) 139 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" associated 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 equation in practice, one has to know for example that ( ∇A1)rrθ = [ ∂θ(A1)rr - (A1)θr - (A1)rθ]/r, a fact that is certainly not immediately obvious (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 or thogonal 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) 140 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) 141 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) 142 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)[ cscθ ∂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) 143 . 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 notati on would be written ∂'je^i = Γ'(Lai) ijke^k , // Lai p 502 (8A.12) whereas "the true" affine connection measures the change of the full tangent base vectors, (∂' jen) = Γ'k jn ek . // Appendix F (a) It is not hard to show that the connection 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 components 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) 144 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 145 Appendix K: Deformation Tensors in Continuum Mechanics Tensor-like objects appear every where in continuum mechanics. As was noted in Section 2 (k), and later in Appendix E (i), continuum mechanics texts ge nerally refer to all objects having indices as being "tensors", whether or not these objects actually transform as tensors with respect to some underlying transformation. The most commonly appearing tensors have two indices and are just 3x3 matrices associated with 3D space. In this category, there are several kinds of stress tensors, 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 equation 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 deformation 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 discussed 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 thr ough this document, such as The arrow represents an underlying generally non-li near transformation 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 dx ' = 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 ma trix objects associated 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 146 Fig 1 Consider for the moment just the bot tom transformation. The transformation F to ≡ 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 conti nuous 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 gets 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 paralle lepiped 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 deformati on 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 tu mbles and stretches and ends up as d x at time t and location x, Fig 2 This differential dumbbell can be regarded as a ma thematical "probe" embedded in the continuous medium. The relationship between d x and dX is given by d x = F(X ,t) dX dx i = FijdXj // Lai p 105 (3.18.13) Appendix K: Deformation Tensors 147 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 dx ' = R( x)dx with R(x) here being F(x,t). The picture above is then the second figure shown in Section 2 (left-right switched). Thus, the deform ation gradient F is the linearized version (at point x) of some fancy non-linear (and unknown) "flow transformation" F. Since one can write dx i = (∂ 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 the upper flow by making these replacements : t 0 → 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' = F t(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 148 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 differen tiation 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 dx ', the translational part of the camera platform motion is irrelevant since dX 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 = dX 1e^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 149 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 = Σ ij 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 Appendi x 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 dx' = 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 150 { 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 no w 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 labeled right on the transformation arrows. Appendix K: Deformation Tensors 151 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 dx = 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 cameramen 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 x 0 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 S 0 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 deformation". 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 ra nk-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 152 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" transform as tensors with respect to Q(t) ? By "the usual derived tensors" we mean B, C, U, V and associated R all defined as follows: B = FF T = the left Cauchy-Green deformation tensor = the Piola deformation tensor 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 = R TVR 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 the left and right stre tch 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 the 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/stre tch/shear affair and one will find that none of R, U, V are unity. One can combine the three equations 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 derive d 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 153 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 "objective". 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 references are to Lai page 336-337 : F * = Q ( t ) F / / L a i ( 5 . 5 6 . 2 1 ) B* = Q(t)BQ(t) T // rank-2 tensor with respect to Q(t) so objective // Lai (5.56.31) C * = C / / L a i ( 5 . 5 6 . 2 8 ) U * = U V* = Q(t)VQ(t) T // rank-2 tensor with respect to Q(t) so objective R * = Q ( t ) R 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 parameter. Thus, for example, one could write the transformation of B as B t0*(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 154 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 impor tant time argument 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 continuo us 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 curre nt 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 objective) 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 155 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 several 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, with references to Lai page 472: F t* = 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 tio n 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 versi on 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 a pproximated 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 from the get-go (except perhaps as an approximation). Appendix K: Deformation Tensors 156 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. Furthermore, 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 a nd 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 constitutive equation and we end up with T = aI + bB + cB2. Since one can replace B3 = αB2 + βB + γ I, 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 current flow (a rh eostat controls electric current), Appendix K: Deformation Tensors 157 // 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 elem ents of matrix -pI are th e 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 de rivatives 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 own set of basis vectors we might call un, u*n, u'n and u*'n. It happens that the picture refers to S and S* at time t, and S' and S'* at time τ, but any basis vectors can be "used" at any time one wants. For example, 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 158 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 course 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 u n 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) = Σ ab T*ab(x*,t) ua ⊗ ub Q(t) = Σ ab Qab(t) ua ⊗ ub . Our operator statement of objectivity then becom es 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 followi ng two equations which we express back in operator notation with the position arguments suppressed (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 constitutive 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 derived versions are described. Appendix K: Deformation Tensors 159 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 160 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) [Jaumann-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 proble m 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 161 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 derivative of the stress tensor are called "rate type constitutive equations". Here are some models for incompressible fluids : T = -pI + S where S + λ So = 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 162 Fluids with stress time derivatives in their constitutive equations exhibit both elastic 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 field damps out (mol asses, 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 str ess, 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 tensors 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 backwa rds-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 163 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 transforms 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 unprimed 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 ca lled 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 dx(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 164 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 ha ppy to have the interpretation above for J U. The Oldroyd convected stresses in developmental and standard notation In the previous section two adjusted 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.