Appendix A.8 on tensor functions

Appendix A.8 on tensor functions! PhL 11.20.15 This is a copy and edit job. It has been installed into Appendix A v4, do not edit! A.8 Application to Tensor Functions We now restate the "generic" results of Sections A.2, A.3 and A.4 for this special case: f(1,2...k) = T(vi,vi....vi) // a "tensor function" T ϵ V*k (A.8.1) Here T is any rank-k tensor function. This f seems perhaps an odd looking "function", but one can consider it to be just an evaluation of this more respectable mapping, f(a,b,c,...q) = T(vi,vi....vi) a,b,c... ϵ {1,2...k} f: {1,2...k}k → Vf*k . (A.8.2) This technical mapping issue is not important because we are just regarding T(vi,vi....vi) as a "carrier" of the labels 1,2,3..k, from the point of view of doing permutations. The actual indices like i1 could be arbitrary objects (labeled pancakes) as far as the permutation theorems are concerned, but in our applications we have in mind that i1 is an integer in the range 1,2....n where n = dim(V) and n is unrelated to the tensor rank k. Here then are some Section A.2,A.3,A.4 results translated according to f(1,2,3...k) = T(vi,vi....vi) . For some of the translations, we show the actual equation from above, then its translation. For others we just state the translated result. In all the results below, one can always specialize to the case i1,i2...ik → 1,2,...k. The resulting equations are then as if our mapping were f(1,2...k) = T(v1,v2....vk). Note then that iP(r) → i(r) in a subscript. (a) Alt Equations (translated from Section A.2 above) The basic Alt definition of (A.2.1) g(1,2...k) = [Alt(f)](1,2...k) ≡ (1/k!) ΣP (-1)S(P)f( P(1),P(2)...P(k) ) (A.2.1) becomes, G(vi,vi....vi) = [Alt(F)](vi,vi....vi) = (1/k!) ΣP (-1)S(P)F(vi,vi....vi) (A.8.3) G = Alt(F) // definition of Alt acting on a tensor Examples: G(vi,vi) = [Alt(F)](vi,vi) = (1/2)[ F(vi,vi) - F(vi,vi) ] (A.8.4) G(vi,vi,vi) = [Alt(F)](vi,vi,vi) = (1/6)[ F(vi,vi,vi) - F(vi,vi,vi+ the other four terms ] In practice, we might more easily write G(va,vb,vc) = [Alt(F)](va,vb,vc) = (1/6) [ F(va,vb,vc) - F(vb,va,vc) + the other four terms ] but when it comes time to prove permutation-related theorems, we use subscripts like i1. Continuing on, R{1,2....k} = {R(1),R(2).....R(k)} of (A.2.3) becomes, with R→ P, P T(vi,vi....vi) = T(vi,vi....vi) (A.2.3) (A.8.5) Fact: Any permutation P is a linear operator, so P( ΣrarTr(vi,vi....vi)) = Σrar(PTr(vi,vi....vi)) (A.2.5) (A.8.6) Definition: A tensor function T(vi,vi....vi) is totally antisymmetric if it changes sign when any two arguments are swapped. (A.2.9) (A.8.7) Example: T(vi,vi....vi) = - T(vi,vi....vi) or T(va,vb....vq) = - T(vb,va....vq) Fact: T(vi,vi....vi) totally antisymmetric P T(vi,vi....vi) = (-1)S(P) T(vi,vi....vi) , where P is any permutation of [1,2..k]. (A.2.10) (A.8.8) Fact: The function T(vi,vi....vi) ≡ [Alt(F)]T(vi,vi....vi) is totally antisymmetric in its labels. (A.2.11) (A.8.9) Fact: Alt is a linear operator, so Alt (ΣjajTj(vi,vi....vi)) = Σjaj [Alt(Tj)](vi,vi....vi). (A.2.12) (A.8.10) Fact: Alt is a projection operator, so Alt(Alt(T)) = Alt(T) . (A.2.13) (A.8.11) Fact: If T is a totally antisymmetric rank-k tensor, then Alt(T) = T . (A.2.16) (A.8.12) (b) Sym Equations ( translated from Section A.3 above) The basic Sym definition of (A.3.1) g(1,2...k) = [Sym (f)](1,2...k) ≡ (1/k!) ΣP f( P(1),P(2)...P(k) ) (A.3.1) becomes, G(vi,vi....vi) = [Sym(F)](vi,vi....vi) = (1/k!) ΣP F(vi,vi....vi) (A.8.12) G = Sym(F) // definition of Alt acting on a tensor Examples: G(vi,vi) = [Sym(F)](vi,vi) = (1/2)[ F(vi,vi) + F(vi,vi) ] (A.8.13) G(vi,vi,vi) = [Sym(F)](vi,vi,vi) = (1/6)[F(vi,vi,vi)+F(vi,vi,vi+ the other four terms ] In practice, we might more easily write G(va,vb,vc) = [Sym(F)](va,vb,vc) = (1/6) [ F(va,vb,vc) + F(vb,va,vc) + the other four terms ] but when it comes time to prove permutation-related theorems, we use subscripts like i1. Definition: A tensor function T(vi,vi....vi) is totally symmetric if it is unchanged when any two superscripts are swapped. (A.3.9) (A.8.14) Example: T(vi,vi....vi) = T(vi,vi....vi) or T(va,vb....vq) = T(vb,va....vq) Fact: T(vi,vi....vi) totally symmetric P T(vi,vi....vi) = T(vi,vi....vi), where P is any permutation of [1,2..k]. (A.3.10) (A.8.15) Fact: The function T(vi,vi....vi) ≡ [Sym(F)]T(vi,vi....vi) is totally symmetric in its labels. (A.2.11) (A.8.16) Fact: Sym is a linear operator, so Sym(ΣjajTj(vi,vi....vi)) = Σjaj [Sym(Tj)](vi,vi....vi). (A.3.12) (A.8.17) Fact: Sym is a projection operator, so Sym (Sym (T)) = Sym (T) . (A.3.13) (A.8.18) Fact: If T is a totally symmetric rank-k tensor, then Sym(T) = T . (A.2.16) (A.8.19) (c) Alt/Sym and Other Equations (translated from Section A.4,A.6 and A.7 above) Fact: The projection operators Alt and Sym are orthogonal, so Alt(Sym(T)) = Sym(Alt(T)) = 0 . (A.4.1) (A.8.20) Fact: A tensor function T(vi,vi....vi) can be decomposed in the following manner: (A.8.21) T(vi,vi....vi) = A(vi,vi....vi) + S(vi,vi....vi) + E(vi,vi....vi) (A.4.3) Alt(A) = A Sym(A) = 0 Alt(E) = 0 (A.4.4) Alt(S) = 0 Sym(S) = S Sym(E) = 0 (A.4.5) where A is totally antisymmetric, S is totally symmetric, and E is whatever is left over. Definition: Tensor product: (TS)(vi,vi....vi) ≡ T(vi,vi....vi) S(vi,vi....vi), where the ranks of tensors T,S are k,k'. (A.2.19) (A.8.22) The following are based on Section A.6 and concern use of the ε tensor with tensor functions. If A is totally antisymmetric, then A(vi,vi....vi) = [ A(v1,v2....vk)] εii...i . (A.6.3) (A.8.22) The tensor function [Alt(T)](v1,v2....vk) can be expressed as, ΣP (-1)S(P) T(vP(1),vP(2)....vP(k)) = Σii...i εii...i T(vi,vi....vi) (A.6.5) (A.8.23) Let T(vi,vi....vi) = (αi αi ..... αi)(vi,vi....vi) so T(v1,v2....vk) = (α1 α2 ..... αk)(v1,v2....vk) Then (A.8.23) gives this way to write (αj ^ αj ^ .....^ αj) : ΣP (-1)S(P) (αP(1) αP(2) ..... αP(k)) (A.6.8) (A.8.24) = Σii...i εii...i (αi αi ..... αi), ir = 1,2...k The following is based on Section A.7. (αj ^ αj ^ .....^ αj) = Alt(αj αj ..... αj) (A.7.5) (A.8.25)