Chapter 8

Chapter 8 initial draft PhL 11.11.15 8. The Wedge Product of k dual vectors : the vector spaces Λk and Λ(V) 1 8.1 Definition of the wedge product of k dual vectors 1 8.2 Properties of the wedge product of k dual vectors 2 8.3 The vector space Λk and its basis 3 8.4 Tensor Expansions for a tensor in Lk 5 8.5 The alt, Alt and Sym operators 6 8.6 Expansions for the wedge product of k dual vectors 7 8.7 Number of elements in Λk compared with V*k. 8 8.8 Multiindex notation 9 8.9 The Dual Exterior Algebra Λ(V) 10 Associativity of the Wedge Product 10 8.10 The Wedge Product of two dual tensors 12 (a) Wedge Product of two dual tensors T and S 12 (b) Commutivity Rule for the Wedge Product of dual tensors T and S 14 (c) Multiindex Notation Rehash of Section 7.10 14 8.11 The Wedge Product of N dual tensors in Λ(V) 16 8. The Wedge Product of k dual vectors : the vector spaces Λk and Λ(V) Comment: This Chapter 8 is a partial copy, paste and edit version of Chapter 7 -- a translation from non-dual to dual. See our similar comment at the start of Chapter 6. Since Chapter 7 is so long, here in Chapter 8 we shall delete all material that is basically unchanged from the non-dual Chapter 7. We also delete most "comments" and all examples. Since the dual world involves functions as well as functionals, a new Section 8.12 is tacked onto the end, similar to Section 6.7 for the dual tensor product. The equation numbers for Chapter 8 match those of Chapter 7, and deletions thus cause "holes" in the sequence for Chapter 8. 8.1 Definition of the wedge product of k dual vectors We wish to define the wedge product of k linear functionals αi ϵ V* (the αi are vectors of V*) , α1^ α2^ .....^ αk . Wedge products of this form (and their linear combinations) inhabit a vector space we call Λk. The wedge product sign-change requirement leads to the following candidate definition for the wedge product of k vectors (functionals) in V* (the jr are vector labels), αj^ αj^ .....^ αj = (1/k!) ΣP (-1)S(P) ( αP(j) αP(j) ..... αP(j)) = (1/k!) [ (αj αj ..... αj) + all signed permutations ] = Alt(αj αj ..... vαj) . (8.1.2) with this simplified form, α1^ α2^ .....^ αk = (1/k!) ΣP (-1)S(P) ( αP(1) αP(2) ..... αP(k)) = (1/k!) [ (α1 α2 ..... αk) + all signed permutations ] . = Alt(α1 α2 ..... αk) = (1/k!) Σii...i εii...i (αi αi ..... αi) ir = 1 to k (7.1.3) where (1/k!) is a normalization factor. 8.2 Properties of the wedge product of k dual vectors 1. The sums in (8.1.2) and (8.1.3) have k! terms. (8.2.1) 2. The wedge product is k-multilinear. (8.2.2) It is by-fiat axiom that the wedge product of k vectors is k-multilinear and therefore satisfies these rules, α1^(α2 + α'2)^α3^.....^αk = α1^α2^α3^ .....^αk + α1^α'2^α3^ .....^αk α1^(sα2)^α3^ .....^ αk = s(α1^α2^α3^ .....^αk) s,s1,s2 = scalar ϵ K or α1^(s1α2 + s2α'2)^α3^.....^αk = s1(α1^α2^α3^ .....^αk) + s2(α1^α'2^α3^ .....^αk) . (8.2.3) Here we show the rules just for the 2 position, but k-multilinear means these rules must apply to all the vector positions. These rules cannot be derived from the similar tensor product rules (5.3.1). 3. The wedge product changes sign if any vector pair is swapped. (8.2.4) Comment: From (8.2.2) the pure vector wedge product α1^ α2^ .....^ αk is k-multilinear in the αi, and so is the underlying tensor product α1α2 ..... αk. 4. Wedge product of vectors vanishes if any two vectors are the same. Given a sign change for any pair swap of vectors in the wedge product, we know that α1^ α2^ .....^ αk = 0 if any two (or more) vectors are the same. (8.2.5) 5. Wedge product vanishes if vectors are linearly dependent. (8.2.6) 6. Wedge product vanishes if k > n . (8.2.7) 7. Components. Equation (7.2.8) reads, (vj^ vj^ .....^ vj)ii...i = (1/k!) ΣP (-1)S(P) ( vP(j) vP(j) ..... vP(j))ii...i = (1/k!) ΣP (-1)S(P) (vP(j))i(vP(j))i ... (vP(j))i (7.2.8) Since functionals don't have components, this equation gets replaced by an evaluation of (8.1.2) at the argument set (ei, ei ....ei) which yields, (αj^ αj^ .....^ αj)(ei, ei ....ei) = (1/k!) ΣP (-1)S(P) ( αP(j) αP(j) ..... vP(j))(ei, ei ....ei) = (1/k!) ΣP (-1)S(P)αP(j)(ei)αP(j)(ei) ..