edit log for tensor paper

Tensor doc editing PhL 11.3.11 Nov 30, 2011 Release 14 March 14, 2012 Release 19 March 31, 2012 Release 20 April 16, 2012 Release 20 June 25, 2012 Release 20 Sept 12, 2012 Release 20 Oct 21, 2012 Release 25 Feb 10, 2014 Release 27 May 6, 2014 Release 96 May 19, 2016 Release 100 I think the monster is finally done as of today 11.3.11, and now comes a long phase of checks. The spelling was done yesterday in three chunks, but new errors have probably crept in during the edits. I could select areas and disable them from spelling, but that may damage the document I fear. And it would take a very long time to do, each equation block at a time. sdfasdf 11.3 Added last section to the intro. 11.3 Reviewed the old 2009 doc and updated tensor doc accordingly. 11.3 5 PM. Verified that all subsection lettering is correct, no missing letters or duplicate ones. 11.3 5:40 PM. Verified all references of the form section N (*) by searching on <section N (> and having a copy tensor doc open on the right side so I could verify. Things seem OK. What about equation numbers, there are some in appendix A at the start. **** Reading on this pass not to check equations, but to look for errors arising from the many edits I have done in other sections that might influence a section, etc. Below a subsection has been read when its letter appears. Appendix A: pre,a,b,c,d,e,f,g,h,i,j DONE Appendix B: pre,a,b,c Something is wrong in the 4-piped section! ( refer to e2 x e3 e4 as the face area, but the notation e2 x e3has not meaning for 4-vectors! Repairs needed!!!! ( new day 11.4.11) All fixed! \ Appendix B: pre,a,b,c,d,e DONE Section 8: overview,a,b,c,d,e,f,g,h DONE. Appendix C: a,b,c,d,e,f,g, h DONE. References: DONE. I now have long 13 sections to review! Section 9: a,b,c,d,e DONE. But reopened and done again with more reliance on the M&S Picture. Section 10: a, b, c, d DONE. Section 11: DONE. Section 12: a,b,c,d,e,f DONE Section 13: a,b,c DONE Section 14: DONE. ________________________________________ Section 1: open, Example 1, Example 2, Cart and quasi , pictures, coord lines, Example 1, Example 2, level surfaces. DONE. Section 2: opening, a, b, c, d, e, f, g, h, i, j, k, DONE Section 3: opening, a, b, c, d, Example 1, DONE. Section 4: DONE. Section 5: opening, a, b, c, d, e, f, g, h, i, j, k, l, Example 1, m, n, DONE! Section 6: a, b, Notes on, c, d, e, f, g, h, DONE. Section 7: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, DONE. Introduction: DONE. All done with this complete edit review. I then did another full spell check at this point and wrote more details about how to do that. I then did a page view and made one adjustment after all the Maple code, a page break. Decided NOT to have a page break at the start of each section. I don't want to mess much here because edits will shift stuff later, and most people don't print anyway and who knows what the PDF will do. Decided to add company name again. I changed the unit-vector expansion coefficients everywhere to be Bn because my former bn did not support lower-case vectors. I did not realize this until I added the last Example 1 section at the end to show a sample curvilinear evaluation in polar coordinates. So once again I think I am done. PDF Production Next problem: it cannot find PDF995. I think I have to reinstall that since I now have XP. I tried the Xchange-Lite pdf printer, it worked fine but failed to build a live TOC! After fiddling a long time (see sofware notes / PDF Production files) I got that thing working again. It then took 17 minutes to process my 150 page document. I page through it a bit, I notice that my overbars don't come out right for some reason, they are too short. as in and sometimes too low as on the V here is how I originally made these things. Pictures are ratty as before. The outline font is a little shabby but OK. Typo 1: lower case b's still present in orthog curl in summary! Maybe I should not release my stuff in PDF's. It makes a lot of work for me. But UNIX people then won't be able to see my doc. That would be very bad. After looking at Gwen's Preview later on, the Mac shows very good pix, so PDF is a MUST! I spent a day on PDF production and solved all problems. The answer is to use Acro 8.0 to create the PDF, and to use xChange only to view it on a PC and not get ratty pix. See folder noted above. So this doc and pdf is ready to be put up on the web site, but I will hold off just a little, I may want to do another reading. Let it cool off a bit. Another edit cycle starting Nov 10. The tensor doc has cooled off for a day or two. I want to scan it again before publication. App C. I made at least 10 changes here, so I guess another round is justified. This appendix I think gives some definite "value added" to the paper because it looks at nitty gritty details in an extremely simple but non-orthogonal coordinate system. Items like g and hn are computed etc etc. App A. Made a few small corrections. I think this appendix is very good and mentions things the reader is not going to find elsewhere, such as the contravariance of a cross product of vectors. App B. Many edits here, trying to make the boilerplates the same, some errors fixed, notation inconsistencies fixed. Up to section "the 3-piped". // Finished about 2 PM, heading to do Jim Thunderbird. This appendix stinks, but I don't want to replace it, I simply abandon it as they say. It does get the points made which I want to make, but the presentation is lousy. I had no previous knowledge of this subject, and suspect readers won't either, so this long winded approach will help them at least. Section 8. Read it, extremely dry, few tiny cosmetic edits. This was the subject that launched my ship. Section 9. This is pretty good I think. Added comment about the lousy B font and why it is only needed in the formal formulas, not in practice. The Christoffel derivation also is pretty good now that I have associated it cleanly with one of my Pictures. Section 10. Seems OK, got rid of some bk in favor of Bk. Added a comment on the phrase "G'n are the covariant components of G in x'-space" . Section 11. Very short indeed. Eliminated redundant N=3 orthogonal example. Section 12. Lots of fixes were needed here, again getting Bk into play. Also had Cn expressed wrongly in many places. Just shows that after first round of edits, you need to let cool then proof again. SNAFU!! Once again, for perhaps the 27th time, I am unclear about curl B as a tensorial vector, so I am off once again in a separate doc to review this once again. This has now advanced to the question of whether ε is a true tensor or not. Weinberg and others say it is, others suggest up and down make no difference. So I need to detangle this, it has been lurking there in hiding and I can no longer sweep it under the rug. So restart the tensor days counter! I am NOT done. I don't understand things like εabc∂bBc which I have been using. Maybe I can make it be a tensor by definition or something like that. Right now it is a confused mess. Hold the presses! Weds Nov 16. Much is happening. Appendix D is under construction. I have two sphericals sections ready to install, and a section on handedness I think for Section 7. But there are still some mysteries yet to be unraveled so I will hold on these installations for a while. I have to somehow clean up my comments about how this transforms as already written in Appendix A section (e) or whatever it is. Also, what about those εabcεab'c' rules? Is there something going on there I need to know about. I only know this in Cartesian x-space. Nov 19, 2011. Am now editing the completely rewritten Section 8. (a) done. (b) done. (c) done. (d) done. (e) done. (f) including both examples sections, done (g) done (h) done (i) done. (j) done -- all done!! I have now installed this new Section 8 and threw out the old one! There is much editing left, and more stuff goes in Appendix D, questions remain open. I am keeping Appendix D separate until it is complete. Section 9 divergence review. OK, here I fixed up the scalarity argument of div B, then replaced J with g' everywhere. It is better and done now. No need to refer to this scalarity in Appendix D, one less section needed there. Two Christoffel sections cleaned up. Used the Γ and Aa;b notations as the main ones since Weinberg does this. And referred explicitly to Weinberg on the covariant derivative. My job was only to explain what it is. Cleaned up the curl section, adding comments about εabc and getting all the ε positions and primes right. Here is a small "do list" 1. Get that Levi paper from the French library and try to see what notation he really used. Quote this thing and add it to the References section. Partially done. 2. Add the covariant εε stuff and εε derivations to Appendix D. DONE. 3. Clean up the vector Laplacian section since things really are vectors now! DONE 4. Repair the main introduction survey as needed. DONE. 5. Install Appendix D. 6. Repair ε δ section 7 (h) and find "tensor" quoted references to ε and fix them. DONE. 7. Repair Appendix C in light of tensor density information etc and the "Cartesian view" stuff. DONE. 8. Update the references with Ricci and Hermann. DONE Web search for Levi paper. Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen (Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201 Found it, wiki had a PDF ready for me. In French which is fine. It matches the above Google English translation, and is submitted 1899, so for sure I can learn all about the notation. R. Hermann, Ricci and Levi-Civita's Tensor Analysis Paper (English translation with comments) (Math Sci Press, Brookline, MA, 1975) 260 pages ! M.M.G. Ricci,T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen (Springer) 54 (1–2): 125–201 (March 1900) 77 pages In this translation Hermann inserts long sections of his own notes relating the authors material to modern mathematical concepts such as rings, modules, tensor products with signs, in the sense that Vij = Vi Vj, moving frames. frame bundles, fiber bundles, congruences, Killing vectors, orbits, The Chrisoffel reference, who knew about covariant and contravariant things I think. This reference on page 138 of Christoffel. Appendix D review. (a) done (b) done (c) done (d) done (e) done (f) done. (g) done. // it is now Nov 22 (h) done. (i) done. (j) done. (k) done. References: done. Reread introduction: done What about that 1869 paper of Chrisoffel and the words covariant etc? What about the idea of a direct product for tensors? “Über die Transformation der homogenen Differentialausdrücke zweiten Grades,” in Journal für die reine and angewandte Mathemarik, 70 (1869), 46–70, 241–245 Here is perhaps another instance of the 1969 paper [GMA] Gesammelte mathematische Abhandlungen, 2 vols., Leipzig-Berlin: Teubner, 1910. [1868a] Beweis des Fundamentalsatzes der Invariantentheorie, J. reine angew. Math., 68 (1868), pp. 246–252; GMA 1, pp. 269–276. [1868b] Theorie der bilinearen Functionen, Ibid., 68 (1868), pp. 253–272; GMA 1, pp. 277–296. [1869a] Ueber die Transformation der homogenen Differentialausdr¨ucke zweiten Grades, Ibid., 70 (1869), pp. 46–70; GMA 1, pp. 352–377. [1869b] Ueber ein die Transformation homogener Differentialausdr¨ucke zweiten Grades betreffendes Theorem, Ibid., 70 (1869), pp. 241–245; GMA 1, pp. 378–382. Notice that we are now up to 1919 in the GMA item first line. I. Original Works. Christoffel’s writings were brought together as Gesammelte mathematische Abhandlungen, L. Maurer, ed., 2 vols. (Leipzig-Berlin, 1910). It is there in google books! I see the word Covarianten on page 370 I often see the word Formen at the same time (look starting at page 347) I was able to download the whole thing as a pdf! But of course it is German. And I just don't see in it what I am looking for, but with Sylvester I do! I have now downloaded Sylvesters collected papers and and doing OCR as we speak. On the general Didn't Christoffel use single-letter vector notation? And wasn't he before 1881? Something is not right! I am going to have to open that can of worms yet again. But first I have to clean up my cov/contra explanation, I have tried many times but it always wobbles and crashes down in the end. Weds Nov 23, 2011 continue I installed a new chunk in Section 7 showing the meaning of covariant and contravariant. I think maybe I finally have it right. I then added a little more showing how covariant applies to rotations, and managed to work in the active and passive views, the only place in this entire tome. Next: what about that last curl section that is currently in blue? While running, I remembered to do the direct product business. I installed that in the rank-n tensor section of Section 7. Now back to the curl idea for N>3. DONE! I nailed it completely I think! Only one item remains: my comments about using a single letter for vector, Section 2 (e). I failed to take notes on my Sylvester scan. Page 200 is where he does all the concomitance stuff. Sylvester says to consider two scalar functions ("transformations") F(Ax) and G(Ax) covariant or concurrent F(Ax) and G(A-1x) contravariant or reciprocal Now consider F(Ax) and G(Ax) covariant F(Ax) and H(Ax) covariant => G(Ax) and H(Ax) covariant of a covariant is a covariant F(Ax) and G(A-1x) contravariant F(Ax) and H(A-1x) contravariant => G(A-1x) and H(A-1x) contravariant of a contravariant is a covariant F(Ax) and G(Ax) covariant F(Ax) and H(A-1x) contravariant => G(Ax) and H(A-1x) covariant of a contravariant is a contravariant http://www.archive.org/stream/117714283#page/4/mode/2up J.W. Gibbs, E.B. Wilson, "Vector analysis" , Yale Univ. Press (1913) E. B. Wilson (notes of J.W. Gibbs), Vector Analysis (Dover, New York, 1960), OK, I think it is once again done! Nov 24, 2011. Cleaned up the Lie section a bit, added Christoffel reference, cleaned up the direct product presentation, it was no good before. PDF made. Bug noted: in the special relativity section, the Ji generators are not covariant, indices seem wrong, look into this please. In this section, analysis should be replaced with Algebra in several places! And rotation group should not be capitalized DONE Maple line missing closing paren! DONE Why big break after the two Maple boost drawings? FIXED. Other notation used in the ε business. Nov 25, 2011 continue review. Section 8 overview done a, done b, done c, done d, done e, done f, done g, done i, done j, done I don't want to proof this ever again it is done! This is somewhat rough sailing for the reader. There are two spaces to worry about, and four objects in each space (including magnitudes) . Notation is a bear, and then we have that annoying visualization x'-space business. It is the way it will be. Section 9 a, done b, done c, done d, done e, done I think Sec 9 is finally stabilized and contains no lies or arm-waving. I can think of nothing to add or remove. Section 10 a, done b, done c, done d, done Section 11 only one section , no letters, seems OK. Section 12 a, done b, done c, done d, done e, done f, done Section 13 a, done b, done c, done Section 14 main body, saw nothing obviously wrong like old symbols. Example 1, done. Nov 26, 2011 continue review. I will now read things ignoring sections that I know have not been altered in a long time. Section 1 done. Section 2 intro = done and fine. a,b,c done d,e done f,g, done h,i done j,k done DONE Section 3 a done b,c done DONE. Section 4 DONE Section 5 a, b done // took a long time just for these two, now we have a better start off c, d done e, f done g, h done i, j done k, l done m, n done Section 6 a,b done c, d done e, f done // problems in section f, I am confused about the overbar on one of 4 expansions! g, h done i DONE Section 7 a,b,c done d,e done f,g done h,i done j done k,l done m,n,o,p,q done r,s done t,u done v done DONE! After this I perused the appendices, then did some print preview fiddlings. Not all views agree with each other. Time to make a copy and then try a PDF. Pagination: Started each section except Refs on a new page, and then did some tuning of orphans (single lines at page bottom) . This way, when a change is later made in one section, it won't throw off all other sections in the orphan sense. I don't so much mind widows, which are single lines at the top of a page which finish a paragraph on the previous page. Nov 27, 2011. (1) Redid the gradient section, no more method 1 and method 2, it is better; (2) added comment about xB for curl notation. Am now doing a manual full orphan scan, it is going well so far. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,A,B,C,D, Refs This is all much improved. Adding page breaks is better than adding blank lines because in normal view nothing bad happens. We have dropped now to 191 pages from perhaps 198 yesterday. Just did a pass through the intro, made a few changes. Nov 30. PDF make takes only 3 minutes now, see Stackel edit log and PDF. Tagging is off. A mere 2 MB! This PDF is not up and the plucht site and the bookmarks are working fine. At this point there are 14 chapters and App A (E vectors) ,B (N-piped),C (ellipt polar), D (tensor densities). There is no covariant Ch 15 yet. Nov 30, 2011 Release ____________________________________________________________________________________ Feb 19, 2012. Since the last entry above, I have added Appendices E and F. (tensor expansion, affine connection) These were constructed externally, visually spell checked, and then installed. I then did some small modifications to both Apps. Today I have decided to fix the Sans Outline problem. Sans Outline Problem and Solution. The problem is that the font has vector perimeters which are rendered too lightly in PDF (anybody's PDF) and just look bad in PDF as a result. Maybe not on a Mac, but on a PC with any PDF viewer. The problem is the font itself. I looked through a lot of free outline fonts online, and realized that there will always be a problem with a font that is "composite" in structure at various font sizes. To view a font you just download it and click on the ttf and you see a summary of what the font looks like at lots of sizes, and whether or not it has lower case characters. So I made this decision: Decision: Replace all Sans Outline occurrences with Script MT bold. I thought of doing this once before but was put off by the lower quality of some lower case letters, one of which I was using a moderate amount vi . But now that I know that these will ultimately be replaced by things like vθ, I am not so concerned. Also, the fact that this font is "bold" makes components look bold like vectors, but on the other hand a script font needs "boldness" to get the fine lines rendered. I made this global change after experimenting with a small file. I saved the previous file as itself with a sans suffix. I then made a full PDF of tensor doc just to check how things look, and they look just fine. Notice taht this script MT bold is now being used for several different purposes: representations like R(2) R(1) R(0) certain areas, volumes and lengths in the transform of same section 8 unit vector components (my current change) I had to manually change the sans outline occurrences inside field codes, only the curl stuff. All done. I have to now make a small consistency change. In Section 6 (f) I changed script coefficients to αn in a place where this is a temp variable only for a few lines. Did not want to use script since that is now going to be associated with unit vector components. Done. Added a det(T) analysis as App F (f) and think this round of changes is now done. Mar 8, 2012. Things have changed pretty heavily in the last three weeks or so. I was not intending to do so, but now all that fancy Weinberg tensor semicolon affine connection stuff is included with every single thing derived by me, and I think I actually understand it for the first time ever. Appendices now go up through H ! I had to split the doc in two because the TOC stopped working around 250 pages and saves were very slow, etc, it was just too much for Word in a single document. Added Section 15 with fast covariant derivations of the diff op results, a fitting final section I think. The vector Laplacian was a bear, but I managed to get the matching results in every single case. I plan to let things cool for at least a few days. Every time I let it cool, I find problems, so that very well may happen again. I will need to recheck paginations in both documents, give them right names, and do PDF production and then release. But cooling for now. Mar 9, 2012. Doing some cleanup. Scanned both docs for red text and handled as necessary. Spelling fails in both docs now, too many errors due to the equations. Their elements are considered misspelled words. I might be able to fix this problem. I am checking pagination on tensordoc1 right now, first 5 sections now OK, had to do some adjusting, viewing in print view and 100%. Then Sections 6 and 7. Section 8 and 9. Sectoin 10,11,12 done. Sections 13 and 14 have bugs in the displayer, things not stable all the time, I have seen this before. But they seem OK. Section 15 and done. This took quite a while! Now starting tensordoc2, the appendices. A really had no changes. Nor did B, both look OK. C is OK, made one change. App D done. App E done. App F done, I adjusted the red letters and fixed some typos. App G had more strangeness, but seems OK. App H, all done as best I can do. Having done this stuff, I will now let the cooling off period continue. Mar 9, 2012 Continue later in the day. I gave docs their final names, and now I want to do some proofing. Played with spell checking, it cannot even do the first 100 pages of the first doc. Spell check: the copy out and scan method. Sections1-5 OK. next chunk OK, corrected two p 100-135 corrected 2 more and rest, OK. When you find one, just search for it in the main doc. Appendix spell check. Γ ' T ' first 30 p , OK none p 30-60 , OK none rest, OK, none I will try to make PDF's now. Done, 7 minutes with tag. After doing this for doc 1 only, I read Section 15 and found 9 (!) errors below! There is no substitute for actual proofing by me! Checking equations is something I have not been doing too much. I think a lot more proofing is needed! I really need to read through the NEW sections most of all. After making this PDF: (1) typo page 155, should say : x' and x'-space objects (prime missing) DONE (2) page 156 "and other x'-space coordinates" does not make sense. DONE (3) page 156 " space, This is so" should be a period. DONE (4) page 157 "B'n are the contravariant coordinates" should be components. DONE (5) The notation Γ' is very hard to read in the PDF, perhaps Γ ' with some space there (all). DONE (6) page 159 "is" should say "would not be" DONE (7) page 159 "and this is what appears in (*)" repeats what was just said. DONE (8) the last equation on p 160 is missing three primes on ∂ . DONE (9) same for first equation page 161. DONE Mar 10, 2012 Continue proofing parts of doc 1 . (1) p 155 the f should be primed in the section 10 review, or should say f ' = f somewhere. DONE (2) global replace f' with some spaced version. (did in doc1; and doc2 ) DONE Did another primes review of Sec 15, all OK. Names and Symbols: (3) remove the phrase "the magnitude of" and replace with "the" DONE p45 need to bold F on F(x). DONE p 47 replaced g is a tensor density with g is a scalar density, page top/ DONE p 50 top, prime looks wrong on x'μ DONE p 51 One can paragraph, need space after a period 2-vectors. DONE p 53. Why does the F = ma equation look like vector = vector if Γ is NOT a rank3 tensor??? Is it m? Don't have room to add much text here due to next page! [ I scanned Weinberg, maybe the velocity is not a vector when Γ ≠ 0, let's not bring this up in tensor doc! ] p 56 say "solid steel". DONE p 56 -- why are the T superscripts so dim? Maybe fix other places too. (the Lai footnote) DONE p 63 -- fix alignment of |un| DONE p 67 heading should say The x-space en coordinate system. DONE p 67 If N is even, handedness stays the same under F = -1. DONE Status: I have now taken a pretty hard edit look at Section 6 which did have changes! p 75 near top replace |J| with J. DONE p 78 delete space before Although, near top. DONE same line, replace once by one. DONE p 79 big problem to fix. In the second case, things are supposed to be "up tilt" !!! DONE p 80. Should point out in Theorem 1 that not valid for dev notation S matrix! This is quoted next P. DO p 82, replace Item (3) with "the third subsection above" DONE p 94. I think the added work on R-1 = RT can make this discussion much better Do it! (no go) DONE Somewhere I wanted to add a simple "rule" for moving R from one side to the other of an equation. This rule could be stated on page 96 but maybe should be elsewhere. Then have to fix pagination! DONE p 98 use words like homogeneous for the Maxwell equations! DONE p 98 replace with lower-index . ???? OK, enough for Section 7. I ferreted out some errors as seen above. I don't have enough energy to go through Section 8 again. I think it is OK now, it saw lots of hours. It is recent and so should not show conflicts. A some point, I might check ALL internal references in some clever way. DONE Section 9: p 120 fix underline on Note: DONE table add M&F where M&S was added! DONE p 127 remove word presumably since I now know it is correct. DONE p 154 maybe increase font size on the script v , check this globally. DONE I did the Intro recently so not again. I have been reading in xchange 130% and everything looks clean and readable except as noted above. I will now peruse doc2 and get an edit list for it as well. No edits fixed yet. But I need to get it into a PDF first. Appendix H maybe delete comment 1, rename comment 2, and really use notation divT everywhere. DONE p 95: T' has the same spacing problem as Γ'. DONE p 98 change to "so the above sequence for divT... " DONE Appendix G p 84 as example: go through headings and make supers and subs be bold. Same for doc 1. Make sure the TOC does not shift one line to another page in doc2 !! If so, undo this fix. DONE p 84 *b( something wrong with prime in second equation. DONE p 88 components of v, v should be bolded. DONE That concludes App G. Since so new, no legacy problems. the Lai book has always had three authors, maybe I should say Lai et al to be fair to other authors, or at least comment on this in the References. Appendix F: p 67 delete space in front of (polar coordinates) DONE same space thin 7 lines later DONE p 68, in (b) say "the identities are ", there are no claims any more since no theorem any more and say "the first identity above" a few lines later. DONE p 69 top not theorems but identities DONE p 70 bottom, identify→identity, may do a global on this. DONE p 77 the b-related terms section is messed up, ↔ probably wrong, and equation is wrong. Fix. DONE p 80 replace N=2 by J=2. DONE p 82 the "two vectors" block has at least two errors, fix DONE end of appendix F. Appendix D p 36 add comma after Section 7 (j), DONE change ordinary to ordinary = regular = standard-issue or some such. DONE with J being the Jacobian noted above. DONE p 37 should say "within a tensor density" DONE p 39 heading" there is really only one: εabc... " DONE p 40: third last line text, add " from (*) DONE p 41 repair s =-1 with a space. DONE p 42 align |g| in item 2 DONE p 43 4th line replace two e by ε DONE OK, I petered out at the end on App D, think all the ε stuff is OK. I am done for this round! How does doc2 pagination look in PDF? Found one problem page 34 pagination something is wrong in the PDF DONE Sun Mar 11. I just added a Inversion Rule and a Cancellation rule to section 7 (c) did some edits on the matrix multiplication section. used the cancellation rule three times to speed up my section (u) proof. I need to use my inversion rule in Appendices where I never do! DONE How about numbering sections in my "further development" deal, since I did this elsewhere. Also do this in Section 8 and then fix references to same in both cases. DONE Somehow tune doc2 TOC to be rid of orphan heading. DONE Reference Checks. (1) I just went through all of doc1 searching for "section" as a single word, and at each stop, I checked the reference against my PDF. That is, I used the PDF in a separate window as a fast check method. So this check handled "internal references" in doc1. I found and fixed about 8 bad references!!! (2) Next check is to put up doc2 in PDF and search doc1 for "appendix" or "app." // That took a long time and again I found maybe 4 errors. I changed the App. to Appendix. (3) Next is to check doc2 for "Appendix" references while viewing doc2 in PDF. Done. (4) final check is search doc2 for "section" while viewing doc1 in PDF. Done. This took a VERY long time, and I did not do all the "inside a section" checks. I won't ever be doing this check again! Repaginate required: A, H, E, B, C, D, F,G all done doc 2. be sure to redo both TOC before being done! DONE make sure all equations start at the same tab. DONE I also did the right Properties. Time I think to make new PDF's!!!! march 11. Done, and I think I