wedge doc in retrospect

Wedge Doc In Retrospect PhL 5.20.16 The edit log covers the time period Sept 24, 2015 to the first release on May 19, 2016, a period of 8 calendar months. During this time I did a Cod October trip, a Hawaii trip, a Cod birthday trip, a Maze trip so with recovery days that would account for 11+10+12+9 = 42 days = 6 weeks = 1.5 months. So I think I can say it took me 6.5 months full time to write this wedge doc paper! I did not plan to treat differential forms, but started that as Chapter 10 on Feb 27, 2016 so that chapter took maybe 3 months all by itself. This chapter is about 1/4 of the doc, but it took me half the time to write. Chapter lengths: 1 8 pages tensor in math 2 48 pages tensor algebra 14% 3 10 pages Kronecker 4 21 pages 2 vector products 5 10 pages tensor 6 7 pages tensor dual 7 33 pages wedge 10% 8 22 pages wedge dual 9 5 pages wedge in math 10 80 pages differential forms 24% A 32 pages permutation 10% B 7 pages direct sum C 11 pages pre-sym theorems D 6 pages unified view E 4 pages kin package F 10 pages volume of n-piped in Rm G 10 pages det(RTR) theorem and measure Refs 1 page 335 pages total Comments: Why did I write this thing? Well, I felt I was pretty good on tensor stuff, but I knew that I was very weak on "integration on manifolds" and in general "tensor integration" which I felt would come up in general relativity. What I knew about was "tensor differential" stuff, ODE and PDE from Stakgold, covariant derivatives and differential operators from tensor doc and Weinberg. Every time I would "look into" manifold integration, I was always faced with the mysterious Stokes' theorem and boundaries ∂M and wedge products and differential forms and it was all a huge black hole for me. That led me to read Buck and Sjamaar who cleared up most of the mysteries. So that wedge product was a big stumbling block! My web perusals revealed strange seemingly unrelated things like geometry and quotient spaces and Spivak's very mysterious "definition of the wedge product". Most people just quoted other people and seemed not to know much. It is like people quote M&F for bipolar coordinates. And a little like the Stackel matrix. So OK, that was my wedge motivation. As I got underway, I realized that tensor products were also "ill defined" and I was going to build my wedge product on top of the tensor product, and I had written a little about tensor products in tensor doc but nothing systematic. So at this point we had Tensor Products and Wedge Products as paper title. So why did I insert the huge Chapter 2 review of tensor doc into wedge doc? The reason is that the things that you "wedge together" or "tensor product together" are in fact "tensors", and people need to know what a tensor really is, and that means Chapter 2. When I decided to add the k-forms chapter, Chapter 2 was even more important because it already discussed tangent base vectors which are key to understanding the TxM tangent space for a manifold. I remember being very confused by this concept, and now it is totally clear. So k-forms is an application of wedge tools and I always like to include an application in things I write. There is payoff for the reader. I think my whole "invention" of tensor functions helps a lot in understanding Spivak and other sources. And using the Dirac notation is the key to this understanding. At the end of Chapter 2 I added the dual space stuff since that is another key idea. In my web perusals on wedge, there was always this dichotomy between space and dual space, always very unclear. It is now clear. I could have gone on to do Clifford Algebras because they too use the wedge product. That was another path that wedge confusions was blocking. But I decided to cut off wedge doc with the k-forms and not add Clifford which would take another 3 months. I am building tools here for myself and others. At least I know how the wedge tools work 100% now. The permutation support stuff was crucial, and I think that is a very strong section. One problem for me was that wedge sources tend to be math sources which are very hard for people like me to read. Jammed with category diagrams and modules, quotient spaces and densepack rigorous Definitions and Propositions. I did scan the web pretty hard before deciding to write my own thing. Sjamaar's main ref was Spivak which is mysterious exactly as noted, being a math source. Arapura is a good example of a source. I just browsed it. It says a lot of stuff, but nothing is systematic. It never mentions dual space, just says wedge integrals equal calculus integrals with no comment. So in 30 pages, he does a general survey, but really adds nothing to an underpinnings discussion. It is sort of a repeat of what everyone else says in the same way. I think my notation dxi is ground breaking in that it forces Honesty. You have to face the questions. Gill Williamson has a 76 page doc which includes much of my stuff. It is Ch 11 of his 1989 book with someone else at San Diego. It has my stuff, yes, but in language that just seems strange to me, too much math, too few words. But a valiant effort. Most of my PDF sources are super-math and super-non-readable by me.