αP(j)(ei) = (1/k!) ΣP (-1)S(P)[αP(j)]i[αP(j)]i ... [αP(j)]i . (8.2.8) Note that [αP(j)]i is a component of the vector αP(j) as in α = Σi(αi)λi . Since this matches the form of (A.4.4) (with all indices down) we conclude from (A.4.5) that Fact: (αj^ αj^ .....^ αj)(ei,ei ....ei) is totally antisymmetric in both the labels jr and the indices ir. (8.2.9) 8. Associative Property of the wedge product. For example, (α1^ α2)^ α3 = α1^ (α2^ α3) = α1^ α2^ a3 . 9. New equations for Chapter 8: evaluations to create functions First we write (8.2.8) evaluated for an arbitrary list of vector arguments, (αj^ αj^ .....^ αj)(vi, vi ....vi) = (8.2.10) = (1/k!) ΣP (-1)S(P)αP(j)(vi)αP(j)(vi) ... αP(j)(vi) . For the wedge product of k basis vectors this becomes (λj^ λj^ .....^ λj)(vi, vi ....vi) = = (1/k!) ΣP (-1)S(P)λP(j)(vi)λP(j)(vi) ... λP(j)(vi) = (1/k!) ΣP (-1)S(P)(vi)P(j)(vi)P(j) ... (vi)P(j) = (1/k!) ΣP (-1)S(P)(vP(i))j(vP(i))j ... (vP(i))j . (8.2.11) The last line is allowed since the previous line has the template form (A.4.4). As a special case of the above, (λj^ λj^ .....^ λj)(v1, v2 ....vk) = (1/k!) ΣP (-1)S(P)(vP(1))j(vP(2))j ... (vP(k))j (8.2.13) Finally, evaluating (8.2.12) at (ei,ei ....ei) one finds, (λj^ λj^ .....^ λj)(ei, ei ....ei) = (1/k!) ΣP (-1)S(P)(ei)P(j)(ei)P(j) ... (ei)P(j) = (1/k!) ΣP (-1)S(P)δiP(j)δiP(j) ... δiP(j) = (1/k!) ΣP (-1)S(P)δP(i)jδP(i)j ..δP(i)j // (A.4.4) (8.2.14) Fact: Since equations (8.2.11,12,14) all fit the template (A.4.4), they are all antisymmetric in both the ir and jr indices according to (A.4.5). (8.2.15) 8.3 The vector space Λk and its basis Λk is the space whose elements are all linear combinations of wedge products of k vectors of V*. (8.3.1) Λk is a vector space. (8.3.2) Basis elements for Λk Consider the following objects in Lk obtained by wedging together k basis elements of V*, where each λi is selected from the set of n available for V (which has dimension n), (λj ^ λj ^ .... ^ λj) . (8.3.3) Of these putative nk objects, only n*(n-1)*...*(n-k+1) = n!/(n-k)! are non-zero by (8.2.5) because all the others have at least two vectors the same. Thus we can assume that all the labels jr are different. By the same argument of Chapter 7 below (7.3.4) we conclude that (λi ^ λi ^ .... ^ λi) where i1 < i2 < ..... < ik basis elements (8.3.6) Components of the basis elements for Lk Now reconsider the basis vectors of the vector space Lk , (λj ^ λj ^ .... ^ λj) . (8.3.3) This wedge product can be expanded using (8.1.2), (λj ^ λj ^ .... ^ λj) = (1/k!) ΣP (-1)S(P) ( λP(j) λP(j) ..... λP(j)) . (8.3.8) Evaluating this functional at (ei, ei ....ei) one gets, (λj ^ λj ^ .... ^ λj)(ei, ei ....ei) = (1/k!) ΣP (-1)S(P) ( λP(j) λP(j) ..... λP(j))(ei, ei ....ei) = (1/k!) ΣP (-1)S(P) λP(j)(ei)λP(j)(ei) ..λP(j)(ei) = (1/k!) ΣP (-1)S(P) δiP(j) δiP(j)...δiP(j) (8.3.9) = (1/k!) ΣP (-1)S(P) δP(i)j δP(i)j... δP(i)j // from (A.4.4), (8.3.10) and from (A.4.5) we conclude that, Fact: (λj^λj^...^λj)(ei,ei ....ei) is totally antisymmetric in both the labels jr and the indices ir. (8.3.11) We saw an example of both antisymmetries for k = 2 back in equation (4.4.20), (λi^ λj)(vr,vs) = - (λi^ λj)(vs,vr) = - (λj^ λi)(vr,vs) // two forms of antisymmetry (4.4.20) Either form (8.3.9) or (8.3.10) can be expressed in our usual informal notation (λj ^ λj ^ .... ^ λj)(ei, ei ....ei) = (1/k!) [ δij δij...δij + signed permutations] (8.3.12) where these permutations can be taken to act on either the jr or the ir. As shown in (A.4.8), the above can be written as a determinant. For example (swapping i↔j). 3! (λi^λi^λi)(ej,ej,ej) = det = det ≡ det ( δjjjiii ) = det(δJI) // in multiindex notation (8.3.13) 8.4 Tensor Expansions for a tensor in Lk Λk can be regarded as the vector space whose "vectors" (rank-k tensors) can be written in this ordered-sum form, T = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi ^ .... ^ λi) . (8.4.1) This is so because the set (λi ^ λi ^ .... ^ λi) with 1 ≤ i1 < i2 < ..... < ik ≤ n forms a complete basis for Λk, as discussed just above. On the other hand, one can also expand this same tensor T in a (redundant) symmetric sum as, T = Σii...i T ii...i (λi ^ λi ^ .... ^ λi) ir = 1,2..n (8.4.3) as in (4.3.5) for k = 2. Here T ii...i are a set of coefficients (this T is in italics). What then is the connection between the Aii...i and the Tii...i