am ready for a release. I did them without tags, and the doc1 takes 4 minutes and doc2 only 2 minutes. Top left button in Word, doing this with Acrobat 8. I will LOG HERE the errata as they accumulate from whatever source after Mar 11, 2012. New link for the free pdfchange viewer http://www.tracker-software.com/pdf-xchange-products-comparison-chart Doc is now in 2 parts and has new Section 15 (covariant) and App run thru App I. March 14, 2012 Release ____________________________________________________________________________________ March 30, 2012. Well, I have made many changes and reissued both docs. Appendix I on the vector Laplacian is new, and several small inserted sections were added throughout. I had something major wrong but forget now what it was! Oh yes, in the vector Laplacian section my V(1) form was completely wrong (and I thought I had verified it in Section 15), that was very bad and is now all cleaned up. I found this when Maple kept giving bad results. Generalized the covariant derivative theorem to tensor densities, and the Leibnitz as well. We are now 168 + 117 = 285 pages long. It is a much better document now that on the previous release. So I will put it out right now at 9 PM. I also made a new PDF of the Maple guide and put it out there today. March 31, 2012 Release ____________________________________________________________________________________ April 16, 2012. Yesterday I read through App B and then App A and then App D. All were OK, though I always make tiny changes on any reading. Today I read App C and it too is OK. I am looking for Big Mistakes, or chances to cross reference to newer Appendices. App C is good because it is such a simple system in which to look at things. // Just finished App E, and the added section (h) is a winner I think. Again, lots of tiny changes being made, getting notation consistent etc. Task 1: (?) Change picture C to show ξ space maybe and make this uniform everywhere. Continuing. App F was very good I think. More tiny fixed, periods after some equations, etc. Then App G is rather workmanlike but gets in some Maple work, reads OK and results have Lai verification. These results might be hard to find on the web. App H is a bit brutal, we just want the results here. I next did a quick read-through on App I on 2B -- you would only want to put this kind of detail in an appendix, it is very messy (but I don't think there is an easier way). Now I will read the intro to the appendices. OK. Well, I have run out of steam on reviewing stuff, at least all the Appendices got done in this pass. So I will prepare for publication. Added copyright statements. I have to paginate again. Did some of that, my patience wears thin. Maybe just make the PDF's and see if any disasters. no blank pages, seems OK, dates are Apr 16. April 16, 2012 Release ____________________________________________________________________________________ June 25, 2012. Today I added a bit to the "motivation" start of Appendix G. I wanted to be sure to show the convective derivative in its usual form, not just the strange Lai form, and then I wanted to motivate the u and v cases more. I added this, made a new TOC, then made the pdf. I noticed that the other doc is May 31, and I have no record of what changes I made there, but I made a new PDF from it as well, and I will today push both out there with these dates. I decided to update these both with today as the official date, otherwise people might wonder if the two files are out of sync! June 25, 2012 Release ____________________________________________________________________________________ Sept 12, 2012. Did some errata to both docs, see errata list. Republishing both today. Sept 12, 2012 Release ____________________________________________________________________________________ Oct 19, 2012. I have been doing lots of tensor doc update work and today I "installed" lots of stuff: new Appendix K on (T) (major) new Appendix J on Deformation Tensors (major) replaced Section 5 (o) on tensors in continnuum mechanics (major) updated App E section (h) concerning unit vector expansions (major) new App E section (i) on mixed-basis tensors (major) new App E section (j) on what is a tensor (major) replaced Section 2 (k) on meaning of words vector etc (major) update of section 5 (a) concerning the invariance of (ds) or lack thereof update of Section 5 (i) ending concerning invariance of (ds) or lack thereof and idea of "two metric tensors" in CM. addition to Section (6) of a few lines on En um = Rnm addition to Section 7 (s) of a few lines on en um = Rnm \ change to Section 8 (c) 7, removed script g's in that section. I have the following still to do repair references to appendix J and K since I just swapped these names (DONE) make sure Section 5 (o) refers to Appendix K on deformation tensors (DONE). update the introduction to reflect all of the above changes (DONE) go through any collected errata repaginate the whole thing carefully, so much has changed (DONE) republish on the web Oct 20, 2012. Task: The comment Concerning the invariance of (ds) in section 2a needs repair, it is now stated wrongly. First go read the later comment in Section 5 (i). Done. Now I have edited the earlier 2a section comment, removing wrong statements and tying it to other sections. I wanted to get an "early word" about the CM issue in this Section 5 on the metric tensor. DONE. Repaginate Section 5. DONE. Three somewhat different applications of tensor analysis are treated concurrently. Our primary concern is the subject of curvilinear coordinates in N dimensions. All the basic expressions for the standard differential operators in general curvilinear coordinates are derived from scratch (in several ways). These results are often stated but not so often derived. The second application involves transformations connecting "frames of reference". These transformations could be spatial rotations, Galilean transformations, the Lorentz transformations of special relativity, or the transformations involving the effects of gravity in general relativity. Beyond establishing the tensor analysis formalism, not much is said about this set of applications. The third application deals with material flows in continuum mechanics. Repaginate Section 2. DONE, nothing to do. Repaginate Section 6. DONE, has 3 lines hanging over onto last page which is OK. . Repaginate Section 7 starting at (s): DONE. Repaginate Section 8 starting with item (c) 7. DONE. Repaginate Appendix E starting with section (h): DONE. Paginate Appendix J: DONE. Paginate Appendix K: DONE. OK, at 1 PM I think I have done everything. Question: is there some simple way to get Section numbers in header or footer? I will practice with a toy document. Answer: Yes, there is a very simple way to do this, see "how to make distinct section headers.doc". I am now going to undertake this on a scratch copy of the main tensor doc and see how it goes! First step is to insert "next page" breaks in all the right places. Keystroke to insert next page break is Then go through and do the headings, very easy, all done. Make new TOC, fix problems in it. Ready for PDF! It seems to hang for 3 minutes while saving bookmarks, then it recovered, total job maybe 5 minues all done. Bug found: page 49 has a big gap. I have to repaginate after this point and then try again! No other messed upages noted in the PDF. Repaginate Section 5 after p 49. DONE. Changed my Lai reference comment a bit. Should I make the header longer, like Section 11: The Transformation 1. Yes, I did it and I like it. Now you VERY quickly know that you are in Section 6, whereas the sol 6 does not quite convey that so quickly. And the same will be true for the appendices. Make a new TOC, clean it up, and go for another PDF. // It is done. _________________________________________________________________________________ Bug: Something is wrong. in Section 5, not all the sections are appearing in the bookmarks! But the TOC works fine on all the little sections. Ouch! This is in my nice PDF viewer. They are missing in Acrobat as well. The first problem occurs in Section 2. The outline view looks good in Word, things not missing. Let's just "do it again", maybe I fumbled something. I ran it again and let it completely alone, but we have the same problem. Which sections have this problem? 2, 5, 7, -- all other sections are OK !!! Let's delete the entire TOC, save the file, then create a brand new TOC. // That did not fix it. How can the TOC links work right but the bookmark display be wrong, if they are based on the same data? There are PDF settings, but I think I have them right. OK, let's make a document with just Section 2 and see what it does. // Very good, same problem, much simpler file, much faster debug cycle! My doc has only section 2. Whatever is wrong is wrong with this much smaller file! Let's knock out section k, the only one reporting, and see what we see. // Now no subsections show at all, that is good. Let's knock out half of the remaining ones. Now we just have abcd. Same problem, none of these show. Now knock out c and d. Problem still there. Now make the two sections a and b very small. Doc is now just one page and still shows the problem! Shrink it more. Remove all above TOC. Problem still there. Now remove stuff above (a) scalars and below TOC. [ there is a space in front of the (a) I just noticed] Now make each of the two sections be just one Done, and here is the ENTIRE document showing this problem Now let's remove that space to make sure it is not doing it. // No, that was not the cause. OK, enough. I was able to create a tiny file showing the problem. I have written this bug up in the PDF production bugs area of software notes. that is to say, software/ PDF production. The fix is to just reheaderize each missing header, and to realize that headers not on separate lines don't show in the PDF bookmarks pane. I now have all the bookmarks appearing in the main tensor doc doc, and I have not worked in the appendices doc yet. ____________________________________________________________________________ I just read some of the recent additions in the PDF form and have a few errata already. But lets to on to the Appendices section and see if there are new problems. We should have section 1 = initial section sectino 2 = Appendix A ok section 3 = Appendix B ok section 4 = Appendix C ok section 5 = Appendix D ok section 6 = Appendix E ok section 7 = Appendix F ok section 8 = Appendix G ok section 9 = Appendix H ok section 10 = Appendix I ok Appendix I: Expand (B) section 11 = Appendix J section 12 = Appendix K add new page separators in Appendices and save. DONE. add new headings, save, and scan in print mode to see if OK. DONE. Change date to Oct 21. Let simmer at 100% till done. Main doc: make new TOC, then fine tune TOC appearance including pagination. DONE. Make first Oct 21 main PDF: done, seems OK Make first Oct 21 app PDF: it has missing bookmarks! So let's reheaderize them now! Found some bogus paste in App doc!! All of Appendix J except its (a) got pasted into App I !! So I fixed this up and repaginated App I. Make new TOC, tune page alignment (non needed), fix blues. DONE Let simmer at 100% then save. New PDF cycle. Snuck in a single line on viscometric fluids. Now new TOC, just did a cycle. Oct 21, 2012 Release ____________________________________________________________________________________ Now 14 months have gone by with no updates. *********************************************************************************** Make tensor doc a single doc, Feb 2014 *********************************************************************************** Feb 10, 2014. I just put out my Transmission lines doc, and it is 341 pages, which is the exact same length as tensor doc's two parts. I will just try to make a single doc today and see what happens. I combined the docs and before doing a lot of cleanup, I just want to see if it will build a PDF or not. // It did it, a single doc, no problems at all! Took 7 minutes. So now I need to do lots of work since this is going to be a rare release. 1. Remove all notes about Appendices are separate. Done. 2. Lets read the entire overview and summary right now. Done, it is all OK. 3. Are there outstanding errata? My errata doc is all red, meaning all fixes have been made. 4. This entire doc has no equation numbers!! And no Figure numbers. That would take a lot of work to add, and I do not want to do it! 5. I think I should add Section Headings, since not very hard to do, and creates some sense of organization in the document. // They were already present, but I messed things up joining the two docs, I think they are straightened out now. This is the only way you know where you are in this monster. Someday I made add equation numbers, but of course there will then be no internal references to them, so seems pointless. 6. Did some pagination checking, it all looks good. Especially at the joining boundary. 7. Editing in the references? I added my usual P. Lucht item. 8. Clean up the TOC. Done, no underline, make it black, etc. The TOC References hangs over onto a separate page. I fixed this by changing bottom margin which seemed only to affect all TOC pages. But now I need a new TOC since this removed a page! No, it seems to have adjusted itself OK. I did a cycle and stared at the PDF, it looks good, so I will release it today. This is my first "single document" release of tensor doc! I took this opportunity to remove the space before .doc that was present in my two-doc versions! This may throw references, so be it. *********************************************************************************** Add equation numbers and other changes March-April-May 2015 *********************************************************************************** Here is a history of this 39+ day editing process: added equation numbers everywhere 3.26 redid entire Section notation from (1),(b) to 5.1,5.2 as in my other docs 3.27 Proof Chapters 1,2,3,4 3.29 Proof Chapter 5, much reorganizing here (ds = ds' issue) 3.30 Proof Appendix K 3.31 Proof Appendix C 4.1 rewrite section 5.16 on cont mechanics and two metric tensors etc 4.2 Proof Chapter 6 and Appendix A 4.3 Proof Appendix B, start into Chapter 7 4.4 Finish Proofing Chapter 7 4.8 Add section D.12 on how determinants transform 4.10 Proof Appendix D and start on Appendix E 4.11 Work on Appendix E, add completeness, clean up A(b) = BABT 4.14 Proof Appendix F 4.15 Proof Appendix G 4.16 More work on Section 5.4 4.17 More on Appendix E and dyadics and the bn basis 4.18 Reproof Appendix E 4.19 delete section E.11 on "what is a tensor" 4.19 fix up Trr = λ [ Err+ Eθθ] stuff 4.19 Reproof Appendix G 4.20 Clean up Γ definition in App F, clean up G.2 4.21 Proof Appendix H and Appendix I and start Appendix J 4.22 Rewrite E.9 regarding M-covariance stuff 4.23 Work on Appendix J 4.24 Proof Chapter 8 4.26 add Section 7.18 basis vector summaries 4.26 Proof Chapter 9 4.26 Proof Chapter 10 and Chapter 11 4.27 Proof Chapter 13 and Chapter 14 4.28 Proof start on Chapter 15 4.29 Rewrite F.1. on the Γ definition again 4.29 discover that Lai uses a different affine connection, delete a section on that 4.29 deal with confusion about εabc...x;α . 