translate Ch 5 to Ch 6 with rules

7. The Tensor Product of k vectors in the dual space : the vector spaces V*k and L*(V) 1.15.16 Here I am trying to translate from non-dual to dual using a set of rules. Chapter 5 is non-dual k tensor, while Ch 6 is dual k tensor. No wedge stuff involved here. I think this is all taken care of. This chapter is a direct clone of Chapter 5 with appropriate edits made: V → V* v → α ei → λi eI → λI T → T Tii....i → Tii....i T(V) → T(V*) a,b,c.. → α,β,γ Is this all there is? What about functions! There are no functions in Chapter 5. objects like (ei ei ..... ei)jj...j would become (λi λi ..... λi)jj...j but these do not exist in the dual world! There is no such thing as (λi)j because a functional does not have components. This gets replaced by λi(vj) which is a function. So tensor components are replaced by functions. Here is an example, T = Σii....i Tii....i (λi λi ..... λi) . k-multilinear functional (5.2.1) This equation is meaningful as a functional. But it does not make sense to take the components of a functional, so we would never write Tii....i . Instead we do this T(v1, v2....vk) = Σii....i Tii....i (λi λi ..... λi)(v1, v2....vk) = Σii....i Tii....i (v1)i (v2)i... (vk)i k-multilinear function ****************************** 6. The Tensor Product of k vectors in the dual space : the vector spaces V*k and L(V*) Our first plan for this chapter was to copy, paste and edit Chapter 5 and put that here. It seems more useful, however, to provide the fairly obvious "translation rules" and let the reader do his or her own translations of the non-dual equations in Chapter 5 to dual-space equations which belong here in Chapter 6. We shall then provide some sample equation translations. So we start with this set of symbol translations. There are certain ambiguities which are discussed in the text comments on the right. . Vk → V*k the tensor product vector space v → α vector → functional vi → αi labeled vector → labeled functional (i = label) ei → λi basis vector → basis functional (i = label) v → α both vectors in V vi → αi components of the above vectors (i=component) a,b,c,d.. → α,β,γ,δ.. misc. vectors (ei ei ..... ei) → (λi λi ..... λi) basis for V*k eI → λI same in multiindex notation T → T tensor Tii....i → Tii....i tensor components T(V) → T(V*) tensor algebra Certain structures do not translate so easily and require replacement. Whereas the basis vectors ei have components (ei)j = δij, the basis functionals λi don't have components (i is a label). But unlike the case of ei, the functional λi can be evaluated at a vector argument to obtain a function λi(ej) = δij. Consider then these translations: (v1v2 .....vk )jj...j = (v1)j(v2)j...(v3)j → (α1α2 .....αk )(v1, v2, ...vk) = α1(v1) α2(v2)... αk(vk) and then (e1e2 .....ek )jj...j = (e1)j(e2)j...(e3)j = δ1jδ2j...δkj → (λ1λ2 .....λk )(vj, vj, ...vj) = λ1(vj) α2(vj)... αk(vj) = (vj)1 (vj)2 .... (vj)k So we add to the above list these items: (ei)j = δij → λi(ej) = δij i = label, j = component (left), label(right) ei vj = (vj)i → λi(vj) = (vj)i i, j = labels (v1v2 .....vk )jj...j → (α1α2 .....αk )(vj, vj, ...vj) (e1e2 .....ek )jj...j → (α1α2 .....αk )(ej, ej, ...ej) = α1(ej) I don't really know what I am doing with this translation table. Let's go back to the original approach. 6. The Tensor Product of k vectors in the dual space : the vector spaces V*k and L(V*) This is a copy, paste and edit version of Chapter 5. Our task is now to generalize the tensor product from V2 to Vk, where Vk ≡ VV .... V . // tensor product of k vector spaces, each one is V (5.1) We are setting up for a parallel treatment in Chapter 6 where becomes ^, so certain rather obvious statements will be made here to allow for comparison later with the wedge product. 6.1 Pure elements, basis elements, and dimension of V*k A generic pure ("decomposable") element of V*k is this tensor product of k functionals, α1 α2 ..... αk . all αi ϵ V* (6.1.1) Since is associative by (2.8.22), one can install parentheses anywhere in (6.1.1) without altering the meaning of the object, for example, α1 (α2 α3) .... αk = α1 α2 a3 .... αk . The basis elements of V*k are λi λi ..... λi . (6.1.2) The subscripts in (5.1.1) and the superscripts in (5.1.2) are labels, not components. The next two equations in Section 5.1 involve taking components of (5.1.1) and (5.1.2), (v1 v2 ..... vk)jj...j = (v1)j (v2)j .... (vk)j (5.1.3) (ei ei ..... ei)jj...j = (ei)j (ei)j .... (ei)j = δij δij .... δij , (5.1.4) but in the dual space functionals don't have components. The analogous equations involve evaluation of the above functionals at a set of basis vector arguments (ej, ej ....ej): (α1 α2 ..... αk)(ej, ej ....ej) = α1(ej)α2(ej) ... αk(ej) = (α1)j(α2)j ... (αk)j (6.1.3) (λi λi ..... λi)(ej, ej ....ej) = λi(ej)λi(ej)... λi(ej) = (ej)i (ej)i ...(ej)i = δij δij .... δij (6.1.4) Note that in expression α1(ej) the α1 is a functional, but in (α1)j the α1 is a vector in V which is associated with the functional α1 according to α1 = Σi(α1)iλi. If n = dim(V*), the total number of such basis elements is nk, so dim(V*k) = nk. (6.1.5) In the full set of tensor-product basis elements shown in (6.1.2), two or more of the λi might be the same. This will always be the case if k > n where n ≡ dim(V*). For example, k = 3 and n = 2, one such element would be λ1 λ1 λ2 ≠ 0.