coefficients? Using the same method as Chapter 7, we find that Aii...i = ΣP (-1)S(P) T P(i)P(i)...P(i) i1 < i2 < ..... < ik = [ T ii...i + all signed permutations ] // k! terms (8.4.12) The Aii...i appear in the expansion (8.4.1) only for index values 1 ≤ i1 < i2 < ..... < ik ≤ n, but we can interpret (8.4.12) as defining Aii...i for all index values. Since (8.4.12) has the general form shown in (A.3.1), we conclude that Fact: Aii...i is a totally antisymmetric tensor. (8.4.13) Comment: Tii...i and Aii...i are both rank-k tensors, see (5.5.3). As in Chapter 7 we find furthermore that Tii...i = (1/k!) Aii...i (8.4.15) T = (1/k!)A (8.4.15a) and from (8.4.13) it follows that T must be a totally antisymmetric rank-k tensor. This shows that the expansions (8.4.1) and (8.4.3) for T are capable only of describing a totally antisymmetric tensor T. This tensor of course is a functional in Λk but we still call it a tensor. Fact: The space Λk is the space of totally antisymmetric rank-k tensor functionals T. (8.4.16) In contrast, the space V*k is the space of all rank-k tensors T, so Λk V*k. See Section 8.7 below. Recall that the pure vector wedge product α1^ α2^ .....^ αk is k-multilinear in the αi, and so is the underlying tensor product α1 α2 ..... αk. For k = 1, (8.4.15) says Ti = Σj T j(λj)i = ΣjT jδji = Ti (8.4.17) so for a vector there is no distinction between Tk and Tk . do this later! We add now an equation not in Chapter 7 : evaluation of (8.4.3) at a set of vector arguments: T(v1,v2....vk) = Σii...i T ii...i (λi ^ λi ^ .... ^ λi)(v1,v2....vk) = Σii...i T ii...i [(1/k!) ΣP (-1)S(P)(v1)P(j)(v2)P(j) ... (vk)P(j)] = (1/k!) ΣP (-1)S(P) Σii...i T ii...i (v1)P(j)(v2)P(j) ... (vk)P(j) fix up where we used (8.2.12) to obtain the second line. (8.4.18) 8.5 The alt, Alt and Sym operators One can regard the sum shown in (8.4.12) as being an operation "alt" performed on the tensor T to produce another tensor A which is totally antisymmetric. If one defines [alt(X)]ii...i ≡ ΣP (-1)S(P) XP(i)P(i)...P(i) (8.5.1) then (8.4.12) becomes this relation between coefficients of the ordered and symmetric expansions of the functional T Aii...i = [alt(T )]ii...i or A = alt(T ) . (8.5.2) This is then the relation between coefficients of the ordered and symmetric expansions of the functional T shown in (8.4.1) and (8.4.3), It is useful to define another version of the alt operator which has a scaling factor 1/k! , [Alt(X)]ii...i ≡ (1/k!) ΣP (-1)S(P) XP(i)P(i)...P(i) (8.5.3) so that Alt(X) = (1/k!)alt(X) . (8.5.4) It is shown in (A.3.1,2) that alt(X) and therefore Alt(X) are totally antisymmetric tensors for any tensor X. If X is already totally antisymmetric, one finds that alt(X) = k! X // since all k! terms in (8.5.1) are the same X = totally antisymmetric (8.5.5) Alt(X) = X . X = totally antisymmetric (8.5.6) We showed in (8.4.15a) that T = (1/k!) A . (8.5.7) so (8.5.2) can also be written T = Alt(T ) . // (1/k!) A = (1/k!) alt(T ) (8.5.8) See Section 7.5 regarding the Sym operator and decomposition of a general tensor into its symmetry pieces. Alt and Sym for Functions (new in Chapter 8) We now temporarily distinguish between a dual tensor T in the tensor product world (Chapter 6) and a dual tensor T in the wedge product world (Chapter 8). We now therefore write (ir= 1,2..n) T = Σii....i Tii....i (λi λi ..... λi) T ϵ Lk (6.2.1) T^ = Σii...i T^ ii...i (λi ^ λi ^ .... ^ λi) T^ ϵ Λk . (8.4.3) Evaluating the second equation at (v1,v2....vk) one finds, T^(v1,v2....vk) = Σii....i T^ii....i (λi^ λi .....^ λi)(v1,v2....vk) = Σii....i T^ii....i (1/k!) ΣP (-1)S(P)(vP(1))j(vP(2))j ... (vP(k))j // (8.2.12) = (1/k!) ΣP (-1)S(P) [ Σii....i T^ii....i(vP(1))j(vP(2))j ... (vP(k))j ] (8.5.9) Now recall (6.2.5), T(vj, vj ....vj) = Σii....i Tii....i (vj)i (vj)i .... (vj)i . (6.2.5) If we assume that both expansions have the same coefficients, so Tii....i = T^ ii...i, then from (6.2.5) we then recognize the quantity [...] in (8.5.9) as T(vP(1),vP(2)....vP(k)), so T^(v1,v2....vk) = (1/k!) ΣP (-1)S(P)T(vP(1),vP(2)....vP(k)) . (8.5.10) Clearly T^ is just the antisymmetrization of T and this suggests the definition of other versions of the Alt and Sym operators which act on functions to create functions