5.1 Proof Chapter 15 5.2 creating this little history, do repair on D.11 5.3 I made a huge number of repairs and upgrades and clarifications, too many really to enumerate. There were serious errors that are now fixed. Doing this stuff just burns days, there is just no way around it! Feb 10, 2014 Release ____________________________________________________________________________________ At this point, 13 months passed by with no releases. Then new work: Update 3/26/15 I am working today in a "practice" doc, seeing what happens when I add equation numbers. I will give figures the same equation numbers to make them easy to find. I don't want to issue too many equation numbers, just want to be reasonable. I find that when I do this, I have to repaginate at the same time so that equation numbers land in good places. I spent all day doing this, and then ... Update 3/27-28/15 Then I decided that was no good, don't want things like (4.174) as an equation number. If you have to add equations, it makes a huge mess! So I decided to redo everything so tensor doc has the exact same system as lines doc! This means renumbering subsections and so on, but I am convinced it is the right way to go, after yesterday! I just did Ch1, Ch2 and Ch3 quickly in this new manner. Notation (2.4.3) seems easy on the eye, no letters mixed in, easy to see ordering and easy to find something. I just fully redid sections 1 through 7. In Section 7, changed numbering scheme to get letters on lowest level, then decided to go to a 4 digit code for eq nums in order to maintain some modularity for future changes! This took a long time of course. As of 1 PM I have now done Sec 1→7 and App A→D. It is very tedious. Second session: Did App E, then F, then G, then H, then I. Just adding and aligning and sequencing, am reading nothing at all here! Done at 9 PM for now. Update 3/29/15 Comments On Tensor Doc and Equation Numbering The whole point of putting all the pieces of tensor doc into one document is to allow for cross references between different parts of the doc. Otherwise I could have them be lots of separate docs. Without equation numbers, my cross reference system was "see Section 5 (g) ". I could not be specific. Plus, it takes a reader forever to FIND section 5 (g) since there are no headers. With everything having sequential equation numbers, it makes it MUCH easier to look up a cross reference. Fancier documents just jump to the cross reference when you click on it, but I am not that fancy. Reader does need an effective split screen by loading the doc twice into a PDF viewer. Comments On Tensor Doc Proofing Order Careful attention is needed here, I will come back to propose a proofing sequence best suited to handling all the cross references. *************************************************************************** Appendix J and K done. Chapter 8 done, very painful. Section 9 went OK. Section 10 easy gradient. Chapter 11 also very easy. Chapter 12 not too bad. Chapter 13 went quick. Chapter 14 and 15 also done. AT the very end, the TOC has problems so I close out word and try again. Yup, sure enough, just as I edit the very last section, the TOC update routine won't run, every time I get the memory complaint followed by a debug complaint. The debugger points to a certain line in my VB code for update TOC. Closing out Word does not fix this problem. Can I just delete the TOC and make a new one? Yes, no errors doing that. Only a problem updating the TOC already existing. Proofing of Chapter 1. opening material OK Example 1: Polar coordinates (N=2) OK Example 2: Spherical coordinates (N=3) OK Cartesian Space and Quasi-Cartesian Space OK Pictures A,B,C and D OK Coordinate Lines OK Example 1: Polar coordinates, coordinate lines OK Example 2: Spherical coordinates, coordinate lines OK Level Surfaces OK numbering for entire chapter OK Comments: This is a very short chapter, and there are no specific references to other points in the paper. There are a few reference to other chapters in general. In any proofing pass, one should always do Chapter 1 first. What section should come next? chapter 2 seems right. tent = tentative, pending cross ref resolution Proofing of Chapter 2. 2.1 Linear Local Transformations 20 OK tent 2.2 Scalars 21 OK tent 2.3 Contravariant vectors 22 OK 2.4 Covariant vectors 22 OK 2.5 Bar notation 23 OK tent 2.6 Origin of the names contravariant and covariant 23 OK tent 2.7 Other vector types? 24 OK tent 2.8 Linear transformations 24 OK, did a rewrite! 2.9 Vectors that are contravariant by definition 25 OK 2.10 Vector Fields 26 OK 2.11 Names and symbols 27 OK tent 2.12 Definition of the words "scalar", "vector" and "tensor" 27 OK tent numbering for entire chapter OK Apart from pagination and cross refs, I am completely happy with Chapters 1 and 2 on this review. Proofing of Chapter 3. 3.1 Differential Displacements 29 OK 3.2 Definition of the en ; the en are the columns of S 30 OK tent 3.3 en as a contravariant vector 32 OK 3.4 A semantic question: unit vectors 32 OK tent Example 1: Polar coordinates, tangent base vectors 33 OK tent Example 2: Spherical Coordinates, tangent base vectors 34 OK 3.5 The inverse tangent base vectors u'n and inverse coordinate lines 35 OK Example 1: Polar coordinates: inverse tangent base vectors and inverse coordinate lines 36 OK numbering for entire chapter OK Apart from pagination and cross refs, I am completely happy with Chapters 3 on this review. Proofing of Chapter 4. // very short chapter OK Update 3/30/15 I think Chapter 5 is the next item to proof in the general logic flow, and it will take lots of time and is perhaps the heart of my document. Proofing of Chapter 2 5.1 The Picture D Context 40 OK 5.2 Definition of the metric tensor 40 OK tent 5.3 Inverse of the metric tensor 42 OK 5.4 A metric tensor is symmetric 43 OK 5.5 det(g) and gnn of a Cartesian-generated metric tensor are non-negative 43 OK 5.6 Definition of two kinds of rank-2 tensors 43 OK 5.7 Proof that the metric tensor and its inverse are both rank-2 tensors 44 OK 5.8 Metric tensor converts vector types 46 OK tent 5.9 Vectors in Cartesian space 47 OK 5.10 Metric tensor: covariant scalar product and norm 47 OK but hairy Part of this section needs more clarification, but is independent of surroundings. 5.11 Metric tensor and tangent base vectors 50 OK tent 5.12 The Jacobian J 51 OK tent 5.13 Some relations between g, R and S in Pictures B and C (Cartesian x-space).55 OK tent Example 1: Polar coordinates: metric tensor and Jacobian 56 OK Example 2: Spherical coordinates: metric tensor and Jacobian 57 OK 5.14 Special Relativity and its Metric Tensor: vectors and spinors 57 OK 5.15 General Relativity and its Metric Tensor 61 OK 5.16 Continuum Mechanics and its Metric Tensors 62 OK very tent numbering for entire chapter OK Ran off the end of the day after 5.15, then finished next day. A lot of progress today on Chapter 5, lots more cross referencing makes it much better I think, you can trace every single claim. Update 3/31/15 Finished off Chapter 5. However, everything regarding continuum mechanics I feel is wobbly and needs to be cleaned up. Why are there two metric tensors, and so on? Then I have to back edit earlier where I talk about ds and ds' not equal. Should I do this now or not? I think yes. Therefore, I will now skip to Appendix K which is my most detailed presentation of continuum mechanics. Proofing of Appendix K. opening material OK tent K.1 A Preliminary Deformation Flow Picture 327 OK tent K.2 A More Complicated Deformation Flow Picture 332 OK tent K.3 Form of a solid constitutive equation involving the deformation tensor 337 OK tent K.4 Some fluid constitutive equations 338 OK tent K.5 Corotational and other objective time derivatives of the Cauchy stress tensor 339 OK tent numbering for entire appendix OK Comments: This appendix K is very "advanced" and deals with stuff near the very end of Lai. I interpret all of the Lai work in terms of "tensor analysis" of tensor doc, and that is the reason Appendix K is present. No mention is made in this entire appendix of any "metric tensors", so no clarification on that subject is provided, though I hoped it would. I feel I need to clean up the contin mech parts of Ch 5 before continuing my overall proofing. 5 PM. I have completely rewritten Section 5.2 to get more laser accuracy on the invariance of ds in different frames. It turns out that in continuum mechanics, ds is not a scalar, so the fact of (5.2.8) that ' = STG S is NOT VALID. Thus, my whole idea of using G = 1 and having ' = STS really has no meaning at all. This combination STS is NOT a metric tensor. The only metric tensor is g = g' = 1 in Lai stuff. My idea of two metric tensors is complete baloney. Maybe a "wannabe" metric tensor? So I am now going to make red lots of tensor doc due to this issue. Well, instead I will just do this -- I remove this entire chunk from Chapter 5 and declare it to be utter nonsense and I archive it right here: In applications in which (ds)2 is NOT regarded as a scalar with respect to transformation F, which includes continuum mechanics flows, things are a little different. We have here, (ds)2 ≡ dx dx ≠ dx' dx' = (ds')2 if (ds') ≡ physical distance in x'-space . (5.10.11) If x'-space has Cartesian coordinates, we have (ds')2 = dx'idx'i = δi,j dx'i dx'j and this is different from the quantity dx' dx' = 'ij dx'i dx'j . One might go so far as to define two different symbols: dx' g dx' ≡ 'ij dx'i dx'j ≠ (ds')2 (5.10.12) dx' 1 dx' ≡ δi,j dx'i dx'j = dx'i dx'i = (ds')2 = physical distance squared (5.10.13) By default, we have = g, so then dx' dx' ≠ (ds')2. In this situation, x'-space in effect has two different "metric tensors". The first is the 'km that appears in the entire discussion of this Chapter 5 and which is involved in raising and lowering indices on a tensor and satisfies ' = ST S (the fact that g transforms as a rank-2 tensor), and so on. The second we might call 'km and this is what determines physical distance in x'-space, (ds')2 = 'km dx'k dx'm . In continuum mechanics with Cartesian x'-space, we then have 'km = δk,m. There are two different "metric tensors" because there are two different "metrics" which are of interest in x'-space : dg2(x,y) = (x-y) g (x-y) = km (x-y)k(x-y)m // raising/lowering metric tensor (5.10.14) dg2(x,y) = (x-y) g (x-y) = km (x-y)k(x-y)m // physical distance metric tensor (5.10.15) For Cartesian coordinates in x'-space, 'km = δk,m, and our two metric tensors may be identified with those of "Cartesian View x'-space" and "Curvilinear View x'-space" as discussed in Chapter 8 and Appendix C (e). Here is a chunk of Section 5.16 that I am now throwing out (but archiving here) As noted in Section 5.16, there are really two metric tensors in this situation called 'km and 'km. The former 'km is the metric tensor which raises and lowers tensor indices in the general tensor formalism (in the Standard Notation of Chapter 7), while the latter 'km is the metric tensor which determines physical distance in x'-space. For the usual Cartesian coordinates in x'-space, 'km = δk,m which is to say ' = 1. The other metric tensor is given by the (5.7.6) relationship ' = ST S where is the metric tensor in x-space. Using Cartesian coordinates there means = 1 and then ' = ST S. Update 4/1/15 I think I have a plan to dig out this "two metric tensors" problem, and have started by using polar coordinate transform in Visio to create an example of a fluid flow, just to have pictures I can look at. I made some progress, but I think the Cartesian View concept is going to arise, so I think I better go off right now and proof Appendix C before continuing. Proofing of Appendix C. C.1 Elliptical polar coordinates 206 OK C.2 Forward coordinate lines 207 OK C.3 Inverse coordinate lines 207 OK C.4 Drawing a contravariant vector V in x-space: the meaning of V'n . 