About the tensor paper

About the Tensor Paper PhL 11.27.11 The "first paper" I wrote on this subject (66p) was written Nov 16, 2009 (two years ago). My intention with the new "tensor doc" was simply to update this older document so it could be added to my web repository. It took longer than expected to do this "update". The up/down g approach. In this older doc, the first 6 sections (12 p) were about a fictitious construct I was calling Tensor Algebra, based entirely up the idea of up and down indices, where gab was regarded as a lowering tensor, and so on, having nothing at all to do with the metric tensor. Then only in Section 7 did I suggest connecting this gab thing with the metric tensor thing. My tensors were indeed based on the derivative of a transformation, but mysteriously I never even mentioned the underlying transformation. I came to realize that this entire approach was baloney, though sort of interesting in its own way. In the new paper, the transformation is at the bottom of everything, and the metric tensor is front and center. In this new doc, g only becomes a raising/lowering thing on page 67 instead of on page 1. In the old doc I did have the same tangent and reciprocal base vector chapters (10 and 11), but the reciprocals were not given any geometrical representation for N > 3. The connection between the papers is that Tab = Rab and so on. I had a section on expansion of the vector V in different ways, though a little confused. A month or two ago (see below) I tried editing this older doc to just fix it up a bit before I realized the major problems noted above. I had what I called my "multi T" formula which is really the new Appendix A Ek thing. But then in the transformation of area section, I could not get things right for general N. I was just doing some in-line hacking that is still there. I could get a pre-hacked version of this doc from storage, but don't really want it! But the point is that I did have a chapter on length, area and volume transformation (section 12). After this I had the same general plan of doing the differential operators one at a time with summary at the end. In the appendices I was confused about raising and lowering indices on Tab and I did a little on the Christoffel stuff. So my basic overall structure has not changed: (1) theory (2) application to curvilinear diff ops. So here is a list of "improvements" that the newer doc has: the bogus g=raiser starting point is completely gone. the underlying F is mentioned at once. the whole theory is developed without up/down notation, my developmental notation idea. there are more examples and basic surface ideas are mentioned (coordinate lines, level surfaces) the dual path of two applications is clearer the origin of the two kinds of vectors is much clearer. I have added a lot of historical references and connections with dates and people's names every object is better defined. the metric tensor chapter is now very good I think, hitting all the key topics the transition to standard notation chapter is also very good. I had to figure out the geometry of N-pipeds in order to get the area stuff done right, Appendix B. the whole alternate representation for the reciprocal vectors of Appendix A is basically new in the diff op sections, attention is constantly paid to covariance and tensor types of things the curl is generalized to N > 3 connections with Weinberg were established, my major modern source. verification of all items with Moon and Spencer is shown tensor densities were never mentioned or known by me in the previous doc. clarified the εabc tensor completely, it was totally mush before derived all the Levi-Civita formulas! specific mention of special relativity stuff, more than just a few passing lines first mention of general relativity stuff! clearer mention of Lie groups and the direct product business. more added on the Christoffel aspect of things, but did not do much since I don't know it yet. the idea of separate R and S I think paid off. new paper has 12 references, the old one had none. I tie in to Moon and Spencer. added the useful idea of Pictures. I doubt there is another "book" quit like this one which starts in a developmental notation and which has the dual applications path. I think that makes it unique. And there are lots of "extras" everywhere. I wish I could have had this thing when I was learning the stuff. Adding all the above stuff look lots of TIME. If I scan through the logs, here is what I find: I finished Stakgold Chap 8 around Aug 6 (2011) and got involved in the notion of separation of variables. That consumed the rest of August. Then Sept 1 I did my little Levi-Citiva A doc where I did the εε stuff for the first time ever in N dimensions. On Sept 2 for some reason I got out the old tensor doc and started wondering about the area transformation in N dimensions, because this is connected to ε because for N=3 you have the cross product for your dA elements. So Sept 3 was when it started! Therefore, I have spent a total of 85 calendar days on this thing, stunning! Well, I did the Cod /Lee trip Sept 12-24 and then did the Needles Bruce trip 27 to Oct 3, so those two items probably took 15 + 6 = 21 days including setup and recovery, so we are then down to 85-21 = 64 work days. List of confusions encountered (from logs) Why do you "need" up and down indices? Why are there two kinds of vectors? Contravariant by definition business. Gave up on summarizing the early theory sections in a summary section, wasted effort. Had to do "we removal" since I had not been writing for publication. On Oct 14 I have 120 pages and think I am done. Is a cross product of vectors a vector? Oct 19 I am doing edit passes. Spelling checks and verify actual results against outside sources. Confusion about x'-space and the two views of the N-piped there, still an unhappy point. Oct 28 complete rewrite of Section 8 on area transformation etc. The x'-space Troubles continue. Oct 29 finish Appendix C. Spell checking problems, more edit passes, more we removal. Nov 6 I think I am done at 151 pages, start PDF production, copy to Gwenn Nov 11 confusion about curl and the ε tensor, start looking at Weinberg/ Nov 14 am doing tensor densities. Nov 16 Appendix D appears. What is the vector nature of dAn ? Nov 19 another rewrite of Section 8. Nov 20 look at Ricci paper. , EabcEabc stuff, looking outside my world. Nov 27 is today. So roughly it took me 64 work days to "update" my old tensor doc. Perhaps some of those days I could now work, but probably average was 10 hours a day on tensor so call it 640 hours! Just amazing. Call it about 3 months calendar time, and 2 months actual work time.