which are either totally antisymmetric or totally symmetric, [Alt(X)](v1,v2....vk) ≡ (1/k!) ΣP (-1)S(P)X(vP(1),vP(2)....vP(k)) [Sym(X)](v1,v2....vk) ≡ (1/k!) ΣP X(vP(1),vP(2)....vP(k)) . (8.5.11) Then (8.5.10) can be written, T^(v1,v2....vk) = [Alt(T)](v1,v2....vk) (8.5.12) which we can interpret as a rule applied to functionals, T^ = Alt(T) // relates T ϵ Lk and T^ ϵ Λk (8.5.13) whereas A^ = alt(T^ ) . // relates two sets of coefficients used to represent T^ ϵ Λk . (8.5.8) Like the Alt operator for functionals, Alt for functions is also a projection operator as we now show: [Alt(Alt(Y))](v1,v2....vk) = (1/k!) ΣP (-1)S(P)[Alt(Y)](vP(1),vP(2)....vP(k)) = (1/k!) ΣP (-1)S(P)(1/k!) [ ΣQ (-1)S(Q)Y (vPQ(1),vPQ(2)....vPQ(k)) ] = (1/k!)2 ΣP { ΣQ (-1)S(PQ) Y (vPQ(1),vPQ(2)....vPQ(k)) } = (1/k!)2 ΣP { ΣQ (-1)S(Q) Y (vQ(1),vQ(2)....vQ(k)) } // rearrangement = (1/k!)2{ ΣP (1)} { ΣQ (-1)S(Q) Y (vQ(1),vQ(2)....vQ(k)) } = (1/k!)2{ k!} { ΣQ (-1)S(Q) Y (vQ(1),vQ(2)....vQ(k)) } = (1/k!) { ΣP (-1)S(P) Y (vP(1),vP(2)....vP(k)) } = [Alt(Y)](v1,v2....vk) (8.5.14) which compare to (7.5.11). 8.6 Expansions for the wedge product of k dual vectors The symmetric expansion is very straightforward. Let Tii...i = (α1)i (α2)i ... (αk)i . (8.6.1) Then the symmetric expansion (7.4.3) gives, T = Σii...i T ii...i (λi ^ λi ^ .... ^ λi) (8.4.3) = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi ^ .... ^ λi) (8.6.2) = [Σi(α1)iλi] ^ [Σi(α2)i λi] ^ .... ^ [Σi (αk)i λi] = α1 ^ α2 ^ ... ^ αk . (8.6.3) This pure tensor T = α1 ^ α2 ^ ... ^ αk is an element of Λk . Expressing α1 ^ α2 ^ ... ^ αk in terms of the ordered expansion is more complicated. One must first compute the tensor A as in (8.4.12), Aii...i = ΣP (-1)S(P) T P(i)P(i)...P(i) // = [alt(T )]ii...i (8.5.1,2) = ΣP (-1)S(P) (α1)P(i)(α2)P(i)...(αk)P(i) (8.6.4) = ΣP (-1)S(P) (α1α2...αk)P(i)P(i)...P(i) // outer product = [alt(α1α2...αk)]ii...i // (8.5.1) (8.6.5) so that, A = alt(α1α2...αk) . (8.6.6) Then the ordered expansion (8.4.1) becomes, α1 ^ α2 ^ ... ^ αk = Σi<i<....<i Aii...i (λi ^ λi ^ .... ^ λi) = Σi<i<....<i [alt(α1α2...αk)]ii...i (λi ^ λi ^ .... ^ λi) . (8.6.7) Since (8.6.4) has the general form (A.4.4) with jr = r, we can write Aii...i = det = det . (8.6.8) We then have these three variations of the vector wedge product expansion: α1 ^ α2 ^ ... ^ αk = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi ^ .... ^ λi) (8.6.2) α1 ^ α2 ^ ... ^ αk = Σi<i<....<i [alt(α1α2...αk)]ii...i (λi ^ λi ^ .... ^ λi) (8.6.7) v1^ v2^ .....^ vk = Σi<i<....<i det (λi ^ λi ^ .... ^ λi). (8.6.9) 8.7 Number of elements in Λk compared with V*k. We know from (6.1.5) and (8.3.6) that, dim(V*k) = nk // number of basis elements of V*k (6.1.5) dim(Λk) = // number of basis elements of Λk . (8.3.6) If the number of elements of field K is N ( N → ∞ for K= reals), then ratio = = = = / nk . (7.7.1) For a given n, this is a strongly decreasing function of k, see graph (7.7.2). 8.8 Multiindex notation In this section, multiindex versions of equations are shown in red. Multiindexing is done in two different ways. First, for the symmetric expansion (8.4.3) : T = Σii...i T ii...i (λi ^ λi ^ .... ^ λi) (8.4.3) T = ΣI TI λI where λI ≡ λi ^ λi ^ .... ^ λi TI ≡ T ii...i and I ≡ {i1, i2,.... ik} with 1 ≤ ir ≤ n = ordinary multiindex, n = dim(V*) . (8.8.1) The more significant notation involves the ordered expansion (8.4.1) which has only one term for each linearly independent basis element. Note our use of Σ'I (prime) to indicate an ordered multiindex summation : T = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi ^ .... ^ λi) . (8.4.1) T = Σ'I AI λI where λI ≡ λi ^ λi ^ .... ^ λi AI ≡ Aii...i and I ≡ {i1, i2,.... ik} with 1 ≤ i1< i2<....< ik ≤ n = ordered multiindex, n = dim(V*) (8.8.2) Here are some unofficial multiindex notations for other equations developed above: T = α1 ^ α2 ^ ... ^ αk T = (^αZ) (8.6.3) Tii...i = (α1)i (α2)i ... (αk)i TI = (αZ)I (8.6.1) with the idea that Z = 1,2...k . Continuing on, T = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi ^ .... ^ λi) T = ΣI (vZ)I λI (8.6.2) A = alt(α1α2...αk) A = alt(αZ) (8.6.6) α1 ^ α2 ^ ... ^ αk = Σi<i<....<i [alt(α1α2...αk)]ii...i (λi ^ λi ^ .... ^ λi) . (8.6.7) (^αZ) = Σ'I alt(αZ)I λI Aii...i = det AI = det [(αZ)I] (8.6.8) α1 ^ α2 ^ ... ^ αk = Σi<i<....