transformation of tensor functions

Transformation of tensor functions PhL 1.13.16 I think this D0 I just wrote below is all wrong, but it is what I was presenting. I keep thinking of ψ(r1, r2...rk) and how you would never mix the different coordinates with a physical rotation. This vector transformation idea seems totally useless and meaningless. D.0 Tensor Fields versus Tensor Functions: how they transform. Back in *** we discussed the transformation of a tensor field. For a rank-k tensor field we find, T'ii...i(x') = RijRij ...RijTjj...j(x), where x' = F(x) or x'i = Fi(x) dx' = Rdx or dx'i = Riadxa Here Tjj...j(x) is a function of x with k covariant component indices j1 through jk and there are implied sums on all k indices i1 through ik. Although Tjj...j(x) is a function of x, we wish to define a radically different object which we shall formally refer to as a "tensor function". Here are the characteristics of a tensor function: It has no component indices, regardless of its rank If it is a rank-k tensor function, it has k vector arguments which we can denote by vi,vi, .... vi . The subscripts here are labels, not components, so vi is a certain vector labeled by i2, and it has contravariant components (vi)a where a = 1,2...n. a rank-k tensor function is k-multilinear as defined in *****. associated with these k vector arguments is a linear transformation with matrix Rij which mixes one set of vectors in x-space to produce another set of vectors in x'-space. Thus we have, v'i = Σj=1k Rij vj i = 1,2....k Once again, vj is a vector labeled by j, it is not the jth component of a vector v. Since we are soon going to drop the bolding of vectors, one must constantly keep in mind that vj refers to a vector labeled by j, and does not refer to a vector v which has component vj. Thus, the above mixing equation is not the rule for the normal transformation of a vector. That rule for the vector vj would be, (vj')a = Σa=1n Rab(vj)b We shall refer to this last rule as "the component transformation of a vector vj", whereas the previous rule is the "vector transformation of a vector". The first mixes the n components of a vector to make a new vector, whereas the second mixes one set of k vectors {vi} to make a new set of k vectors {v'i}. the transformation rule for a "tensor function" g(vi,vi, .... vi) is based on the fact that this function is k-multilinear, and on the fact that v'i = Rij vj (implied sum on j). We then have, g(v'i,v'i, .... v'i) = g (Rijvj, Rijvj, .... Rij vj) = RijRij ...Rij g (vj, vj, .... vj) This tensor function transformation rule has a similar "look" to the transformation rule for a rank-k tensor just noted above T'ii...i(x') = RijRij ...Rij Tjj...j(x) However, in this last tensor transformation the implied sums run from 1 to n, where n is the dimension of the vector space which is defined by vector coordinate x. In the "tensor function" transformation rule, the implied sums run 1 to k. ******************************************************** Start over on the tensor function transformation concept. I have already shown that T(vi,vj) = Σab (vi)a (vj)bTab So what does this look like in a rotated (primed) space? I don't have to do much work to answer that. Since the vectors vi and vj are contracted with tensor T, the result is a scalar, so we get T'(v'i,v'j) = T(vi,vj) and this is my familiar answer where the underlying transformation is x' = F(x). So why do I ever need a different "rule" for this "tensor function"? I feel the need top insert this different rule below 2.11.7 but why do I feel that need? I cannot recall right now. I think it is much deeper into my paper that this comes up. I think it is in Chapter 8 which is not installed yet. But I don't see the "tensor function transformation rule" coming up anywhere in Chap 8 Nov 20. But in that doc I have not dealt with "tensor functions" very much. The word transformation does not even appear there. Where do I talk about wedge products and tensor functions? That would be in the dual space wedge rank-k discussion, which is in fact this Chapter 8. But even through 8.9 it says nothing about those tensor functions! My Chap 8 Nov 20 goes through Section 8.9. But I see tensor function comments mixed into the discussion in Chap 8 Nov 20. I am lost in the mess I left on Nov 25 or so. Major cleanup is needed. Lots of my docs can be put into Obs and so on.