Ch 8 Nov 20

Chapter 8 Nov 20 PhL 11.20.15 This was installed on1.26.16 do not edit. 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 vectors 1 8.2 Properties of the wedge product of k vectors 2 8.3 The vector space Λk and its basis 3 8.4 Tensor Expansions for a tensor in Λk 4 8.5 Expansions for the wedge product of k vectors 6 8.6 Number of elements in Λk compared with V*k. 8 8.7 Multiindex notation 8 8.8 The Exterior Algebra Λ(V) 9 Associativity of the Wedge Product 10 8.9 The Wedge Product of two or more tensors in L(V) 12 (a) Wedge Product of two tensors T^ and S^ 12 (b) Special cases of the wedge product T^^ S^. 16 (c) Commutivity Rule for the Wedge Product of two tensors T^ and S^ 16 (d) Wedge Product of three or more tensors 18 (e) Commutativity Rule for product of N tensors 19 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 examples. Since the dual world involves functions as well as functionals, a new Section 8.12 is tacked onto the end, similar to Section 6.6 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. [ rewrite this paragraph] 8.1 Definition of the wedge product of k vectors We wish to define the wedge product of k dual vectors αi ϵ V*, α1^ α2^ .....^ αk . // <α1| ^ <α2| ^ .... ^ <αk| Wedge products of this form (and their linear combinations) inhabit a vector space we call Λk. We now impose the requirement that this wedge product must change sign when any two vectors are swapped. This property is injected into the wedge product theory, it does not fall out from it. This sign-change requirement leads to the following candidate definition for the wedge product of k vectors 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 ..... αj) . (8.1.2) For the purposes of this section, we simplify things by taking jr → r so (7.1.2) becomes, 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 (8.1.3) where (1/k!) is a normalization factor. 8.2 Properties of the wedge product of k 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,r= scalar ϵ K or α1^(rα2 + sα'2)^α3^.....^αk = r(α1^α2^α3^ .....^αk) + s(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) 4. Wedge product of vectors vanishes if any two vectors are the same. Given a sign change (7.2.4) 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. From (7.1.2) we find (αj^ αj^ .....^ αj)(ei,ei....ei) // = (αj^ αj^ .....^ αj)ii...i = (1/k!) ΣP (-1)S(P) (αj αj ..... αj)(ei,ei....ei) = (1/k!) ΣP (-1)S(P) (αj)(ei)(αj)(ei) ... (αj)(ei) // outer product form = (1/k!) ΣP (-1)S(P) αj(ei) αj(ei) ... αj(ei) // (A.1.19) Mab = (αj)i = (1/k!) det[ (αj)i] . //(A.1.19) **(8.2.8) Similarly, (αj^ αj^ .....^ αj)(vi,vi....vi) = (1/k!) ΣP (-1)S(P) (αj)(vi)(αj)(vi) ... (αj)(vi) = (1/k!) det[ (αj)(vi)] . (8.2.8a) Fact: ( αj^αj^...^αj)(vi,vi....vi) is totally antisymmetric in both the labels jr and the labels ir. **(8.2.9) 8. Associative Property of the wedge product. For example, (α1^ α2)^ α3 = α1^ (α2^ α3) = α1^ α2^ a3 . 