<i det (λi ^ λi ^ .... ^ λi) . (8.6.9) (^αZ) = Σ'I det [(αZ)I] λI 8.9 The Dual Exterior Algebra Λ(V) We now construct the graded algebra Λ(V) in analogy with that of T(V) in (5.4.1). Define a large vector space of the form ( this is "the dual exterior algebra on V" ) Λ(V) ≡ Λ0 Λ1 Λ2 Λ3 + .... // Λ(V) = Σk=0∞ Λk (8.9.1) Here Λ0 = the space of scalars, Λ1 = the space of vectors, Λ2 = Λ ^ Λ V*2 the space of antisymmetric rank-2 tensors (8.4.3), and so on. The most general element of the space Λ(V) would have the form X = s ΣiTi λi Σij Tij λi^λj Σijk Tijk λi^λj^λk + ..... or X = s ΣiTi λi Σi<j Aij λi^λj Σi<j<k Aijk λi^λj^λk + ..... (8.9.2) Associativity of the Wedge Product See below (7.9.2). The conclusions are: Fact: The wedge product of k vectors (functionals) α1^ α2^ .....^ αk can be "associated" in any manner without altering the meaning of the product. By this we mean that parentheses can be added in any manner without altering the object. (8.9.4) Fact: The wedge product of k tensors T1^ T2^ .....^ Tk can be "associated" in any manner without altering the meaning of the product. By this we mean that parentheses can be added in any manner without altering the object. (8.9.7) We now resume the discussion of Λ(V). This large space Λ(V) is in fact itself a vector space. (8.9.8) We know this is true since Λ(V) = Σk=0∞ Λk and we showed in (8.3.2) that each Λk is a vector space. For example, the "0" element in Λ(V) is the direct sum of the "0" elements of all the Λk. See Appendix B for more detail. To show that Λ(V) is an algebra, we must show that it is closed under both addition and multiplication. It should be clear to the reader that Λ(V) is closed under addition and has the right scalar rule. For example, if k1 and s are scalars, k1 α β^κ ρ^σ^η = sum of 4 elements of Λ(V) = an element of Λ(V) s(k1 α β^κ ρ^σ^η) = (sk1) (sα) (sβ)^κ ρ^(sσ)^η = element of Λ(V) (8.9.9) This additive closure is of course necessary for Λ(V) be a vector space. The space is also closed under the multiplication operation ^. For example (β^κ)^(ρ^σ^η) = β^κ^ρ^σ^η = ϵ Λ5 = ϵ Λ(V) . // (β^κ) ϵ Λ2 (ρ^σ^η) ϵ Λ3 (8.9.10) Here we have used the associative property (8.9.4). This closure claim is stated more generally below (8.10.6). One then makes the following definitions with regard to the space L(V), where n = dim(V) : (the objects in this table are multilinear functionals) Object Name any blade lincomb: Grade(rank): Space s 0-blade scalar ϵ K 0 Λ0 α 1-blade dual vector 1 Λ1 α^β 2-blade dual bivector 2 Λ2 α^β^γ 3-blade dual trivector 3 Λ3 α^β^γ^δ 4-blade dual quadvector 4 Λ4 ..... α^β^γ^δ^.... k-blade dual k-vector k Λk .... α^β^γ^δ^.... n-blade dual n-vector n Λn arbitrary element of Λ(V) dual multivector mixed Λ(V) (8.9.11) Since Λ(V) is closed under the operations + and ^, it is "an algebra" (the space Λk alone is not an algebra because it is not closed under ^). The Λ(V) algebra is different from that of the reals due to its definition as a sum of vector spaces. The elements of Λ(V) have different "grades" as shown above, so Λ(V) is a "graded algebra". Sometimes Λ(V) is called "the dual exterior tensor algebra" over V. A k-blade is a pure wedge product of k vectors, whereas a k-vector is any linear combination of k-blades. A multivector is any linear combination of k-vectors for any values of k. Note that s1( α^β) s2(γ^δ^ε) = (s1α)^β (s2γ)^δ^ε = (α'^β) (γ'^δ^ε) so it is also correct to say that a k-vector is any sum of k-blades since any linear combination can be written as a sum as shown in the above example. Unlike in the tensor product world, in wedge world the above list (8.9.11) is finite for a given n = dim(V). For k = n there is exactly one linearly independent basis vector which is the ordered wedge product of all the basis vectors of V*. For k > n, all wedge products vanish since the vectors in the wedge product are linearly dependent, see (7.2.6). The dimensionality of the space Λ(V) is as follows, based on (7.9.1) and (B.10)', dim[Λ(V)] = dim[Λ0 Λ1 Λ2 Λ3 + ....] = dim(Λ0) + dim(Λ1) + dim(Λ2) + dim(Λ3) + ... but for dim(V*) = n this series truncates with Λn and we find from (7.3.6), dim[Λ(V)] = 1 + n + + + ... + = Σk=0n = 2n = a finite number (8.9.12) 8.10 The Wedge Product of two dual tensors (a) Wedge Product of two dual tensors T and S Consider these Eq. (7.4.1) ordered expansions of general tensors T and S of rank k and k', T = Σi<i<....