208 OK C.5 Drawing a contravariant vector V' in x'-space: two "Views" 209 OK C.6 Drawing the specific contravariant vector dx in x-space and x'-space 212 OK C.7 Study of how dx transforms in the mapping between x-space and x'-space 212 OK C.8 Derivation of the Jacobian Integration Rule 214 OK number ordering: OK Update 4/2/15 I rewrote section 5_16 trying to understand the "two metric tensors" and making a triple connection: the Example shown there from polars, the discussion of Appendix C on ellipticals, and the Chapter 8 generic drawing. The result seems better than last time I did it-- perhaps. Update 4/3/15 I am wondering how I can make my flow using polars go smoothly from t = 0 to t = t, which ought to be a requirement of any flow. I have x = Ycos(X) y = Ysin(X) but where is time t? And how could I ever get x = X at t = 0? Maybe we have x = F-1(X, t) as follows : x = X(1-t) + t Ycos(X) X = θ Y = r X = x' y = Y(1-t) + t Ysin(X) OK, that is at least viable. It has the right limits at t = 0 and t = 1. What would S be for this transformation? Sik(x') ≡ (∂xi/∂x'k) S11 = (∂x/∂X) = (1-t) + tY [-sin(X)] S12 = (∂x/∂Y) = t cos(X) S21= (∂y/∂X) = tY cos(X) S22 = (∂y/∂Y) = (1-t) + tsin(X) S(X,t) = S(X,0) = S(X,1) = // looks like earlier result with r,θ Perhaps I have to give up on this "example". to make it viable, I have to add time, and then it is no longer polar coords. OK, I used the above t transformation in a Comment and I think I have "bailed out" my Example and things have not collapsed. Status: I am happy with my forward transformation line by line table. and Sik = (∂xi/∂x'k) ↔ Fik = (∂xi/∂Xk) New problem: The B tensor does not appear anymore when I try to describe the inverse flow. I think I will simplify and just completely omit this second table, I archive it here after I modified it. Alternately, we can consider the above flow going backwards in time and then x = x is the starting position and x' = X is the ending position. For this inverse flow, we let F have the same meaning as in the forward flow, dx = F dX, and thus end up with this translation table: continuum mechanics our document ( Inverse Flow x → X ) X, x ↔ x', x X = X(x,t) ↔ x' = F(x,t) // this F is the inverse of the forward flow F-1 dX = F-1 dx ↔ dx' = R dx // as in (2.1.6), same F as in forward flow F-1 ↔ R F ↔ S // S = R-1 x = Cartesian ↔ = 1 X = Cartesian ↔ ' = 1 // Cartesian View B = FFT ↔ ' = RRT // this is wrong, because F = S, not R !! C = FTF ↔ ' = STS // as in (5.13.2) // Lai p121 (3.25.2) C-1 = F-1(F-1)T ↔ g' = RRT // since (')-1 = g' and S-1 = R (5.16.2) In both tables g goes with X-space and g' and g' go with x-space. For the inverse flow, the raising/lowering metric tensor ' = STS appears as C = FTF which is the right Cauchy-Green deformation tensor (again manifestly symmetric, so a viable metric tensor). OK, it is now 12:30 and I think I am completely done with Section 5.16 (in a separate doc), apart from a few cross references in red which I cannot resolve yet. I need next to look at other continuum mechanics sections of tensor doc and make sure there are no conflicts, in particular I know there are some g' somewhere. But now at 2:30PM I really want to take F → F-1 to clean up my notation. To do this, I am now attempting a third draft of Section 5.16 with this change. I added a new "Picture A" which I think helps. Adjusted all numbering. I think it is now done. Now let's see if this works with Appendix K notation. // I had to make a few changes at the start of App K only. I had to replace F = R with F = S in the two-flow picture and in the text in a few places, but after that all reference is to F, so I don't worry about R,S being wrong. I think it is OK now, still pending cross references. // I just resolved a lot of these cross references, but ones still red are for equations in sections I have not yet proofed, so they stay red for now. Where other than Section 5.16 and Appendix K do I carry on about continuum mechanics? Well, lots of places, but I don't think I have any issues with F versus F-1 and R versus S anywhere left. I will be hitting these other locations of CM as proofing continues. Browsed App C and glued in a few xrefs. I am very gradually improving the linkage between different parts of this huge tensor doc. I will now install my version 3 rewrite of Section 5.16. Done. Time 5:30 PM 4.2.15. Still working int he "practice" tensor doc which of course is now the real one! Can never go backwards. Update 4/3/15. Reread Chapter 5 since did heavy editing on it of late. // I just did a full 100% proofing of all of Chapter 5, since this is the absolute CORE of my document. On this pass I found some wrong eq nums and made a slew of editorial changes to make things clearer. It took all morning, but I think it was the right thing to do. I don't want to have lots of errors in a possible Researchgate release of tensor doc! Reread Appendix K, and again made various cosmetic repairs. Now 2 PM. Time now to "move on": Proofing of Chapter 6. 6.1 Definition of the En 70 OK 6.2 The Dot Products and Reciprocity (Duality) 71 OK 6.3 Covariant partner for En 73 OK 6.4 Summary of the basic facts: 74 OK 6.5 Repeat the above for the inverse transformation: definition of the U'n 74 OK 6.6 Expanding vectors on different sets of basis vectors 75 OK 6.7 Another way to write the En 77 OK 6.8 Comparison of e¯n and En 79 OK 6.9 Handedness of coordinate systems: the en , the sign of det(S), and Parity 80 OK full check of entire chapter numbering: OK This is more complex that I thought, there is a lot here which will probably be quoted in Section 7. This section connects to Appendix A, so maybe that is where I should go next. I am always improving the cross referencing as I go. Proofing of Appendix A. A.1 Introduction 188 OK tent A.2 Definition of En 188 OK A.3 Simpler notation 189 OK A.4 Generalized Cross Product of N-1 vectors of dimension N 189 OK A.5 Missing Man Formation 191 OK A.6 Apply this Notation to E 191 OK A.7 Compute Em • en 192 OK A.8 Compute En • Em 193 OK A.9 Summary of relationship between the tangent and reciprocal base vectors 193 OK A.10 Another Cross Product Notation and another expression for E 194 OK full check of entire chapter numbering: OK That went pretty easily. I think Appendix B has to come next, it will be painful. Update 4/4/15. Proofing of Appendix B. B.1 Introduction 196 OK tent B.2 Preliminary: Equation of a plane in N dimensions 196 OK B.3 N-pipeds and their Faces in Various Dimensions 197 OK (a) OK (b) OK (c) OK (d) OK B.4 The question of inward versus outward facing normal vectors 201 OK B.5 The Face Area and Volume of N-pipeds in Various Dimensions 202 OK (a) OK (b) OK (c) OK (d) OK B.6 Summary of Main Results of this Appendix 209 OK full check of entire chapter numbering: OK For a while I was messed up on the rule for An and En direction, but I have it right now and I remove the blue text below which was incorrectly added. Note for later. Notice that e1, e2, e3 form a right-handed coordinate system in this drawing. The out-facing normal of the top face is in the direction of n = e2 x e1 . If we now take e3 → -e3, the system becomes left-handed and the top generator face would in fact be the bottom face of the new 3-piped, although it would still be a "near face" touching the origin. The normal vector n = e2 x e1 which was an out-facing normal for face 3 has now become an in-facing normal for face 3. Doing e3 → -e3 basically causes the N-piped to flip inside out, so what were out-facing normals become in-facing normals. These area vectors are out-facing provided det(R) > 0, which is the same as det(S) > 0, which means that the base vectors e1, e2, e3 form a right-handed coordinate system as described in (6.9.8). For a left hand coordinate system the three Ai will instead be out-facing for the near faces face 1, face 2 and face 3. Note: If you take e3→ -e3 on the 3-piped, E3 changes direction because det(R) changes sign and the cross product e1x e2 does not change sign. But that is what you want to have E3 maintains its status of being out-pointing for face face3p. As for the other two cross products, E1 and E2 do NOT change sign because their cross products do change sign but detR also changes sign. Thus, E1 and E2 which started off being outfacing for face 1p and face 2p remain that way! For example, the right wall of the 3-piped maintains its orientation under e3→ -e3, it is still the right wall (and still the far wall). So saying that the whole thing flips inside-out is not correct! OK, what's next?? Have done A,B,C,K and have done 1,2,3,4,5,6. I vote for Appendix D next. No, because Appendix D is in Standard Notation, so need to so Section 7 first! In fact ALL the other appendices are in Standard Notation, so Section 7 must be next. And it is going to take a long time! Proofing of Chapter 7. opening text OK 7.1 Outer Products 83 OK 7.2 Mixed Tensors and Notation Issues 83 OK 7.3 The up/down bell goes off 84 OK tent 7.4 Some Preliminary Translations: raising and lowering indices on a vector with g 85 OK 7.5 Contraction of a Pair of Indices 87 OK 7.6 Dealing with the matrix R 87 OK 7.7 Repeat the above section for S 88 OK 7.8 About ε and δ 88 OK tent 7.9 Matrix Multiplication, the meaning of RT, and Rotations in Standard Notation. 89 (a) OK tent (b) OK tent (c) OK tent (d) OK tent (e) OK tent (f) Ok tent (g) NOT OK. Something seems fishy about this long section. Go on hold. (h) Pause. I think I should tell the reader not to use the "transpose" operator in Standard Notation, it just leads to massive confusion. I don't like the fact that SST = 1 in SN and not in DN. I am also uncomfortable about the inverse thing like S-1. I think I am now facing some major rewriting. My presentation is it now stands is horrible. The problem begins only at 7.9, all is OK before that point. Maybe I can clean up my massive ordering problem at the same time? I should just concentrate on how to translate equations and not get all involved in other stuff. Update 4/5/15. I am working on a new Section 7.9. Here is something I wrote and now seems not useful. Although the subject never came up in Chapters 1-6, we can now ask the question: Is det(A) a scalar? To find out, we consider the determinant for the special case N = 3. Then M'ab = Raa'Rbb'Ma'b' [det(A)]' = εabcA'1aA'2bA'3c = εabc ( R1a'Rab'Aa'b')( R2a"Rbb"Aa"b")( R3αRcβAαβ) = εabc Rab'Rbb"Rcβ R1a' R2a"R3α Aa'b'Aa"b"Aαβ I think the answer is NO, this is not a scalar. But if we used the Levi-Cevita ε tensor, I suspect that det(A) would transform as a scalar density of some sort. I don't think this subject is relevant to me because I don't think I will ever encounter det(A) of a tensor. I will only encounter det(R) and det(S) which have already been dealt with. Today I replaced sections 7(g) and 7(h) with new sections that cover different material, and certain subjects which made a mess were removed (transpose in SN and weird matrix mult forms). I need to continue with 7 (i) onward and clean things up. Maybe now some of the 7 subsections can be removed, such as the tilt rule, which might no longer be needed. Maybe a major reorganization is going to be needed. I remember the threading being very delicate when I first did this. Off to Romine Easter. But first, g' = R g RT g'ab = Raa'Rbb'ga'b' // g is a contravariant rank-2 tensor ' = ST S 'ab = STaa'STbb'a'b' // is a covariant rank-2 tensor (5.7.6) Write this in Standard Notation g'ab = Raa'Rbb'ga'b' Lower index a, g'ab = Raa'Rbb'ga'b' = Raa'Rbb'ga'b' or δab = Raa'Rbb'δa'b' or δab = Raa'Rba' or δab = Rac Rbc and there is one of the orthog rules. Swap a↔b to get δba = Rbc Rac = Rac Rbc = δab so that δab = Rac Rbc which is a second version of the orthog rule Update 4/6/15. Am in AFIB 11AM, see notes, but able to resume here. Thinking of making a Chapter 7 logic flow picture so I can better see what order to put things in. // 7 PM, still in AFIB, I ended up doing a very major rewrite and reordering of Chapter 7. It now has 16 Sections instead of 23. It no longer has subsections (a),(b) ... with that horrible equation numbering system. It no longer talks about the transpose T operator in Standard Notation, all gone. I think things are in a better and more logical sequence. Where I do have sub subsections I just call them 1,2,3... without making eq numbering follow. This work consumed this entire AFIB day, went pretty well. Update 4/8/15. Day off yesterday of shocking. Will proof the new Chapter 7 now from scratch. Proofing of new Chapter 7 opening text OK 7.1 Outer Products 86 OK 7.2 Mixed Tensors and Notation Issues 86 OK 7.3 The up/down bell goes off 87 OK 7.4 Some Preliminary Translations; raising and lowering tensor indices with g88 OK 1. OK 2. OK 3. OK 4. OK 5. OK 7.5 Dealing with the matrices R and S ; various Rules and Theorems 93 OK 1. OK 2. OK 3. OK 4. OK 5. OK 6. OK 7.6 Orthogonality Rules, Inversion Rule, Cancellation Rule 97 OK 1. OK 2. OK 3. OK 7.7 About δ and ε 99 OK tent 7.8 Covariance and Matrix Multiplication 100 OK 7.9 Matrix Inverse, Transpose and Determinant 103 OK 7.10 Tensors of Rank n, direct products, Lie groups, symmetry and Ricci-Levi-Civita104 OK 7.11 The Contraction Tilt-Reversal Rule 107 OK 7.12 The Contraction Neutralization Rule 109 OK 7.13 The tangent and reciprocal base vectors and expansions on same 111 OK 7.14 Comment on Covariant versus Contravariant 114 OK 7.15 The Significance of Tensor Analysis 115 OK 7.16 The Christoffel Business: covariant derivatives 119 OK 7.17 Expansions of higher order tensors 120 OK eq nums for all sections verified at the very end OK Removed fromn Section 7.4: Still, one cannot just write det(g) because det(g) is representation dependent, so one must say something like det(gab) or det(g**) to denote a particular determinant. Time is 4/8/15 8:30 PM. I have finally finished a complete proofing of the newly rewritten Section 7, it is a true Monster Section with lots of tentacles. I think it really is much better than before, ordering is better, and of course has proper eq nums. All linkage to other Sections is done, with a few red items to do. I am not quite sure what should come next. Maybe some of those Appendices I keep referencing. Have done A,B,C,K and have done 1,2,3,4,5,6,7. Update 4/9/15. Proofing of Appendix D D.1 Definition of a tensor density 223 OK D.2 A few facts about tensor densities 224 OK D.3 Theorem about Totally Antisymmetric Tensors: there is really only one: εabc...226 OK D.4 The contravariant ε tensor 227 OK D.5 Some facts about the ε tensor 228 OK D.6 The covariant ε tensor : repeat section (d) as if its weight were not known 230 OK D.7 Generalized cross products 231 OK D.8 The tensorial nature of curl B 231 OK D.9 Tensor E as a weight 0 version of ε : three conventions 232 OK D.10 Representation of ε, εε and contracted εε as determinants 235 OK 1. OK 2. OK 3. OK 4. OK 5. OK 6. OK D.11 Covariant forms of the previous section results 241 // did first cut Update 4/10/15. I want to know how a determinant transforms, so started a separate little doc on that subject. I might add my conclusion as a last section in Appendix D. // Got this all figured out, it has been on my list for a long time. It is all installed as a new Section D.12. Will continue App D proofing tomorrow. This was a nice addition I think. I said nothing about it anywhere. Update 4/11/15. Since time has gone by, I will do a quick reading, not checking EQ numbers, then I will do a full proofing of the last two sections with equation number checks and updates: Proofing of Appendix D D.1 Definition of a tensor density 224 OK D.2 A few facts about tensor densities 225 OK D.3 Theorem about Totally Antisymmetric Tensors: there is really only one: εabc...227 OK D.4 The contravariant ε tensor 228 OK D.5 Some facts about the ε tensor 229 OK D.6 The covariant ε tensor : repeat Section D.4 as if its weight were not known 231 OK D.7 Generalized cross products 232 OK D.8 The tensorial nature of curl B 233 OK D.9 Tensor E as a weight 0 version of ε : three conventions 234 OK D.10 Representation of ε, εε and contracted εε as determinants 237 OK D.11 Covariant forms of the previous section results 244 OK D.12 How determinants of rank-2 tensors transform 246 OK full eq num check OK Time is 11:45AM. This is really an amazing appendix, I think any interested reader will be pretty happy to see all this stuff put in one place, and "for free", and tied in with the general tensor theory. The appendix is 26 pages which is not TOO bad for an appendix. Proofing status is now: Have done A,B,C,D,K and have done 1,2,3,4,5,6,7. I think E should be next. Proofing of Appendix E no opening material E.1 Direct Product Notation 250 OK E.2 Tensor Expansions and Bases 250 OK tent E.3 Polyadic Notation 253 OK E.4 Dyadic Products 254 OK tent E.5 Transpose notation for dyadics 255 OK E.6 Large and small dots used with dyadics 256 OK E.7 Operators and Matrices for Rank-2 tensors 258 OK E.8 Expansions of tensors on unit tangent base vectors 262 E.9 Orthogonal curvilinear coordinates application 265 E.10 Tensor expansions in a mixed basis 270 E.11 What is a tensor? 