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 Λk 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) There are independent basis elements for Λk and they all have this form (λi ^ λi ^ .... ^ λi) where i1 < i2 < ..... < ik basis elements (8.3.6) Fact: (λi ^ λi ^ .... ^ λi) = Alt(λi λi .... λi) λ^I = Alt(λI) (8.3.8) Components of the basis elements for Λk . Now reconsider the basis vectors of the vector space Λk , (λj ^ λj ^ .... ^ λi) . (8.3.3) The components are given from (8.2.8) as [ since λj(ei) = δji ] , (λj ^ λj ^ .... ^ λi)(ei,ei....ei) // = (λj ^ λj ^ .... ^ λi)ii...i = (1/k!) ΣP (-1)S(P) δjiδji .... δji = (1/k!) ΣP (-1)S(P) δjiδji ... δji = (1/k!) det[ δji] . (λ^J)(eI) = (1/k!) det(δJI) **(8.3.9) Similarly, (λj ^ λj ^ .... ^ λi)(vi,vi....vi) = (1/k!) ΣP (-1)S(P) λj(vi)λj(vi) .... λj(vi) = (1/k!) ΣP (-1)S(P) (vi)j(vi)j .... (vi)j = (1/k!) det[ (vi)j] (λ^J)(vI) = (1/k!) det(vIJ) (8.3.9a) which tells us that Fact: (λj ^ λj ^ .... ^ λi)(vi,vi....vi) is totally antisymmetric in both the labels jr and the labels ir. **(8.3.10) 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 ^ .... ^ λi)(ei,ei....ei) = (1/k!) [ δji δji...δji + signed permutations] **(8.3.11) Example: [ see (7.3.12) ] 3!(λj ^ λj^ λj)(ei,ei,ei) = 3! λ^J(eI) = det(δJI) = det ( δjjjiii ) **(8.3.12) 8.4 Tensor Expansions for a tensor in Λk Recall now the tensor expansion for a most-general tensor T in V*k is, T = Σii....i Tii....i (λi λi ..... λi) . T ϵ V*k (6.2.1) (8.4.1) where Tii....i are some general coefficients Consider then the similar-looking most-general object in Λk, evaluated at (vj,vj....vj), Σii....i Tii....i (λi^ λi .....^ λi)(vj,vj....vj) = Σii....i Tii....i AltI(λi λi ..... λi)(vj,vj....vj) // (7.3.8) = Σii....i Tii....i AltJ(λi λi ..... λi)(vj,vj....vj) // (8.2.9) and (A.8.26) = AltJ[ Σii....i Tii....i (λi λi ..... λi)(vj,vj....vj) ] // (A.5.10), Alt is linear = AltJT(vj,vj....vj) = [AltJ(T)](vj,vj....vj) = [Alt(T)](vj,vj....vj) // (8.4.1) ≡ T^(vj,vj....vj) (8.4.2) where we define this functional notation, T^ ≡ Alt(T) (8.4.3) which is really this statement about tensor functions, T^(vj,vj....vj) ≡ [Alt(T)] (vj,vj....vj) . (8.4.3a) From (8.4.2) we then have the following fully general element of Lk, T^ = Σii....i Tii....i (λi^ λi .....^ λi) . (8.4.4) We refer to this type of expansion as a symmetric expansion, and we know it is redundant since the symmetric sum includes each true basis vector k! times. According to (A.8.9), we know from (8.4.3a) that Fact: T^(vi,vi....vi) is a totally antisymmetric tensor function. (8.4.5) Therefore, Fact: The space Λk is the space of all totally antisymmetric rank-k tensors T^. To say that T^ is totally antisymmetric means that T^(vi,vi....vi) is a totally antisymmetric tensor function. (8.4.6) In contrast, the space V*k is the space of all rank-k tensors T, so Λk V*k. See Section 7.7 below. Since the set (λi^ λi .....^ λi) with 1 ≤ i1 < i2 < ..... < ik ≤ n forms a complete basis for Λk, as discussed below (7.3.3), it must be possible to express T^ in the following manner T^ = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) . (8.4.7) What then is the connection between the Aii...i of (8.4.7) and the Tii...i of (8.4.4)? The discussion goes exactly as in Chapter 7 and the result is, Aii...i = ΣP (-1)S(P) Tii...i i1 < i2 < ..... < ik = [ Tii...i + all signed permutations ] // k! terms = k! [Alt(T)]ii...i . // (A.5.3) def of Alt or A = k!Alt(T) = k! T^. // (7.4.3) (8.4.16) where T^ = Alt(T) is shown in (7.4.4), not to be confused with tensor function T^ ≡ Alt(T) in (8.4.3). Since A = k! T^ , (7.4.5) shows that Fact: Aii...i and T^ii...i are both totally antisymmetric tensors. (8.4.17) Vector Case. For k = 1, we find that T = ΣiTi λi // (6.2.1) T^ = ΣiTi λi // (8.4.4) T^ = T (8.8.19) so for a vector there is no distinction between T^ and T(and in fact V*1 = Λ1). . 