<i Aii....i λi^ λi .....^ λi rank k, T ϵ Λk S = Σj<j<....<j Bjj....j λj^ λj .....^ λj rank k', S ϵ Λk' (8.10.1) where from (7.5.2), Aii...i = [alt(T )]ii...i or A = alt(T ) Bjj....j = [alt(S )]jj....j or B = alt(S ) . (8.10.2) The corresponding symmetric expansions (7.4.3) of T and S are given by, T = Σii....i Tii....i λi^ λi .....^ λi. rank k, T ϵ Λk S = Σjj....j Sjj....j λj^ λj .....^ λj . rank k', S ϵ Λk' . (8.10.3) We form the wedge product of these two tensors in a manner similar to the equations leading to (5.6.6) : T^S = [ Σii....i Tii....i λi^ λi ...^ λi] ^ [ Σjj....j Sjj....j λj^ λj ...^ λj] (a) = Σii....i Σjj....jTii....i Sjj....j(λi^ λi ...^ λi) ^ (λj^ λj ...^ λj) (b) = Σii....ijj....jTii....i Sjj....j(λi ^ λi ...^ λi ^ λj ^ λj ...^ λj) (c) = Σii....iii....i[Tii....i Sii....i] (λi^ λi ......^ λi) (d) = Σii....i[Tii....i Sii....i] (λi^ λi ......^ λi) (e) = Σii....i[T S ]ii....iii....i(λi^ λi ......^ λi) (8.10.4) where, as in (5.6.7), we use in the last line the standard outer product notation, [T S ]ii...iii...i = Tii....i Sii...i . (8.10.5) Notice that the (8.9.4) associativity of ^ is used going from (a) to (b). Eq. (8.10.4) shows that the product T^S is an element of Λk+k' with the following tensor components, T^S = Σii....i (T^S )ii....i(λi^ λi ......^ λi) (T^S )ii....i = [T S ]ii....i (8.10.6) Thus we have strengthened the claim made in (8.9.10) that Λ(V) is closed under the operation ^ : the wedge product of an Λk tensor with an Λk' tensor lies in Λk+k' which is in Λ(V). It is easy then to show that this is true for the wedge product of any two multivectors as defined below (8.9.11). The next step is to express (8.10.4) as an ordered sum rather than a symmetric sum. Momentarily replace T S by the symbol T,' so that (8.10.4) reads, T^S = Σii....iT ' ii...i(λi^ λi ......^ λi) . (8.10.7) This symmetric sum can be replaced by the ordered sum (8.4.1) T^S = Σi<i<....<i A' ii...i(λi^ λi ......^ λi) (8.10.8) where, according to (8.4.12) and then (8.5.2), A' ii...i = ΣP (-1)S(P) T ' P(i)P(i)...P(i) = [alt(T ' )]ii...i . // A ' = alt(T ') (8.10.9) Restoring T ' = T S gives this ordered expansion for the wedge product of tensors T and S, T^S = Σi<i<....<i [alt(T S)]ii...i (λi^ λi ......^ λi) . with (8.10.10) [alt(T S)]ii...i = ΣP (-1)S(P) TP(i)P(i)...P(i)S P(i)P(i)..P(i) where ΣP is over all permutations of (the subscripts of) {i1, i2, ....ik+k'}. (b) Commutivity Rule for the Wedge Product of dual tensors T and S This proceeds exactly as in Section 7.10 (b) with these by-now obvious changes Tii....i → Tii....i Sjj....j → Sjj....j ei → λi giving this final result S^T = (-1)kk'T^S ranks of the two tensors are k and k' . (8.10.20) The wedge product of two tensors commutes if kk' is even, and anticommutes if kk' is odd. In the "rehash" below we show this derivation in full in multiindex notation. (c) Multiindex Notation Rehash of Section 7.10 For "practice" and for use in the next section, we give an abbreviated copy, paste and edit rehash of Section 8.10 above using the multiindex notation. Whenever there is a confusion, one must write things out in detail. Equation numbers from above are shown in italics. __________________________________________________________________________________ Consider these Eq. (7.4.1) ordered expansions of general tensors T and S of rank k and k', T = Σ'I AI λI I = {i1, i2, .. ik} λI ≡ (λi^ λi ...^ λi) rank k S = Σ'J BJ λJ J = {j1, j2, .. jk'} λJ ≡ (λj^ λj ...^ λj) rank k' (8.10.1) where from (7.5.2), AI = [alt(T )]I BJ = [alt(S )]J . (8.10.2) The corresponding symmetric expansions (7.4.3) of T and S are given by, T = ΣITI λI // symmetric expansions rank k, T ϵ Λk S = ΣJSJ λJ. rank k', S ϵ Λk' (8.10.3) We form the wedge product of these two tensors in a manner similar to the equations leading to (5.6.6), T^S = [ ΣITI λI] ^ [ ΣJSJ λJ] (a) = ΣI ΣJTI SJ (λI^λJ) (b) = ΣI,JTI SJ (λI^λJ) I = {i1, i2, .. ik} (c) = ΣI,I'[TI SI'] (λI^λI') I ≡ {i1...ik+k'}, I' ≡ {ik+1...ik+k'} so I I' = I (d) = ΣI [TI SI'] λI λI ≡ (λi^ λi ......