272 Day ended in the middle of Section E.8 where repairs are needed. Update 4/12/15 Continuing starting with E.8 on proofing App E, using chart above. In fact, today I reproofed all of the Sections E.1 through E.7 and in fact made many clarifications. Here are notes on that process. I am removing the following section which was at the end of Section 2. If it is needed and I understand it better, I might restore it ________________ Expansion of tensor-like objects If Aijk is some "tensor like" object having three indices (such as ∂iTjk) one can still do the three expansions shown above, but (E.2.11) would have to be restated this way: A = Σijk... αijk... (bibjbk...) αijk... = Aabc... (bi)a (bj)b (bk)c... A = Σijk... Aijk... (uiujuk...) Aijk... = x-space components of A A = Σijk... A'ijk... (eiejek...) A'ijk... = RiaRjbRkcAabc = x'-space components of A (E.2.21) Here A is not a tensor, Aijk... are not contravariant tensor components in x-space, and A'ijk... are not contravariant components in x'-space, all relative to the transformation x' = F(x). BUT, how can this be if the bi transform as vectors and we have an outer product of them? For ∂iTjk is there really an expansion of the above form??? I think this is wrong and I am removing this section, which luckily is at the very end of Section E.2. _______________ More replaced text from below (E.7.3). _______________ The idea is that A as an operator has an abstract meaning distinct from the matrix Aij. In the above equation the symbol A is this abstract operator and (bT)n A bm has a meaning distinct from our interpretation of it in terms of the matrix combination of three objects. In the matrix interpretation, one writes (bT)n A bm = [(bT)n]i Aij [bm]j = [bT]i Aij [bm]j and only then does A become a "matrix". This matrix happens to be the contravariant Aij matrix because we happened to select the un basis to write the components like [bm]j = uj bm. _________________ I am now trying to add completeness to the entire document, but I find that Σn (En)j(en)i = Σnc [ g'ncSjc] [Sin] = Σnc Sjc g'cn (ST)ni = [Sg'ST]ji = gji and I was expecting to find Σn (En)j(en)i = δi,j which seems to say that we have completeness of { en, En } ONLY if x-space is Cartesian. This comes as a big surprise to me, so I have to debreak and find out what is going on here. I think I have things wrong! It is {en} that forms a complete set. In developmental notation that says Σn (en)j(en)i = δj,i So I have bogged down on completeness, made a separate little doc. Update 4/13/15 Continuing on completeness. My mistake yesterday was not doing the dot product with g, stupid. so Here is what I have done so far today: 1. I rewrote the duality discussion of Section 6.2 adding new equation numbers. This means any equation number cross references might be wrong above (6.2.7) , so have to check on this. I have this section entirely in dev notation, there is no mention now of Standard Notation results. This section now has a discussion of the completeness relation, never mentioned in previous tensor doc. 2. I added the completeness relation's translation in (7.13.1) at the very end. 3. I created a new Section 7.18 in which I show how the dual stuff with bn Bn and wnm translates to Standard Notation. I can then quote SN equations from this new section. I will now take another running stab at proofing Appendix E. I will continue starting with Section E.7. Did some work on Section 7.9 regarding that Transpose stuff, and showed how to recognize a rotation matrix in Standard Notation, something that has never appeared before in tensor doc. Proofing of Appendix E earlier 4/13 4/14 no opening material E.1 Direct Product Notation 250 OK E.2 Tensor Expansions and Bases 250 OK E.3 Polyadic Notation 253 OK E.4 Dyadic Products 254 OK E.5 Transpose notation for dyadics 255 OK E.6 Large and small dots used with dyadics 256 OK E.7 Operators and Matrices for Rank-2 tensors 258 OK (3:30 PM) E.8 Expansions of tensors on unit tangent base vectors 262 OK tent E.9 Orthogonal curvilinear coordinates application 265 E.10 Tensor expansions in a mixed basis 270 E.11 What is a tensor? 272 So today I got a total of one subsection proofed: Section E.7 and started on Section E.8. Update 4/14/15 Continuing on E.8, will temp store some text here which I may remove: Since the metric tensors are diagonal, one could write H = , but we continue to use H. In what follows, the plan is simply to exercise both the developmental and standard notations with regard to the M and N matrices. To this end, we first collect the following facts using (7.5.9), Rab = Rab' gb'b = Rab Rab = g'aa'Ra'b' gb'b = g'aa'Ra'b = h'a2 Rab = h'a2 Rab Rab= g'aa'Ra'b = h'a2 Rab or Rab = Rab Rab = Rab = h'a2 Rab Rab = h'a-2 Rab . (E.9.3) Also from (7.5.6) and gij = δij and (7.5.13) that Sij = Rji, g'ab = Raa'Rbb'ga'b' = Raa'Rba' = RacRbc g'ab = Sa'a Sb'b ga'b' = Raa' Rbb'ga'b' = Raa' Rba' = RacRbc or g'ab = RacRbc and g'ab = RacRbc . (E.9.4) *********************88 What we really mean is that in developmental notation M is a real-orthogonal matrix, MMT = 1. Since N=M-1, N is then also a rotation. To prove that M is a rotation in developmental notation, the standard notation equation Mab ≡ h'a Rab can be reverse-translated to Mab = h'aRab . Then (now g' = RRT from Chapter 5 (l) ) (MMT)ac = MabMcb = h'aRab h'cRcb = h'ah'c RabRTbc = h'ah'c(RRT)ac = h'ah'cg'ac = h'ah'c [ h'a–2 δa,c] = δa,c => MMT = 1 . (E.9.5) Proving the same thing directly in standard notation requires showing that Mab Mcb = δa,c ( see the end of Chapter 7 (i) ) Mab Mcb = h'a Rab h'c Rcb = h'a h'c Rab Rcb = h'a h'c Rab (h'c-2 Rcb) = (h'a/h'c) (Rab Rcb) = (h'a/h'c)δac = δac = δa,c (E.9.6) where use was again made of the orthogonality rule of Chapter 7 (r), Rab Rcb = δac. ************ which can be written, using the Standard Notation transpose shown in (7.9.3), [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm . // Mni = h'n(en)i . (E.8.17) In earlier notation, we can define our generic basis vector bn for this situation as : bn = (1/h'n)en = n and bm = h'mem bm bn = ( h'm/h'n) em en = ( h'm/h'n) δmn = δmn // verifying (6.2.11) wmn= bm bn = (1/h'n)(1/h'm) em en = (1/h'n)(1/h'm) g'mn // from (7.13.6) [A()]nm = [A(b)]nm According to (E.7.5) the tensor wmn can be used to lower indices on [A()]nm . We don't need to know Why is this blue stuff of any use??? Let's just delete it soon. Lowering just the m index and then reversing the j tilt gives [A()]nm = Mni A ij Mmj = Mni A ij (MT)jm = Mni A ij (MT)jm (E.8.18) and we replicate the section (g) result with B = M : A() = M A MT // that is, [A() = M A MT ]SN,dt . (E.8.19) next section needs a little more work to clarify the Picture shown. ************ A" = M A MT // rank-2 tensor, matrix notation I have bogged down on a small issue. I show in E.8 that A" = MAMT and I then in passing try to make this compatible with A(b) = BABT or more precisely [A(b) = BABT ]SN,dt (E.7.11) but something is not right if I select bn = n and g' ≠ diagonal. It needs g' diagonal to work, but I have not assumed that yet at this point in Appendix E. I have done something wrong somewhere and it needs to be fixed so as not to propagate. // I have resolved this and documented it in tensor doc as a gray comment. Rank-1 tensors. For a vector A = V, the coefficient relation (E.8.8) is written [V()]i = Mij Vj or V() = M V and V = N V() (E.8.14) Vector V has two familiar expansions, the first being (7.13.12) and the second (E.3.8) for N = 1, and for this expansion the coefficient is given by (E.8.5), V = Vnun = [V()]n n [V()]n = Vx' // for example [V()]1 = Vr . (E.8.15) Rank-2 tensors. Here the coefficient relation (E.8.8) is [A()]ij = Mii'Mjj'A i'j' (E.8.16) Question: What is metric tensor g" ? I never state this anywhere. I have finally finished Sections E.8 and E.9, these were for me VERY difficult sections, even after I clarified everything I could clarify. I think they are a lot better now, and has Lai applications which agree with Lai. I threw out things as shown in blue above that seemed not useful. So here we are Proofing of Appendix E earlier 4/13 4/14 no opening material E.1 Direct Product Notation 250 OK E.2 Tensor Expansions and Bases 250 OK E.3 Polyadic Notation 253 OK E.4 Dyadic Products 254 OK E.5 Transpose notation for dyadics 255 OK E.6 Large and small dots used with dyadics 256 OK E.7 Operators and Matrices for Rank-2 tensors 258 OK E.8 Expansions of tensors on unit tangent base vectors 262 OK E.9 Orthogonal curvilinear coordinates application 265 OK E.10 Tensor expansions in a mixed basis 270 OK E.11 What is a tensor? 272 OK eq num full check OK I question the usefulness of this section. I think it is just junk extra stuff. Alternatively, one could relate this coefficient to the coefficients expanded on uiujuk, [ A(b,e,u)]ijk = A (biejuk) = Aabc (bi)a (ei)b (ui)a (E.10.6) as was shown near the start of section (b). ****************8 Mixed-basis expansion of the rank-2 unity operator One can expand the identity operator in this way, since ui uj = (ui)Tuj = δij = <ui| uj> , 1 = Σj uj uj = Σj uj(uj)T = Σj ujuj = Σj | uj>< uj| . (E.10.10) direct product matrix dyadic bra-ket The ui are related to the ei according to uj = Rij ei . (E.10.11) Proof: (uj)a = Rij (ei)a δja = Rij Ria since (en)i = Rni in (7.13.1). But δja = Rij Ria is just orthogonality rule #2 of (7.6.4), QED. It follows that 1 = Σij Rij ei uj = Σij Rij ei(uj)T = Σij Rij eiuj = Σij Rij | ei><uj| (E.10.12) direct product matrix dyadic bra-ket where Rij = (ei)T 1 uj = ei uj = eiuj = <ei | uj> // as in (7.13.7) (E.10.13) matrix dot dyadic bra-ket This then is a mixed-basis expansion of the identity operator. Another form would be 1 = Σij Rij ei uj = Σij [1(e,u)]ij ei uj (E.10.1) following the notation discussed above, leading to this rather obscure way of writing Rij , Rij = [1(e,u)]ij . (E.10.14) With this stuff knocked out, I have finished proofing Appendix E. I will now do a full eq num check. Here is our ne Proofing status is now: Have done A,B,C,D,E,K and have done 1,2,3,4,5,6,7. Next in line is Appendix F I suppose. F.1 Definition and Interpretation of Γ : Γcab = ec • (∂aeb) = Rci(∂aRbi)281 OK F.2 Identities of the form (∂aRdn) = – Ren Rdm (∂aRem) 282 OK F.3 Identities of the form (∂cgab) = – [gan Γ bcn + gbn Γacn] 283 OK F.4 Identity: Γdab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab] 284 OK F.5 Picture D1 Context 286 OK F.6 Relations between Γ and Γ ' 287 OK F.7 Statement and Proof of the Covariant Derivative Theorem 288 OK F.8 Rule for raising any index on a covariant derivative of a covariant tensor density.293 OK F.9 Examples of covariant derivative expressions 294 OK F.10 The Leibniz rule for the covariant derivative of the product of two tensor densities297 OK full eq num scan: OK Will resume tomorrow with F.5. Update 4/15/15 Continuing on App F as per above OK chart. Just now I fixed my little wizmouse problem, see maintenance folder. // Now doing Section F.7, lunch break. did MRL stuff as per MWL. It is 9 PM and I just finished Appendix F! It is now very much improved, I did this in full detail. I wonder where else this is explained in my manner? Much better with equation numbers! Proofing status is now: Have done A,B,C,D,E,F,K and have done 1,2,3,4,5,6,7. Not sure what should come next. Maybe get all these annoying appendices nailed down. Each of them looks very long and painful! Update 4/16/15 OK, today I will start into Appendix G on the v stuff. G.1 Continuum Mechanics motivation 301 OK G.2 Expansion of v on eiej by Method 1: Use fact that vb;a is a tensor.301 G.3 Expansion of v on eiej by Method 2: Use brute force. 303 G.4 Expansion on eiej and e^ie^j 305 G.5 Orthogonal coordinate systems 306 G.6 Maple evaluation of (v) in several coordinate systems 306 Question: I show this in Appendix E: A = Σijk... αijk... (bibjbk...) αijk... = Aabc... (bi)a (bj)b (bk)c... (E.2.11) A = Σijk... Aijk... (uiujuk...) Aijk... = contravariant components of A in x-space A = Σijk... A'ijk... (eiejek...) A'ijk... = contravariant components of A in x'-space How can I rewrite these with up indices on the basis vectors? (stack level -1) I think I have no definition of ui at this point, although I do have a definition of ei. In fact, I seem to lack much info on the ui and ui properties. I need the above tilted the other way because I USE such things in Section G.2 which I am trying to proof! I then tried clearing up the "facts" of all my kinds of basis vectors (stack level -2) : (partial) _______________________________________________________________________________ en tangent base vectors (in x-space) En reciprocal base vectors (in x-space) en em = 'nm => |en| = = h'n (scale factor) En = g'ni ei En em = δn,m en = 'ni Ei En Em = g'nm => |En| = . (6.2.4) ______________________________________________________________ e'n axis-aligned basis vectors in x'-space E'n reciprocal base vectors (in x'-space) e'n e'm = 'nm => |e'n| = = h'n (scale factor) E'n = g'ni e'i E'n e'm = δn,m e'n = 'ni E'i E'n E'm = g'nm => |E'n| = . // I think!!! (6.2.4) ___________________________________________________________ un' inverse transformation's tangent base vectors in x'-space Un' inverse transformation's reciprocal base vectors in x'-space U'n= gnm u'm ? un' um'= nm => |un'| = = hn (scale factor) Un' um' = δn,m Un' Um' = gnm => |Un'| = . (6.2.4) ___________________________________________________________ un axis-aligned basis vectors in x-space Un reciprocal base vectors (in x-space) un um = nm => |un| = = h'n (scale factor) Un um = δn,m Un Um = g'nm => |Un| = . // I think!!! (6.2.4) ___________________________________________________________-- While pondering how to present all these basis vector results together, I starting thinking about the various covariant partner vectors like n . (stack level - 3) I did some partial work on such vectors in Section 6.3 which today I expanded to be more complete. Then when I started listing off my catalog of "results" such as the dot products en em = 'nm, I asked for the first time ever: What about n m ? (stack