8.5 Expansions for the wedge product of k vectors The symmetric expansion is very straightforward. Let Tii...i = (α1)i (α2)i ... (αk)i = (α1 α2 ..... bk)ii...i or T = (α1 α2 ..... αk) . // α1 ϵ V*, T ϵ V*k (8.5.1) Then the symmetric expansion (8.4.4) gives, T^ = Σii...i Tii...i (λi ^ λi .....^ λi) (8.4.4) = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi .....^ λi) (8.5.2) = [Σi (α1)iλi] ^ [Σi (α2)i λi] ^ .... ^ [Σi (αk)i λi] = α1 ^ α2 ^ ... ^ αk . (8.5.3) 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 (8.4.16), (1/k!) A = Alt(T) = Alt(α1α2.....αk) = T^ = (α1 ^ α2 ^ ... ^ αk) ϵ Vk (8.5.4) Then the ordered expansion (8.4.7) can be written in a battery of ways, α1 ^ α2 ^ ... ^ αk = // α1 ^ α2 ^ ... ^ αk ϵ Λk (a) = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) //(8.4.7) (b) = Σi<i<....<i k! [Alt(α1α2...αk)]ii...i (λi ^ λi .....^ λi) . // (8.5.4) (c) = Σi<i<....<i k! [α1 ^ α2 ^ ... ^ αk]ii...i (λi ^ λi .....^ λi) . // (8.1.3) (d) = Σi<i<....<i det[ (α*)i] (λi ^ λi .....^ λi) . // (8.2.8) with jr → r (e) = Σi<i<....<i det (λi ^ λi .....^ λi) . (f) = Σi<i<....<i det (λi ^ λi .....^ λi) . (g) = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi .....^ λi) // (8.5.2) (8.5.5) where we throw in the symmetric sum at the end. Remember that, since generally dim(V) = n > k, the determinant in (f) is a full-width minor of matrix M = [α1, α2.....αk]. If k = n, the minor is the full matrix. Example : Suppose k = n = 3. Then the following sum (form (f)) has only one term, α1 ^ α2 ^ α3 = Σ1≤i<i<i<3 det[α1, α2, α3] (λi ^ λi ^ λi) = det[α1, α2, α3] (λi ^ λi ^ λi) (8.5.6) as quoted in (4.4.15). Here are the above expressions for k = 2 and general n ≥ k : (a) α1 ^ α2 = Σi<i Aii (λi ^ λi) (b) = Σi<i 2! [Alt(α1α2)]ii (λi ^ λi) (c) = Σi<i 2! (α1 ^ α2)ii (λi ^ λi) (d) = Σi<i det[ (α*)i] (λi ^ λi) (e) = Σi<i det (λi ^ λi) (f) = Σi<i det (λi ^ λi) = Σi<i[(α1)i(α2)i- (α2)i(α1)i](λi ^ λi) (g) = Σii (α1)i (α2)i (λi ^ λi) = [Σi(α1)iλi] ^ [Σi(α2)i λi] = α1 ^ α2 (8.5.7) Result (f) matches that shown in (4.4.12), α ^ β = Σij αiβj (λi ^ λj) = Σi<j (αiβj- αjβi) (λi ^ λj) = Σi<j Aij (λi ^ λj) = Σi<j det (λi ^ λj) Aij = (αiβj- αjβi) = det . (4.4.12) 8.6 Number of elements in Λk compared with V*k. We know from (5.1.5) 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 . (8.6.1) For a given n, this is a strongly decreasing function of k, see graph in (7.6.2). 8.7 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 (8.4.4) : T^ = Σii....i Tii....i (λi^ λi .....^ λi) (7.4.4) T^ = ΣI TI λ^I where λ^I ≡ λi ^ λi ^ .... ^ λi TI ≡ Tii...i and I ≡ {i1, i2,.... ik} with 1 ≤ ir ≤ n = ordinary multiindex, n = dim(V*) . (8.7.1) The more significant notation involves the ordered expansion (8.4.7) 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) . (8.4.7) T^ = Σ'I AI λ^I where λ^I ≡ λi ^ λi ^ .... ^ λi AI ≡ Aii...i and I ≡ {i1, i2,.... ik} with 1 ≤ i1< i2<....