^ λi) (e) = ΣI (T S )I λI // symmetric expansion of T^S (8.10.4) where, as in (5.6.7), we use in the last line the standard outer product notation, (T S )I = TI SI' . (8.10.5) Eq. (8.10.4) shows that the product T^S is an element of Λk+k', T^S = ΣI (T^S )I λI with (T^S)I = (T S )I . (8.10.6) The next step is to express (8.10.4) as an ordered sum rather than a symmetric sum. Momentarily replace (T S ) by the symbol T,' so that (8.10.4) reads, T^S = ΣIT 'I λI . (8.10.7) This symmetric sum can be replaced by the ordered sum (8.4.1) T^S = Σ'I A' I λI (8.10.8) where, according to (8.4.12) and then (8.5.2), A' I = ΣP (-1)S(P) T ' P(I) = [alt(T ' )]I . // A ' = alt(T ') (8.10.9) Restoring T ' = (T S ) gives this ordered expansion for the wedge product of tensors T and S, T^S = Σ'I [alt(T S )]I λI. (8.10.10) [alt(T S )]I = ΣP (-1)S(P) TP(I)S P(I') where ΣP is over all (k+k')! permutations of (the subscripts of) {i1, i2, ....ik+k'}. Commutivity Rule for the Wedge Product of two tensors T and S Recall the expansion of T^S from (8.10.4)(b), T^S = ΣI,J TI SJ (λI ^ λJ) (8.10.17) Swapping T↔S , T↔S, k↔k' and I ↔ J gives the following form for the wedge product S^T , S^T = ΣJ,I SJTI (λI ^ λJ) = ΣI,J TISJ (λI ^ λJ) (8.10.18) Equations (8.10.17) and (8.10.18) are identical except for the last factor involving the basis vectors. (λJ ^ λI) = [(-1)k']k (λI ^ λJ) // slide λJ left through λI , see (7.10.19) (8.10.19) S^T = (-1)kk'T^S ranks of the two tensors are k and k' . (8.10.20) ________________________________________________________________________________ 8.11 The Wedge Product of N dual tensors in Λ(V) First, we mimic the multiindex development just above to obtain the wedge product of three tensors: T = ΣITI λI rank k I = {i1, i2, .. ik} S = ΣJSJ λJ rank k' I' ≡ {ik+1...ik+k'} R = ΣKRK λK rank k" I" = {ik+k'+1, ik+k'+2, .. ik+k'+k"} (8.11.1) T^S^R = (ΣITIλI) ^ (ΣJSJλJ) ^ (ΣKRKλK) = ΣI,J,K TISJRK (λI^λJ^λK) . // symmetric expansion = ΣI,I',I" TISI'RI" (λI^λI'^λI") = ΣI(TSR)I λI (TSR)I = TISIRI" (8.11.2) = Σ'I [alt(TSR)]IλI // ordered expansion where [alt(TSR)]I = ΣP (-1)S(P) TP(I)SP(I')RP(I") (8.11.3) Sample reordering rule: S^T^R = ΣI,J,K TISJRK (λJ^λI^λK) = ΣI,J,K TISJRK [ (-1)kk'(λI^λJ^λK)] = (-1)kk' T^S^R (8.11.4) Systematic Tensor Products To develop a more systematic approach, consider the first three tensors in a product sequence, T1 = tensor of rank k1 I1 = {i1, i2.....ik} T2 = tensor of rank k2 I2 = {ik+1, ik+2.....ik+k} T3 = tensor of rank k3 I3 = {ik+k+1, ik+k+2.....ik+k+k} . (8.11.5) Define the following "cumulative ranks", κ1 = k1 κ2 = k1+ k2 κ3 = k1+ k2 + k3 ... κN = k1 + k2 + ... + kN = Σi=1N ki . (8.11.6) Then rewrite (7.11.5), T1 = tensor of rank k1 I1 = {i1, i2.....iκ} T2 = tensor of rank k2 I2 = {iκ+1, iκ+2.....iκ} T3 = tensor of rank k3 I3 = {iκ+1, iκ+2.....iκ} ... TN = tensor of rank kN IN = {iκ+1,iκ+2.....iκ} . (8.11.7) Define, (T1)P(I) ≡ (T1)P(i)(T1)P(i)...(T1)P(i) (T2)P(I) ≡ (T2)P(i)(T2)P(i) ....(T2)P(i) (T3)P(I) ≡ (T3)P(i)(T3)P(i) ....(T3)P(i) ... (TN)P(I) ≡ (TN)P(i)(TN)P(i) ....(TN)P(i) . (8.11.8) We can now write out the product of any number of tensors. In each case we show the symmetric expansion first, then the ordered expansion. Wedge Product of 2 Tensors T1^T2 = ΣII (T1)I(T2)I (λI^ λI) = ΣI (T1T2)I λI T1^T2 = Σ'I [alt(T1T2)]I λI λI ≡ λi^ λi .....^ λi where [alt(T1T2)]I = ΣP (-1)S(P) (T1)P(I)(T2)P(I) (8.11.9) Wedge Product of 3 Tensors T1^T2^T3 = ΣIII (T1)I(T2)I(T3)I (λI^λI^λI) = ΣI (T1T2T3)I λI T1^T2^T3 = Σ'I [alt(T1T2T3)]I λI λI ≡ λi^ λi .....^ λi where [alt(T1T2T3)]I = ΣP (-1)S(P) (T1)P(I)(T2)P(I)(T3)P(I) (8.11.10) Wedge Product of N Tensors T1^T2^...^TN = ΣII...I (T1)I(T2)I....(TN)I (λI^λI ... λI) = ΣI (T1T2 ....TN)I eI T1^T2^...^TN = Σ'I [alt(T1T2...TN)]I λI λI ≡ λi^ λi .....^ λi where [alt(T1T2...TN)]I = ΣP (-1)S(P) (T1)P(I)(T2)P(I)...(TN)P(I) . (8.11.11) Sign Rule for swapping two tensors See Section 7.11 for a derivation of the following rule (and examples) : Fact: In a product of tensors T1^T2^T3.... of rank k1, k2, k3 ... , if two tensors are swapped Tr ↔ Ts (with r < s), the resulting tensor incurs the following sign relative to the starting tensor, sign = (-1)m where m = (kr+1+kr+2 ...