level - 4) This led me to wonder if there is any meaning to as opposed to A B. ? This then launched me off into a new world of pain, and so I have to shut down the proofing process until I can resolve this world of pain. (stack level -5 ) I am totally confused now about whether there is only one vector V, or whether there are two distinct vectors V and . In standard notation there is in effect only one V. It is dangerous for my to have such a low level confusion in tensor doc!! Update 4/17/15 Continuing in my side doc on meaning of V, I think I have a plan. Edited Section 5.4 to make it better, need to check on equation number linkage ******* . done OK Rewrote Section 5.10 just now to get this fact down on paper A B ≡ Aii = iBi = B = A = I install this rewrite right now, and again need to check eq num linkage ****** done OK. Do the any 5.4 linkage repairs: DONE Do the any 5.10 linkage repairs: DONE Now let's get back to all those basis vector dot products! I see nothing to change prior to Section 6.3. I just updated the e/E basis vector "package" in Section 6.4 which has nothing but that! Reread the duality notes, all still good. I am wondering now about my completeness thing Σn( n)a (bn)b = δa,b Σn n bnT = 1 As I now realize, in standard notation these become Σn(bn)a (bn)b = δa,b Σn bn bnT = 1 I created a tempest in a teapot for about an hour or two not realizing the above translation!! The key idea I forgot is this: Remember from (7.13.1) that (n)i →(en)i , so the label goes up, the index stays down, and the bar goes away. I may have made an error again with the notation ion Appendix E . Lets review E.1 All OK, used covariant notation. E.2 OK down to (E.2.11), and I retain my red commend about trying to tilt the other way, put that off for the moment. Orthonormal Basis section: seems OK as is. I think all of E.2 is OK. E.3 Polyadic notation. Seems all OK, not dot products appear. E.4 OK, not dot products, keeping up and down indices in right places. E.5. Perhaps first signs of trouble. First problem is in (E.5.6) where I say aTb = a b = (a1 a2) = a1b1 + a2b2 = a scalar (if a and b are vectors) . (E.5.5) This is NOT the covariant dot product!!! Beware!! I can fix this section simply by replacing the above line with this aTb = (a1 a2) = a1b1 + a2b2 . (E.5.5) Yes, that rescues the entire Section E.5 so install that change right now. E.6 Now more big trouble with dot notation! I have highlighted in red chunks that are problematical, but most of it survives for now. E.7 Let's try to redo the bra-ket thing. Start with column vector a on the lef a → |a> // vector aT → <a| // transpose vector For example, we can then write (both sides of → are Standard Notation!!!! ) bi → |bi> (bi)T → <bi| bi → |bi> (bi)T → <bi| Next, we have abT (E.5.3) → |a><b| // a matrix (outer product) aTb (E.5.5) → <a | b> // a number (inner product) My problem here is that I have not way to express a b in this little matrix notation abT . I have to back up to the dyadic section and ponder this. E.5 Matrix notation for dyadics For a vector V, certainly Vi = (VT)i, meaning the object in the ith row of V is the same as the object in the ith column of VT . Therefore one can express the dyadic product in this more down-to-earth manner, (ab)ij ≡ aibj = ai (bT)j = (abT)ij (E.5.1) or ab = abT . (E.5.2) But this is with all indices contravariant! How would I deal with this object (ab)ij ≡ aibj = ai (bT)j = (abT)ij (E.5.1) I guess I could argue that notation abT stands for all four rank-2 tensor forms. But then we come to abT = (b1 b2) = = a matrix // same as matrix ab = ab (E.5.3) and THIS is only good if one of the vectors is covariant, otherwise not a b. In standard notation I might say this: bT = (b1 b2) = a b How would you translate the above line to Standard Notation? I have to keep the bar!!! ************************** Suppose I restrict sections E.5, E.6 and E.7 to Cartesian space. Then g = 1 in Picture A. We do have en em = g'nm and we are NOT assuming that g' = 1. Status 7:15. I am workin in the "edit section E.5" doc. I have slightly new versions of E.5 and E.6 with the g = 1 restriction, and they seem OK. Time is 7:45 and I think I have "rescued" sections E.5 and E.6 and E.7 with new text in "edit section E.5" . I will hold off on installing this until I get happy with E.8 and the basis vector stuff and the expansion with reverse tilt stuff. Back now to the basis vectors. I have updated Section 6.4 and changed equation numbers. So cross ref check soon *******. OK. Hopefully I have enough " un data" to move forward in Appendix G.2 where I came to a halt. I have to get tilt reversal on the expansion, so I am now sitting at (E.2.11) getting ready to reverse tilts somehow. Start with A = Σijk... αijk... (bibjbk...) **************************** Let bi be an arbitrary complete set of basis vectors in x-space. As shown in the comments leading up to (6.2.11) there exists a unique set of dual ("reciprocal") basis vectors bi (also in x-space) such that bi bj = δij , where we now use the Standard Notation bn equations of Section 7.18. Consider then the following expansion of a rank-3 tensor A, A = Σijk αijk (bibjbk) where (bibjbk ...)abc = (bi)a (bj)b (bk)c . Aabc = Σijk αijk (bi)a (bj)b (bk)c (E.2.1) Reinforcing the previous section, if the bi transform as tensorial vectors under some transformation x' = F(x), we know that any outer product of such vectors transforms as a tensor. Since A is a linear combination of such outer products, A transforms as a rank-3 tensor whose contravariant components are Aabc. The coefficients αijk can be obtained by dotting both sides with (bi'bj'bk') and using (bi'bj'bk') (bibjbk) = bi' bi bj' bj bk' bk = δii'δjj'δkk' . (E.2.2) The result is then (unpriming indices) αijk = A (bibjbk) = Aabc (bibjbk)abc = Aabc (bi)a (bj)b (bk)c . (E.2.3) where Aabc are the contravariant components of tensor A in x-space, and (bi)a are the covariant components of vector bi in x-space. Update 4/18/15 Want to reverse the tilts on the tensor expansion. But lots to do first. 12:45PM. I have a new stable Section 7.18 which tells you "everything you ever wanted to know" about the three vector families en, u'n and bn . Using this data for bn I hope to show that I can tilt things one at a time in my general tensor expansion of Appendix E. Feeding time. Done and resume 1:30. A = Σijk... αijk... (bibjbk...) αijk... = Aabc... (bi)a (bj)b (bk)c... (E.2.11) Question 1: Is αijk a tensor? To find out, we would try this, starting in x'-space α'ijk... = A'abc... (b'i)a (b'j)b (b'k)c... Now use A'abc... = Raa'Rbb' ......Aa'b'c'... // contra, here we assume A is a rank-3 tensor (b'i)a = Raa"(bi)a" // we assume b is a rank-1 tensor Then α'ijk... = A'abc... (b'i)a (b'j)b (b'k)c... = Raa'Rbb' ......Aa'b'c'...Raa"(bi)a" Rbb"(bj)b"Rcc"(bk)c" .... = ( Raa'Rab")(Rbb' Rbb") .Aa'b'c'..."(bi)a"(bj)b"(bk)c" .... = δa'a" δb'b".... Aa'b'c'...(bi)a"(bj)b"(bk)c" .... = Aa'b'c'...(bi)a'(bj)b'(bk)c' ..... = αijk... This tells me that αijk transforms more like a scalar than a tensor! BUT, in App E I show that if b = e, then αijk... = A'ijk which is a tensor. Well, this just shows that the equation Aa'b'c'..."(bi)a'(bj)b'(bk)c' .... is covariant. What I really want to know is how α transforms. Can I show that α'ijk... = Rii'Rjj' ..... αi'j'k'. ?? or that A'abc... (b'i)a (b'j)b (b'k)c... = Rii'Rjj' ..... Aabc... (bi')a (bj')b (bk')c... But above I processed the LHS already and so I have to show that Aa'b'c'...(bi)a'(bj)b'(bk)c' ..... = Rii'Rjj' ..... Aabc... (bi')a (bj')b (bk')c... (*) Well, I can now use this on the right side so (b'i)a = Raa"(bi)a" so RHS (*) = Rii'Rjj' ..... Aabc... Raa"(bi)a" Rbb"(bj)b" ...... = Rii'Rjj' Raa"Rbb" .... Aabc... (bi)a"(bj)b" ... install the α to get A = Σijk... Aabc... (bi)a (bj)b (bk)c... (bibjbk...) now let's try playing with the j index only. STOP. Something is amiss with Section E.2! I get contradictions as noted there in red. Note. Added new EQ nums for examples like (2.9.5). Cross link this as needed ******. // Too hard to trace, I will just do this on new readings! Update 4/19/15 I think I have a way out and will provide this in a little preamble. // This is all done. The upshot is that I have done a very heavy rewrite of Section E.2, Lots of eq nums have changed, I now have the reverse tilt expansion which I can quote later in Appendix G.2.3, This has been such a long-winded change, and since I see lots of red in the rest of Appendix E, I am going to have to do a whole new recheck. I think I failed to install new text for Sections E.5, E.6 and E.7 just because I was holding off to get the above paradox stuff figured out. Where is this text? It is at the very end of the doc "edit section E.5". I will do the install right now. DONE. New TOC, new date and save. New Proofing of Appendix E no opening material E.1 Direct Product Notation 250 OK E.2 Tensor Expansions and Bases 250 OK E.3 Polyadic Notation 253 OK E.4 Dyadic Products 254 OK E.5 Transpose notation for dyadics 255 OK E.6 Large and small dots used with dyadics 256 OK E.7 Operators and Matrices for Rank-2 tensors 258 OK E.8 Expansions of tensors on unit tangent base vectors 262 OK E.9 Orthogonal curvilinear coordinates application 265 OK E.10 Tensor expansions in a mixed basis 270 OK E.11 What is a tensor? 272 gone! eq num full check OK! I have run into some trouble now in E.7. This line has a problem in (E.7.4), Σi bi biT = 1 (7.18.6) → Σi |bi><bi| = 1 = Σi |bi><bi| // completeness Can I replace this with Σi(bi)n(bi)m = δn,m (7.18.6) → Σi |bi><bi| = 1 = Σi |bi><bi| // completeness How can I make this work better. Σi |bi><bi| = 1 Σi <un|bi><bi| um> = <un|um> = δnm <un|bi> = unT bi = Σk (unT)k(bi)k = Σk (un)k(bi)k = Σk δnk (bi)k = (bi)n <bi| um> = bi um = Σk (biT)k(um)k = Σk (bi)k(um)k = Σk (bi)kδmk = (bi)m OK, I have cleaned up Section E.7, it has survived OK. I shall continue with Section E.8. Oops, back to (E.2.15), we may have a problem Houston. Consider, i ≡ ei/h'i i j = δi,j What can be said about these unit vectors: i = ei / |ei| ? We know from (7.18.1) that |en| = . But g'nn = h'n2 since we have assumed diagonal g'** and therefore g'nn = h'n-2 and |en| = = h'n-1 and then i = ei / |ei| = ei / ( h'i-1) = ei h'i so compare these two facts and beware. i ≡ ei/h'i i = ei h'i ei = g'ijej = h'i2ei i = ei h'i = [h'i-2ei] h'i = h'i-1ei = h'i-1[h'ii] = i So yes, it is true that i = i whereas ei ************************** Confusion: Orthonormal basis ought to mean that bi bj = δi,j so the basis vectors are indeed orthonormal. And bi bj = δi,j is true for any bi basis. If the bi really are unique as I say, then we must indeed have bi = bi . Now can you say bi = i where i ≡ ei / |ei| ??? bi = i ? bi = OK, I wrote it up, no problem, it is really true. ***** I like the idea of saying verified in Lai in place of just Lai I think I am going to delete section E.10 on "what is a tensor". It really adds nothing new. I don't really like the idea of tensor as operator anyway. And I have already said that continuum mechanics has a broader definition of tensor, so this section is just a repeat. App E is already horribly long. I will archive section E.10 right here! E.11 What is a tensor? We are now in a better position to examine some possible answers to this question. (1) A tensor is an operator like A which lives inside a direct product Hilbert Space. We can associate with this tensor A a large variety of up-indexed objects such as [A(b,e,u)]ijk in (E.10.4) above. If one has at hand K different bases of interest, then for a tensor of rank n there would be Kn possible up-indexed objects. In the case of rank 2, these K2 different indexed objects are matrices. Given the un and en bases used throughout this document, there are two special up-indexed objects of rank 3: [A(u,u,u)]ijk and [A(e,e,e)]ijk which we abbreviate as [A(u)]ijk and [A(e)]ijk or as [A]ijk and [A']ijk . The indexed object [A]ijk is a set of N3 contravariant components of a rank-3 tensor in x-space, and [A']ijk is a set of N3 contravariant components of the same rank-3 tensor in x'-space, where these two spaces are linked by a transformation x' = F(x) which has a linearized form dx' = R dx at a point x. The two sets of contravariant components are related by [A']ijk = Rii'Rjj'Rkk' [A]i'j'k' . (E.11.1) These sets of components "transform as a rank-3 tensor with respect to the underlying transformation x' = F(x)." The all-up indexed tensor of rank n is just one of a family of 2n tensors where the indices take all possibly up and down positions, and each such tensor has a corresponding transformation rule, such as [A']ijk = Rii'Rjj'Rkk' [A]i'j'k' for A = Σijk [ A']ijk (eiejek) . (E.11.2) For this definition of "tensor" as an abstract operator A, there are many possible sets of components (indexed objects where the values of all indices are set in all possible ways) which do NOT transform as a rank-3 tensor with respect to F as just described. The component sets that do transform as tensors are those for which the components are coefficients of an expansion of A on a direct product basis where the individual basis vectors are all selected from ei or ei, or are all selected from ui or ui. If some other generic basis vector bi appears, then the component set does not transform as a tensor under F. Definition (1) is basically the definition of "tensor" used in this document. (2) Another definition of tensor might be: a tensor is any of the indexed objects mentioned in (1) above. The set of components like [A(b,e,u)]ijk is called a "tensor" because it is a possible coefficient set that can be obtained by expanding the "tensor operator" A on suitable basis vectors. Just as a matrix is sometimes written without its indices, so this tensor object might be represented just as A(b,e,u). The components of this particular indexed object do not transform as either end of the transformation rule stated above, so this tensor is a tensor, but does not transform as a tensor. (3) A third possible definition: a tensor is any indexed object each of whose indices ranges from 1 to N where N is the dimension of one's space of interest. By this definition, any NxN matrix Aij would be a tensor of rank 2. It would be very unlikely that a random matrix like this would be part of the transformation rule [A']ij = Rii'Rjj' [A]i'j' so this matrix Aij is then a tensor, but it probably doesn't transform as a tensor. In fields of physics involving relativity, the first definition is normally used, and an indexed object is called a tensor only if it transforms the way a tensor should transform with