< ik ≤ n = ordered multiindex, n = dim(V*) (8.7.2) Here are some unofficial multiindex notations for other equations developed above: T^ = α1 ^ α2 ^ ... ^ αk T^ = (^αZ) (8.5.3) Tii...i = (α1)i (α2)i ... (αk)i ≡ (αZ)I TI = (αZ)I (8.5.1) with the idea that Z = 1,2...k . Continuing on, T^ = Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi .....^ λi) T^ = ΣI (αZ)I λ^I (8.5.2) A = k! Alt(α1α2.....αk) A = k! Alt(αZ) (8.5.4) α1 ^ α2 ^ ... ^ αk = Σi<i<....<i k! [Alt(α1α2...αk)]ii...i (λi ^ λi .....^ λi) . (8.5.5b) (^αZ) = Σ'I k! Alt(αZ)I λ^I Aii...i = det AI = det[(αZ)I] (8.5.5a+f) α1 ^ α2 ^ ... ^ αk = Σi<i<....<i det (λi ^ λi .....^ λi). (8.5.5f) (^αZ) = Σ'I det[(αZ)I]λ^I 8.8 The 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 (8.8.1) Here Λ0 = the space of scalars, Λ1 the space of dual vectors, Λ2 = Λ ^ Λ V*2 the space of antisymmetric dual rank-2 tensors (8.4.6), 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 + ..... (8.8.2) The direct sum is described in Appendix B. Associativity of the Wedge Product See discussion near (7.8.3) and replace ei→ λi and v→α. One then concludes that, Fact: The wedge product of k vectors 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. (8.8.4) What this in effect does is define an array of new objects to be the same as α1^ α2^ .....^ α6. For example, (α1^ (α2^ α3 ^ α4) ^ (α5^ α6) ≡ α1^ α2^ α3^ α4^ α5^ α6 α1^ (α2^ (α3^ α4)^ α5)^ α6 ≡ α1^ α2^ α3^ α4^ α5^ α6 (8.8.5) Since tensors like T^ can be expanded on (λi ^ λi ^ .... ^ λi), and since one may associate this wedge product arbitrarily as claimed in (8.8.4), one easily shows that : Fact: The wedge product of N general tensors A^^B^^C^ .... 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. (8.8.7) This fact then extends the claim (8.8.4) made for N vectors, and is exactly analogous to the similar axiomatic statement for associativity made in (2.8.22). Example: In (8.7.1) multiindex notation, consider three Λ(V) tensors T^,S^,R^ of rank k,k',k" : (T^ ^ S^) ^ R^ = ( (ΣITIλ^I) ^ (ΣJSJλ^J) ) ^ (ΣKRKλ^K) = ΣITIΣJSJ { ( λ^I ^ λ^J) ^ (ΣKRKλ^K) } // rules (8.2.3) = ΣITIΣJSJΣKRK (λ^I ^ λ^J) ^ (λ^K) // rules (8.2.3) again = ΣIJKTISJRK (λ^I ^ λ^J ^ λ^K) // detail shown above (7.8.8) with ei→λi = (ΣITIλ^I) ^ (ΣJSJλ^J) ^ (ΣKRKλ^K) // rules (8.2.3) again = T^ ^ S^ ^ R^ . Our example shows that for arbitrary Λ(V) tensors, (T^ ^ S^) ^ R^ = T^ ^ S^ ^ R^. Fact: The large space Λ(V) is in fact itself a vector space. (8.8.8) 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) (8.8.9) 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 (8.8.10) Here we have used the associative property (8.8.4). This closure claim is stated more generally below (8.10.6). One then makes the following definitions with regard to the space Λ(V), where n = dim(V*) = dim(V): (the objects in this table are multilinear functionals) Object Name any blade lincomb: Grade(rank): Space s dual 0-blade scalar ϵ K 0 Λ0 α dual 1-blade dual vector 1 Λ1 α^β dual 2-blade dual bivector 2 Λ2 α^β^γ dual 3-blade dual trivector 3 Λ3 α^β^γ^δ dual 4-blade dual quadvector 4 Λ4 ..... α^β^γ^δ^.... dual k-blade dual k-vector k Λk .... α^β^γ^δ^.... dual n-blade dual n-vector n Λn arbitrary element of Λ(V) dual multivector mixed Λ(V) (8.8.11) 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 mixed values of k. Note that s1( α^β) s2(γ^δ) = (s1α)^β (s2γ)^δ = (α'^β) (γ'^δ) // 2-blades s1( α^β) s2(γ^δ^ε) = (s1α)^β (s2γ)^δ^ε = (α'^β) (γ'^δ^ε) // multivector so it is also correct to say that a k-vector is any sum of k-blades, and a multivector is any sum of k-vectors. That is, any linear combination can be written as a sum as shown in the above examples. Unlike in Tensor World, in Wedge World the above list (8.8.11) 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 (8.8.12) 8.9 The Wedge Product of two or more tensors in Λ(V) (a) Wedge Product of two tensors T^ and S^ Rather than translate the many details of this section, we will skip these details and state the conclusions. The details may be obtained from Section 7.9 by making these simple replacements: ei → λi eI → λI e^I → λ^I Tii....i → Tii....i , TI → TI , T^ → T^ Sii....i → Sii....i , SI → SI , S^ → S^ In subsection (d) below on the product of three tensors, more details are provided. Here then are selected results: Tensor product of two tensors: T^^ S^ = ΣI (TS)I λ^I I ≡ I, I' = i1,i2...ik+k', λ^I ≡ (λi^ λi .....^ λi) . (8.9.a.5) Closure: T^ ϵ Λk and S^ ϵ Λk' T^^ S^ ϵ Λk+k' Λ(V) . (8.9.a.6) Basis relation: (λi^ λi .....^ λi) = Alt(λi λi .... λi) λ^I = Alt(λI) (7.3.8) TS == ΣI (TS)I λI I ≡ I, I' = i1,i2...ik+k', eI ≡ (ei ei .... ei) . (5.6.5) Alt(TS)(vJ) = AltJ(TS)(vJ) = ΣI (TS)I AltJ (λI(vJ)) // (5.6.5) and (A.5.10) that Alt is linear = ΣI (TS)I AltI (λI(vJ)) // use (A.8.26) since λI(vJ) is totally antisymmetric in I and J = ΣI (TS)I λ^I(vJ) // (8.3.8) = (T^^ S^)(vJ) // (8.9.a.5) so T^^ S^ = Alt(TS) . (8.9.a.7) The "components" (tensor functions) are (T^^ S^)(vJ) = Alt(TS)(vJ) = ΣP(-1)S(P) (TS)(vP(J)) = ΣP(-1)S(P) T(vP(J)) S(vP(J')) (8.9.a.8) where J ≡ j1, j2...jk J' ≡ jk+1, jk+2, ....jk+k' J ≡ J, J' = j1,j2...jk+k' (7.9.a.4) This last line is an explicit instruction for computing the "components" of the tensor T^^ S^ . We have added this new notation, T(vP(J)) ≡ T(vj, vj... vj) for J ≡ j1, j2...jk (8.9.a.9) Example: Let S and T both be rank-2 tensors so k = k' = 2 . Then (T^^ S^)(vI) = (T^^ S^)(v1,v2,v3,v4) = (1/4!) ΣP(-1)S(P)T(vi, vi)S(vi, vi) = (1/24) [ T(vi,vi)S(vi,vi) - T(vi,vi)S(vi,vi) + T(vi,vi)S(vi,vi) + 21 more terms] (8.9.a.10) Here as elsewhere we show in red the indices to be swapped to make the next term. From (8.9.c.6) below, T^^ S^ = (-1)2*2 S^^ T^ = S^^ T^. (8.9.a.11) (b) Special cases of the wedge product T^^ S^ Same as section 7.9 (b) with T^→ T^ and S^→ S^ . Here are the conclusions : T^^S^ = κ^S^ = S^^T^ = S^^κ = κS^ if T^ = κ ϵ V*0 T^^S^ = T^^κ' = S^^T^ = κ'^T^ = κ'T^ if S^ = κ' ϵ V*0 T^^S^ = κ^κ' = S^^T^ = κ'^κ = κκ' if T^,S^ = κ,κ' ϵ V*0 (8.9.b.3) (c) Commutivity Rule for the Wedge Product of two tensors T^ and S^ Same as section 7.9 (c) with T^→ T^ and S^→ S^ and e→λ . Here are some of the translated conclusions: (λ^J ^ λ^I) = (-1)kk' (λ^I ^ λ^J) . (8.9.c.5) S^^ T^ = (-1)kk'T^^ S^ ranks of the two tensors are k and k' . (8.9.c.6) (d) Wedge Product of three or more tensors For this section we do a full translation of Section 7.9 (d) : T^^S^^R^ = [ΣITIλ^I]^[ΣJ SJλ^J]^[ΣK RKλ^K] (a) = ΣI,J,K TISJRK (λ^I) ^ (λ^J) ^ (λ^K) (b) = ΣI,J,K TISJRK (λ^I ^ λ^J ^ λ^K) // associative of ^ used here (d) = ΣI,I',I" TISI'RI" (λ^I ^ λ^I' ^ λ^I") // rename multiindices J→I',K→I" I ≡ i1, i2...ik I' ≡ ik+1, ik+2, ....ik+k' I" ≡ ik+k'+1, ik+k'+2, ....ik+k'+k" λ^I ≡ (λi^....