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Chapter 8 Equation Translations
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Working document by Phil (PhL, 11.20.15) in the Wedge World tensor wedge project, listing substitution rules such as e_i to lambda_i and T_{i...i} to T(v_i,...,v_i). It then rewrites Chapter 7 equations side by side as Chapter 8 versions. Topics include components of wedge products as determinants, antisymmetry, the symmetric and ordered expansions of alpha1^...^alphak, the k=2 case, and multiindex notation.
AI-written summary; may contain errors. This description is approximate.
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Chapter 8 Equation Translations PhL 11.20.15
Rules:
ei → λi
v → α
Tii...i → T(vi,vi....vi) some places
Tii...i → T(ei,ei....e i) other places
(AB)I,J = AIBJ → (AB)(vI, vJ) = A(vI)B(vJ)
Example: Function version of the outer product rule (see generic at **** )
(AB)iiii = AiiBii →
(AB)(vi,vi,vi,vi) = A(vi,vi)B(vi,vi)
Sample Translation:
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7. Components. From (7.1.2) we find
(vj^ vj^ .....^ vj)ii...i = (1/k!) ΣP (-1)S(P) (vj vj ..... vj)ii...i
= (1/k!) ΣP (-1)S(P) (vj)i(vj)i ... (vj)i // outer product form
= (1/k!) ΣP (-1)S(P) (vj)i(vj)i ... (vj)i // (A.1.19) Mab = (vj)i
= (1/k!) det[ (vj)i] //(A.1.19) (7.2.8)
becomes
(αj^ αj^ .....^ αj)(vi,vi....vi)
= (1/k!) ΣP (-1)S(P) (αj αj ..... αj)(vi,vi....vi)
= (1/k!) ΣP (-1)S(P) (αj)(vi)(αj)(vi) ... (αj)(vi) // outer product form
= (1/k!) ΣP (-1)S(P) αj(vi) αj(v) ... αj(v) // (A.1.19) Mab = (αj)i
= (1/k!) det[ (αj)i] //(A.1.19) (7.2.8)
That took a lot of effort on my part, ouch!
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Fact: (vj^vj^...^vj)ii...i is totally antisymmetric in both the labels jr and the indices ir. (7.2.9)
Fact: ( αj^αj^...^αj)(vi,vi....vi) is totally antisymmetric in both the labels jr and the labels ir. ________________________________________________________________
(7.2.9)
(ej^ ej^ .....^ ej)ii...i
= (1/k!) ΣP (-1)S(P) δjiδji .... δji
= (1/k!) ΣP (-1)S(P) δjiδji ... δji
= (1/k!) det[ δji] (e^J)I = (1/k!) det(δJI) (7.3.9)
becomes
(λj ^ λj ^ .... ^ λi)(ei,ei....ei)
= (1/k!) ΣP (-1)S(P) δjiδji .... δji
= (1/k!) ΣP (-1)S(P) δjiδji ... δji
= (1/k!) det[ δji] (e^J)I = (1/k!) det(δJI) (7.3.9)
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The symmetric expansion is very straightforward. Let
Tii...i = (v1)i (v2)i ... (vk)i = (v1 v2 ..... vk)ii...i
or
T = (v1 v2 ..... vk) . (7.5.1)
BECOMES
T(vi,vi....vi) = (α1 α2 ..... αk)(vi,vi....vi)
= (α1)(vi) (α2)(vi) ... (αk)(vi) // see (A.8.22)
or
T = (α1 α2 ..... αk)
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Then the symmetric expansion (7.4.3) gives,
T^ = Σii...i Tii...i (ei ^ ei ^ .... ^ ei) (7.4.3)