+ks-1)(kr+ks) + krks . (8.11.13) If the sum of the ranks of the two swapped tensor is even, in effect m = krks . 8.12 The Tensor Product of two or more functions in Λ(V) We now resume the distinguishing notation discussed below (8.5.8) where T ϵ Lk and T^ ϵ Λk, T = Σii....i Tii....i (λi λi ..... λi) T ϵ Lk (6.2.1) T^ = Σii...i T^ ii...i (λi ^ λi ^ .... ^ λi) T^ ϵ Λk . (8.4.3) (8.12.1) Consider now wedge product of two dual tensors shown in (8.10.4b), T^^S^ = Σii...ijj...jTii....i Sjj....j (λi ^ λi ...^ λi ^ λj ^ λj ...^ λj) . (8.10.4b) (8.12.2) Evaluate this functional at (v1,v2....vk, vk+1....vk+k'), (T^^S^)(v1,v2....vk, vk+1....vk+k') = Σii...ijj...jTii....i Sjj....j (λi ^ λi ...^ λi ^ λj ^ λj ...^ λj)(v1,v2....vk, vk+1....vk+k') . (8.12.3) The basis function is provided by (8.2.13) to give (T^^S^)(v1,v2....vk, vk+1....vk+k') = Σii...ijj...jTii....i Sjj....j (1/k!) ΣP (-1)S(P)(vP(1))i(vP(2))i ... (vP(k))i (vP(k+1))j(vP(k+2))j ... (vP(k+k'))j (8.12.4) = (1/k!) ΣP (-1)S(P) [ Σii...iTii....i (vP(1))i(vP(2))i ... (vP(k))i ] * [ Σjj...jSjj....j (vP(k+1))j(vP(k+2))j ... (vP(k+k'))j ] . From (6.2.5) we recognize the square brackets and the end result is (T^^S^)(v1,v2....vk, vk+1....vk+k') = (1/k!) ΣP (-1)S(P) T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) . (8.12.5) As shown in (C.4.1), one may replace T→ T^ and S→ S^ in the above "for free" to get (T^^S^)(v1,v2....vk, vk+1....vk+k') // T^ ϵ Λk, S^ ϵ Λk' (T^^S^) ϵ Λk+k' = (1/k!) ΣP (-1)S(P) T^(vP(1), vP(2) ....vP(k))S^(vP(k+1), vP(k+2) ....vP(k+k')) . (8.12.6) The intuitive reason this works is that it makes no difference if one or more of the individual functions in an overall antisymmetrization are "pre-antisymmetrized", as shown for several cases in Appendix C. We have arrived then at an expression for the wedge product of T^ ϵ Λk and S^ ϵ Λk'. Since both (8.10.5) and (8.10.6) have the template form shown in (A.3.1), we know that the function (T^^S^) is totally antisymmetric in its k+k' arguments. This function is also manifestly (k+k')-multilinear due to the form of (8.10.6) and the fact that T^ is k-multilinear and S^ is k'-multilinear. Thus, (T^^S^) ϵ Λk+k'. In analogy with (5.6.7,8) (TS)ii....i = Tii....i Sii....i . (5.6.7) (TSR)ii....i = Tii....i Sii....iRii....i . (5.6.8) we define the tensor product of functions of ranks k and k' in this manner (TS)(v1,v2,...vk+k') ≡ T(v1,v2,...vk) S(vk+1,vk+2,...vk+k') (8.12.7) and adding a third tensor of rank k", (TSR)(v1,v2,...vk+k'+k") ≡ T(v1,v2,...vk) S(vk+1,vk+2,...vk+k')R(vk+k'+1,vk+k'+2,...vk+k'+k"). (8.12.8) In the systematic notation of (7.11.5) → (7.11.7) we can then generalize to get, (T1T2...TN)(v1,v2,...vκ) = T1(v1,v2,...vκ)T2(vκ+1,vκ+2,...vκ) ..... TN(viκ,vκ,...vκ) (8.12.9) Appendix C proves the following general theorem (C.5.11), Alt[(T1)a(T1)a .....(T1)a ] = Alt[(T1)b(T1)b .....(T1)b], ai,bi ϵ {,^} (C.5.14) where each label can be set independently. For example, Alt[TS] = Alt[T^S] = Alt[TS^] = Alt[T^S^] Alt[TSR] = Alt[TS^R] = Alt[T^SR^] = Alt[T^S^R^] etc. (8.10.8) Recall that T^ = Alt(T) // relates T ϵ Lk and T^ ϵ Λk (8.5.13) (8.10.9) and similarly for S^ and R^ . Combining the last three equations leads to a plethora of strange-looking recursive "Alt equations". For example, this line of equal objects Alt[TS] = Alt[T^S] = Alt[TS^] = Alt[T^S^] becomes this line, Alt[TS] = Alt[Alt(T)S] = Alt[TAlt(S)] = Alt[Alt(T)Alt(S)] Similarly this line Alt[TSR] = Alt[TS^R] = Alt[T^SR^] becomes this line Alt[TSR] = Alt[TAlt(S)R] = Alt[Alt(T)SAlt(R)] Applying (8.10.9) to T = TS gives a strange looking equation, (TS)^ = Alt[ (TS)] which is (C.2.2) with T → TS and changing the argument range (TS)^(v1,v2....vk+k') = (1/k!) ΣQ (-1)S(Q)(TS)(vQ(1),vQ(2)....vQ(k+k')) (C.2.2) = (1/k!) ΣQ (-1)S(Q)T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) where (TS)^ = ΣP(-1)P (TS) = ΣP(-1)PTS Stop. I need to talk clearly about a lot more forms. Do all these make sense? (TS^)^ ? (T^S^)^ ? and so on, should be addressed somewhere. Alt[TSR] = Alt[(Alt(T)SR^] = Alt[(Alt(T)Alt(S)Alt(R)] Here is Spivak's little friend which I still have not reached, I need to show that Alt[Alt(TS)R] = Alt[TSR] but I don't yet know how to write this. I suspect what I just wrote is valid but I need more horsepower to prove it.