^ λi) λ^I' ≡ (λi^...^λi) λ^I" ≡ (λi^...^λi) (e) = ΣI (TSR)I λ^I λ^I ≡ (λi^....^ λi) I ≡ I, I',I" = i1,i2...ik+k'+k" (8.9.d.1) The outer product form is TISI'RI" = (TSR)I,I',I" = (TSR)I . The conclusion is this: T^^S^^R^ = ΣI (TSR)Iλ^I I ≡ I, I',I" = i1,i2...ik+k'+k" , λ^I ≡ (λi^....^ λi) (8.9.d.2) Since the λ^I are basis vectors in Λk+k'+k", we have shown that: T^ ϵ Λk and S^ ϵ Λk' and R^ ϵ Λk" T^^S^^R^ ϵ Λk+k'+k" Λ(V) . (8.9.d.3) Recalling the Chapter 6 result, TSR = ΣI (TSR)I λI I ≡ I, I',I" = i1,i2...ik+k'+k" λI ≡ (λi.... λi) (6.6.5) and (8.3.8) that λ^I = Alt(λI) we find, Alt(TSR) = ΣI (TSR)I Alt(λI) // Alt is linear = ΣI (TSR)I λ^I = T^^S^^R^ so T^^S^^R^ = Alt(TSR) (8.9.d.4) and then [T^^S^^R^](vI) = [Alt(TSR)](vI) = ΣP(-1)S(P) (TSR)(vP(I)) // (A.5.3) = ΣP(-1)S(P) T(vP(I))S(vP(I'))R(vP(I")) (8.9.d.5) which gives implicit instructions for how to compute the "components" of T^^S^^R^ . Using the systematic notation outlined in (5.6.10) through (5.6.12), and generalizing the above development for the wedge product of three tensors, we find the following expansion for the wedge product of N tensors of Λ(V), (T1)^^(T2)^^...^(TN)^ = ΣI (T1)I(T2)I .... (TN)I λ^I = ΣI (T1T2....TN)I λ^I (8.9.d.6) where λ^I = λi^ λi .....^ λi = λi^ λi .....^ λκ and (T1T2....TN)I = (T1)I(T2)I .... (TN)I . The rank of this product tensor is then κ = Σi=1N ki and the tensor is an element of Λκ Λ(V). Recall (6.6.16), T1T2...TN = ΣI (T1IT2I .... TNI) λI = ΣI (T1T2....TN)I λI . (6.6.16) Applying Alt to both sides again with λ^I = Alt(λI) shows that (T1)^^(T2)^^...^(TN)^ = Alt(T1T2...TN) . (8.9.d.7) "Components" (the tensor function) of this tensor are computed as follows: [(T1)^^(T2)^^...^(TN)^](vI) = [Alt(T1T2...TN)](vI) = ΣP(-1)S(P) (T1T2...TN)(vP(I)) // (A.5.3) = ΣP(-1)S(P) T1(vP(I))T2(vP(I)) ...TN(vP(I)) (8.9.d.8) where T1(vP(I)) ≡ T1(vi,vi...vi) for I1 = i1, i2...iκ T2(vP(I)) ≡ T2(vi, vi...vi) for I2 ={iκ+1, iκ+2.....iκ} etc. // see (5.6.10 thru 12) for details (e) Commutativity Rule for product of N tensors The argument of Section 7.9 (e) can be repeated with e→λ. Here we just quote the conclusion: 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 . (8.9.e.6) If the sum of the ranks of the two swapped tensor is even, in effect m = krks . Example: T1 ^ T2 ^ T3 = (-1)m T3 ^ T2 ^ T1 r = 1 s = 3 m = (k2)(k1+k3) + k1k3 = k1k2 + k1k3 + k2k3 (-1)m = (-1)kk+kk+kk (8.9.e.8) (f) Theorems from Appendix C : pre-antisymmetrization makes no difference We showed above that one can form wedge products of elements of Λ(V) in this manner, T^^ S^ = Alt(TS) . (8.9.a.7) T^^S^^R^ = Alt(TSR) (8.9.d.4) (T1)^^(T2)^^...