= Σii...i (v1)i (v2)i ... (vk)i (ei ^ ei ^ .... ^ ei) (7.5.2)
= [Σi(v1)iei] ^ [Σi(v2)i ei] ^ .... ^ [Σi (vk)i ei]
= v1 ^ v2 ^ ... ^ vk . (7.5.3)
BECOMES
T^ = Σii...i Tii...i (λi ^ λi .....^ λi) (7.4.3)
= Σii...i (α1)i (α2)i ... (αk)i (λi ^ λi .....^ λi) (7.5.2)
= [Σi (α1)iλi] ^ [Σi (α2)i λi] ^ .... ^ [Σi (αk)i λi]
= α1 ^ α2 ^ ... ^ αk . (7.5.3)
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Expressing v1 ^ v2 ^ ... ^ vk in terms of the ordered expansion is more complicated. One must first compute the tensor A as in (7.4.16),
(1/k!) A = Alt(T) = Alt(v1v2.....vk) = T^ = (v1 ^ v2 ^ ... ^ vk) (7.5.4)
Then the ordered expansion (7.4.7) can be written in a battery of ways,
v1 ^ v2 ^ ... ^ vk
(a) = Σi<i<....<i Aii...i (ei ^ ei ^ .... ^ ei) // (7.4.7)
(b) = Σi<i<....<i k! [Alt(v1v2...vk)]ii...i (ei ^ ei ^ .... ^ ei) . // (7.5.4)
(c) = Σi<i<....<i k! [v1 ^ v2 ^ ... ^ vk]ii...i (ei ^ ei ^ .... ^ ei) . // (7.1.3)
(d) = Σi<i<....<i det[ (v*)i] (ei ^ ei ^ .... ^ ei) . // (7.2.8) with jr → r
(e) = Σi<i<....<i det (ei ^ ei ^ .... ^ ei) .
(f) = Σi<i<....<i det (ei ^ ei ^ .... ^ ei) .
(g) = Σii...i (v1)i (v2)i ... (vk)i (ei ^ ei ^ .... ^ ei) // (7.5.2) (7.5.5)
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 = [v1, v2.....vk]. If k = n, the minor is the full matrix.
BECOMES
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)
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Here are the above expressions for k = 2 and general n ≥ k :
(a) v1 ^ v2 = Σi<i Aii (ei ^ ei)
(b) = Σi<i 2! [Alt(v1v2)]ii (ei ^ ei)
(c) = Σi<i 2! (v1 ^ v2)ii (ei ^ ei)
(d) = Σi<i det[ (v*)i] (ei ^ ei)
(e) = Σi<i det (ei ^ ei)
(f) = Σi<i det (ei ^ ei) = Σi<i[(v1)i(v2)i - (v2)i(v1)i](ei ^ ei)
(g) = Σii (v1)i (v2)i (ei ^ ei) = [Σi(v1)iei] ^ [Σi(v2)i ei] = v1 ^ v2 (7.5.7)
Result (f) matches that shown in (4.3.12),
a ^ b = Σij aibj (ei ^ ej) = Σi<j (aibj- ajbi) (ei ^ ej) Σi<j Aij (ei ^ ej)
= Σi<j det (ei^ej) Aij = (aibj- ajbi) = det . (4.3.12)
BECOMES
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)
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Here are some unofficial multiindex notations for other equations developed above:
T^ = v1 ^ v2 ^ ... ^ vk T^ = (^vZ) (7.5.3)
Tii...i = (v1)i (v2)i ... (vk)i ≡ (vZ)I TI = (vZ)I (7.5.1)
with the idea that Z = 1,2...k . Continuing on,
T^ = Σii...i (v1)i (v2)i ... (vk)i (ei ^ ei ^ .... ^ ei) T^ = ΣI (vZ)I e^I (7.5.2)
A = k! Alt(v1v2...vk) A = k! Alt(vZ) (7.5.4)
v1 ^ v2 ^ ... ^ vk = Σi<i<....<i k![Alt(v1v2...vk)]ii...i (ei ^ ei ^ .... ^ ei) . (7.5.5b)
(^vZ) = Σ'I k! Alt(vZ)I e^I
Aii...i = det AI = det(vZI) (7.5.5a+f)
v1^ v2^ .....^ vk = Σi<i<....<i det (ei ^ ei ^ .... ^ ei). (7.5.5f)
(^vZ) = Σ'I det(vZI) e^I
BECOMES
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
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