tensor wedge doc log1

Tensor/Wedge Document Log PhL 9.24.15 I have been working on this stuff for several weeks now, and I think an "thought log" would be helpful, so that is starting right now at 3 PM on 9.24.15 Issue #1. In my section 2.1 I suddenly start talking about metric tensors and differential distance and dot products etc. I write (ds)2 = Σij ijdxidxj. But the reader can ask, "what is dxi" ? I have been talking about vector space V, not vector space Rn. What is the meaning of "distance" in generic vector space V? It has no distance, so why am I bring it up? The reason is that I want to talk about whether or not the basis vectors ei are orthogonal or not. I keep thinking I need the notion of an a b dot product in order to talk about ei ej = δij. Where do "components" come into the picture? I say V = Σi viei so I can refer to the coefficient there as a "component". So "component" seems like a well-defined concept. Then can I say ek = Σi (ek)iei ? Why not, since ek is like any other vector. Then I have components of the vector ek. Once I have such components, I can then at least talk about this quantity fij ≡ Σn (ei)n(ej)n I can then write this same equation in matrix/vector notation as fij ≡ ( (ej)1. (ej)2....(ej)n') One could then speak of cases where fij ≡ Σn (ei)n(ej)n = δi,jgi // orthogonal fij ≡ Σn (ei)n(ej)n = δi,j // orthonormal Key point: One can do all this stuff for a pure vector space with no metric and no dot product! Nothing I have done above talks about norms, metrics, or inner products. But one could certainly DEFINE an inner product a b = Σi aibi. And this could in turn be used to define a norm and a metric, and then you have a Hilbert space. So any vector space becomes a Hilbert space by this definition. But this then opens to door to consider other metric tensors. I could have a vector a ϵ V and talk about a small change in that vector da, and I could then say (da)2 = Σij ijdaidaj is the square "length" of this differential vector. What has one done in introducing this last equation? I have opened many cans of worms be talking about differential vectors. Suppose you just define a dot product by a b = Σijijaibj and say nothing about distances or metrics. Think of this as () (a,b) = Σijijaibj, then the function () is a bilinear function since then all the usual rules would apply! No reason for ij to be symmetric. Plan A: Suppose I just stop talking after getting to these lines Σn (ei)n(ej)n = δi,jgi // orthogonal Σn (ei)n(ej)n = δi,j // orthonormal Don't mention metric tensors, or norms, or dot products, or non-Euclidean spaces, or Cartesian spaces. Just leave it at the above point. Don't say anything about "covariant notation" or up and down indices. I could say that en is just "some other basis" which is true. OK, let's now apply this plan to section 2.1 and just remove stuff that is not relevant. OK, I have redone section 2.1 to avoid all mention of dot products. I guess I have to go back and remove these from earlier sections. Have to do that soon. // It is done, and I think thereby improved. As of 7 PM 9/24/15 I have finally woven the fabric between my opening Tensor Sections, and the Wedge Sections I wrote earlier. It took yet another full day. I had a lot of trouble combining these two documents into one, had to decide what order to put things, etc etc. Can't repeat things, can't have things out of order, on and on. 9.25.15. I have now battled my way through Section 2.2 on the V*W* space. There were many changes. I am finally ready to move into the first wedge section which is now 2.3. 9.26.15. I am not going through Section 8, things look good. Maybe I should replace my multilinear rule with the statement everyone else uses, rather than insist on my two liner? Not sure of this. The two liner makes things clearer, but one might miss the combined idea.