^(TN)^ = Alt(T1T2...TN) (8.9.d.7) where the operator Alt acts on the vector arguments which are not displayed in the above compact functional notation. For example T^^ S^ = Alt(TS) means, in multiindex notation, (T^^ S^)(vI) = AltI [(TS)(vI)] = AltI [ T(vI)S(vI')] = ΣP(-1)P T(vP(I))S(vP(I')) . A very simple case is the following (recall for vectors that α = α^ ) (α ^ β)(vi,vi) = Alt (αβ)(vi,vi) = Alt α(vi)β(vi) = ΣP(-1)P α(vi) β(vi) = (1/2) [ α(vi)β(vi) - α(vi)β(vi)] = (1/2)[ (αβ)(vi,vi) - (βα)(vi,vi) ] = { (1/2) [(αβ) - (βα) ]}(vi,vi) which replicates our Chapter 4 statement that α ^ β = [αβ- βα]/2 . (4.4.1) The objects here are functionals in Λ(V) which, when closed with a vector set, become tensor functions in Λ(V). We could describe this set of tensor functions by some other name like Λf(V) but there is then a one to one correspondence between Λ(V) = Λf(V). For example, as was shown in (2.11.f.7), T^ = <T| ϵ Λk // functional in Λk T^(v1, v2....vk) = <T| v1, v2....vk> // the corresponding tensor function of Λk . Appendix C uses the rearrangement theorem in three separate Theorems to show that T^^ S^ = Alt(TS) = Alt(T^S) = Alt(TS^) = Alt(T^S^) . (8.9.f.1) Theorem One Theorem Two Theorem Three These three theorems are derived in a generic space with vectors |1,2...k> and so apply to both tensors and tensor functionals, TI and T(vI). Recall that T^ ≡ Alt(T) (8.4.3) so that T^ is a totally antisymmetric tensor functional. What (f.1) above says is that Alt(TS) provides total antisymmetrization on all the (undisplayed) vector argument indices, so pre-antisymmetrizing either or both tensors makes no difference. A similar statement applies to working with totally symmetric tensors. So we have, Alt[TS] = Alt[T^S] = Alt[TS^] = Alt[T^S^] where T^ = Alt(T) S^ = Alt(S) (C.4.1) Sym[TS] = Sym[TsS] = Sym[TSs] = Sym[TsSs] where Ts = Sym(T) Ss = Sym(S) . (C.4.2) These can of course be rewritten as Alt[TS] = Alt[Alt(T)S] = Alt[TAlt(S)] = Alt[Alt(T)Alt(S)] (C.4.3) Sym[TS] = Sym[Sym(T)S] = Sym[TSym(S)] = Sym[Sym(T)Sym(S)] . (C.4.4) Similarly Appendix C shows that T^^S^^R^ = Alt(TSR) = Alt(T^SR) = Alt(TS^R) = Alt(TSR^) = Alt(T^S^R)= Alt(T^SR^)= Alt(TS^R^) = Alt(T^S^R^) . (7.9.f.2) Adding ^ subscripts inside an Alt expression changes nothing. Here is another example: T^^S^^R^ = Alt(TSR) = Alt((TS)R) = Alt((TS)^R) = Alt(Alt(TS)R) . (7.9.f.3) which appears in Spivak p 80 as (C.4.8) (g) Spivak Normalization We won't repeat the discussion of Section 7.9 (g), but the reader can do the translation with the usual rules v → α ei → λi TI → TI . Here are the results, where factors shown in red show changes caused by the Spivak notation in which the normalization factor in (8.1.2) is changed from (1/k) to 1, αj^ αj^ .....^ αj = 1 ΣP (-1)S(P) ( αP(j) αP(j) ..... αP(j)) = 1 [ (αj αj ..... αj) + all signed permutations ] = k! Alt(αj αj ..... αj) . (8.1.2)S The affected equations are these: (λi^ λi^ .....^ λi) = k! Alt(λi λi ..... λi) or λ^I = k!Alt(λI) (8.3.8)S T^ ≡ k!Alt(T) and S^ ≡ k'!Alt(S) . (8.4.3)S T^^ S^ = (k+k')! Alt(TS) . (8.9.a.7)S T^^ S^ = Alt(T^S^) T^ ϵ Λk and S^ ϵ Λk' . (8.9.g.1) T^^ S^^ R^ = Alt(T^S^R^) T^ ϵ Λk , S^ ϵ Λk', R^ ϵ Λk" . (8.9.g.2) These now correspond exactly with Spivak's wedge product definition for tensor functions, page 79 page 80 See (7.9.g.3) for how our notation is related to that of Spivak.