respect to a transformation of interest (our "tensorial tensor"). For example, the affine connection Γcab is never called a tensor because it does not transform as a tensor. In most other areas of physics definition (3) seems more common, where any matrix is a rank-2 tensor, also known as a second-order tensor. The whole subject of second order tensors is then identified with linear algebra where the operators are matrices. It may turn out that a particular matrix is in fact a tensor by definition (1) with respect to rotations. This is the case for basic matrices involved in equations which must be covariant. In continuum mechanics, which generally uses definition (3), there are many tensors which do not transform as tensors under rotations or other transformations. In that field, when a tensor in fact transforms as a tensor, it is called an objective tensor (sometimes an indifferent tensor). Equations which are covariant in the sense of Section 7.15 are called "frame indifferent". See Appendix K for examples. This I can ignore that section in my proofing and I am finally done with Appendix E. I would like to say a little more about (E.9.26) with some Lai connection. Look at Lai p 264 and Err = a Trr - bTθθ a = (1/E)(1-v2) b = (1/E)v(1+v) Eθθ = a Tθθ - bTrr Erθ = (b/v) Trθ Assuming my equations are valid Trr = λ [ Err+ Eθθ] + 2μ Err = (λ+2μ)Err+ λEθθ Trθ = 2μ Erθ . (E.9.26) I conclude first of all that 2μ = (v/β) = (v/b) Now solve the first two equations for Trr and Tθθ in Maple cTrr = a*Err+ b*Eθθ cTθθ = b*Err+ a*Eθθ c = (a2-b2) So I then have these two equations cTrr = c(λ+2μ)Err+ cλEθθ cTrr = a*Err+ b*Eθθ We then find that c(λ+2μ) = a cλ + 2μc = a cλ = b Second equation tells us that λ = b/c First equation then says cλ + 2μc = a b + (v/b)c = a b2 + vc = ab Yes this is true! So what are my conclusions on this. λ = b/c = -Ev/(2v2+v-1) // says Maple 2μ = (v/b) = E/(1+v); So I could say that my equations are consistent with Lai p 264 (5.27.7) with λ = -EYv/(2v2+v-1) = -EYv/[(v+1)(2v-1)] 2μ = EY/(1+v). This is a pretty lousy verification, let's look online! // I found something much better and installed it. I am done for now with Appendix E, it took all day!! Update 4/20/15 Removed duplicate section in E.9 and just said "recall". I then had to renumber E.9 and E.2 because I added numbers (E.2.15) and (E.2.19). Since this altered E.2 above (E.2.18) I will scan now for xrefs: OK. // Resume at 7:30 PM. I have reviewed Appendix F and examined all (E. references in it, there are only a few, and all are OK. So I think I am ready to restart on Appendix G. Proofing of Appendix G G.1 Continuum Mechanics motivation 314 OK G.2 Expansion of v on eiej by Method 1: Use fact that vb;a is a tensor. 314 OK G.3 Expansion of v on eiej by Method 2: Use brute force. 316 OK G.4 Expansion on eiej and e^ie^j 318 OK G.5 Orthogonal coordinate systems 319 OK G.6 Maple evaluation of (v) in several coordinate systems 319 OK full eq num check OK Why do I say that Γ = 0 in Cartesian Space below (G.2.1)? In terms of Picture B, I guess that would be true if g' were Cartesian, since then en em = δn,m? I have been wondering about my Cartesian references of late. If g = 1 and g'=1, what do we know about x' = F(x)? Start here tomorrow! Update 4/21/15 Resolve this Γ problem. // Did so by adding lots of new text at the start of Appendix F. Yes, I really have redefined the symbols en and I had not made that clear (even to me!). Appendix F is now much better. Section G.2 took a LONG TIME to clean up, and it is now OK'd above, Many improvements! Did some work on (G.3) and it is now OK. // Done with all of Appendix G at 1:30 PM! Proofing status is now: Have done A,B,C,D,E,F,G,K and have done 1,2,3,4,5,6,7. I still have App H,I,J to go, and they are not small. Proofing of Appendix H. H.1 Introduction 324 OK H.2 Continuum Mechanics motivation 324 OK H.3 Expansion of divT on en by Method 1: Use fact that Tab;α is a tensor.324 OK H.4 Expansion of divT on en by Method 2: Use brute force 325 OK H.5 Adjustment for T expanded on (e^ie^j) and divT expanded on e^a 327 OK H.6 Maple: divT in cylindrical and spherical coordinates 328 OK full eq num check OK and this concludes Appendix H. I still have Appendices I and J to go! Proofing status is now: Have done A,B,C,D,E,F,G,H,K and have done 1,2,3,4,5,6,7. Proofing of Appendix I. I.1 Introduction 331 OK I.2 The first method : a review 331 OK I.3 The first method in spherical coordinates: Maple speaks 333 OK I.4 The first method in spherical coordinates: putting results in traditional form 33 OK I.5 The second method : Part I 338 OK I.6 The second method : Part II 339 OK I.7 The second method in spherical coordinates: Maple speaks again 340 OK I.8 Results for Cylindrical Coordinates from both methods 343 OK full eq num check OK Update 4/22/15 Continuing on App I above. Ouch! I found a big chunk of text at the very end of Appendix I which I think belongs at the end of Appendix J !! I will park it below for the moment, dTij(x,t)/dt = ∂Tij/∂t + (∂Tij/∂xk) (∂xk/dt) = ∂Tij/∂t + (∂Tij/∂xk) vk or dtTij = ∂tTij + (∂kTij) vk . // dt ≡ d/dt, ∂t ≡ ∂/∂t, ∂k ≡ ∂/∂xk (I.8.13) We take this as a useful prototype equation to work with for two reasons. First, it contains our object of interest, which is the gradient of a rank-2 tensor. Second, this equation plays a role in the continuum mechanics of non-Newtonian fluids as discussed below in section (f). One can define, in Cartesian coordinates, a (T) object : (T)ijk ≡ ∂kTij (I.8.14) so that dtTij = ∂tTij + (T)ijk vk . (I.8.15) Comment: Our convention has been to bold vectors and not to bold other tensors. In line with this idea, we shall write T where the grad is bolded and the T is not bolded. In order to express the above equation in curvilinear coordinates, it must be "tensorized" in the sense of Chapter 15 (b) so that the equation is covariant. Thus, (T)ijk must be regarded as components of a (mixed) rank-3 tensor which, in Cartesian coordinates, are equal to ∂kTij. Since vk are the components of a tensorial vector, (T)ijk vk transforms as a rank-2 tensor, and then all terms in the above equation are rank-2 tensors and the equation is then covariant and therefore appears this way in x'-space, dtT'ij = ∂tT'ij + (T)'ijk v'k . (I.8.16) In our usual formalism, x'-space is the space of some generic curvilinear coordinates x'n (not necessarily orthogonal) and then the above equation tells us the form taken by the time derivative equation in curvilinear coordinates, and it remains only to compute the objects (T')ijk. For convenience, we can lower the tensorial ij indices on the above tensor equations to get dtTij = ∂tTij + (T)ijk vk (T)ijk ≡ ∂kTij // Cartesian coordinates (I.8.17) dtT'ij = ∂tT'ij + (T)'ijk v'k // curvilinear coordinates (I.8.18) and then we can deal with the pure covariant tensor components (T)'ijk . This error is in my current release, horrors! I will come back to this when doing Appendix J below. I have now finished Appendix I, there were errors, I removed stuff, it is much better now. Proofing of Appendix J J.1 Total time derivative as prototype equation 348 OK J.2 Computation of components (T)'ijk 349 OK J.3 Tensor expansions of T on the un and en base vectors 350 OK J.4 Tensor expansions of T on the e^n base vectors 350 OK J.5 Total time derivative equation written in unit-base-vector curvilinear components352 OK J.6 Shorthand notations and a continuum mechanics application 353 OK J.7 Maple computation of the (T)'ijk components in spherical coordinates 355 J.8 Maple computation of the (T)'ijk components in cylindrical coordinates 359 (v) B T div(T) reciprocal base vectors En // drew a Visio block diagram of tensor doc. I had a major breakthrough I think on Section J.5 above. It is now much better than it was. I worked in the big idea of covariance. I just finished J.6 and will resume manana with J.7. This is all good stuff, I must admit. Update 4/23/15 I did some cleanup on the inversion and cancellation rules in Chapter 7. Here is the old Consider now an equation which one wants to invert for the object on the right, [----a-----] = Rab [--------b------] // before (7.6.5a) The bracketed objects are two arbitrary tensors which might have many up-down indices which are indicated by dashes. The inversion rule for moving R to the other side of the equation is to reflect R's two indices in the horizontal line between the indices so that Rab [----a-----] = [--------b------] // after (7.6.5b) Proof: Rename b→b' in (7.6.5a), apply Rab to both sides and sum on a, then use orthogonality rule #1 : Rab [----a-----] = Rab Rab' [--------b'------] = δbb' [--------b'------] = [--------b------] QED Recall from the last section that reflection in the vertical index line has a different application, Sab = Rba (7.5.13) Example: V'a = RabVb RabV'a = Vb or Va = Rca V'c The inversion rule is also valid with indices tilted the other way (similar proof): [----a-----] = Rab [--------b------] Rab [----a-----] = [--------b------] ****** Next, consider a different generic equation ( bracketed objects are again arbitrary tensors) Rab [----b-----] = Rab [--------b------] // before (7.6.6a) The cancellation rule says the equation is still valid if identical contracted R factors are canceled on both sides such that the contraction index becomes a free index, [----b-----] = [--------b------] // after (7.6.6b) Proof: Rename b→b' in (7.6.6a), apply Rab to both sides and sum on a, then use orthogonality rule #1 : RabRab' [----b'-----] = RabRab' [--------b'------] δbb' [----b'-----] = δbb' [--------b'------] [----b-----] = [--------b------] QED Also tried out a silver bullet M-method to compute things, but was no silver bullet, separate doc. The M-method is good only for covariance things. I have added the "M covariance theorem" more clearly in Section E.9 which is the right place to put it, after reader has suffered with all the M and N stuff. Then I can just quote it later! I want to check off the appendices with regard to this change App E my cont mech examples in E.9: OK App G for (v): OK App H for div(T) OK App I for B App J for (T) The matrix A is not a differential operator since the derivative does not act on the vector standing to the right of A, but one is still often interested in expressing A in curvilinear coordinates. This is done in the following sections, where we replace the generic vector A by generic vector v and this v has nothing to do with the velocity v mentioned above. Will start on the B after lunch. Be sure to remove the Big Comment which can be replaced by a tiny comment (once I find it) ****. (still pending Bug: I think I have inconsistent notation in different places. 1. The first place I use script letters is in Section 7.13 where I write V'n = h'nV'n first appears in (7.13.12), notice that there is a prime on the scripted variable here. V = V'n en = V'n n vr = h'r (V')1 // prime is dropped on vr once variable name is used (I think) 2. Another place I do this is in Section 13.1 where I have B = B'nen = B'n n // still have a prime on the scripted variable I then change to the Moon and Spencer picture which is like Picture B, but forget that for now. 3. After giving the summary Chapter 14, I use an example there of "practical notation". But there I have Bn = hnBn v1 = vθ Here there is no prime on Bn or hn or v1 because I am using Picture M&S 4. Another place is Section E.8 where I write [A()]2213 = A2213 = Aθθrφ . (E.8.6) This is wrong! Because I am in Picture A, there should be a prime on A'2213 I have repaired my prime scripted variable problem now up to Appendix I. Continue after break starting with Appendix J which still contains the "big comment" I want to reduce. This also needs some scripted equations installed! Update 4/24/15 After a round of edits to previous appendices to get script equations stated and things primed, I am back once again at the start of Appendix J. Proofing of Appendix J J.1 Total time derivative as prototype equation 348 OK J.2 Computation of components (T)'ijk 349 OK J.3 Tensor expansions of T on the un and en base vectors 350 OK J.4 Tensor expansions of T on the e^n base vectors 350 OK J.5 Total time derivative equation written in unit-base-vector curvilinear components352 OK J.6 Shorthand notations and a continuum mechanics application 353 OK J.7 Maple computation of the (T)'ijk components in spherical coordinates 355 J.8 Maple computation of the (T)'ijk components in cylindrical coordinates 359 where A" ijk... ≡ [A()]ijk... ≡ A' ijk... end of (E.8.20) Qadc = HabTbc Bd x-space (general), Picture A or M&S (1.11) Q'adc = H'abT'bc B'd x'-space (general), Picture A or B (1.11) Q'adc = H'abT'bc B'd x"-space (orthogonal) , Picture A or B (1.11) Qadc = HabTbc Bd x"-space (orthogonal) , Picture M&S (14.1.1) (E.9.13) The following now removed from the end of Section J.5. Comment on Covariance: Equation (J.5.6) dt T'ij = ∂t T'ij + (T)'ijk v'k (J.5.6) displays a remarkable similarity to the starting equation (J.5.1) and it's x-space counterpart, dtT'ij = ∂tT'ij + (T)'ijk v'k x'-space (J.5.1) dtTij = ∂tTij + (T)ijk vk x-space . (J.5.9) This similarlity is no coincidence. First, recall that the two equations in (J.5.9) have the same form because they are "covariant" under x' = F(x), see Section 7.15. Now consider these two equations that we just noted are very similar, dtTij = ∂tTij + (T)ijk vk x-space (J.5.9) dt T'ij = ∂t T'ij + (T)'ijk v'k . (J.5.6) (J.5.10) Let's rewrite the second equation in our alternate notation used above, dt [T()]ij = ∂t [T()]ij + [T()]ijk [v()]k (J.5.6) (J.5.11) and rewrite it once again imagining that the () objects belong to a certain x"-space dt T"ij = ∂tT"ij +T"ijk v" k . (J.5.12) Now restate the equation pair of (J.5.10) as dtTij = ∂tTij + (T)ijk vk (J.5.9) dt T"ij = ∂tT"ij + T"ijk v" k . (J.5.12) The reason these two equations "look the same" is because they are "covariant" under a certain transformation FM(x) = HF(x) which links x-space to x"-space, as explained in Section E.8, see in particular (E.8.20). We therefore could have deduced the validity of (J.5.6) by just staring at the pair of equations (J.5.10), and skipped all the work done above to derive (J.5.6). An example was given in Section E.9 where the equations (E.9.21) and (E.9.24) have the same form. Hopefully these examples give the reader an appreciation for the usefulness of the concept of covariance. ************** archiving my derivation of a fact: In orthogonal coordinates this says δi,k(∂j(hi-2)) = – [hi-2 Γ kji + hk-2 Γijk] = δi,k (-2)hi-3 (∂jhi) or [hi-2 Γ kji + hk-2 Γijk] = 2 hi-3 (∂jhi)δi,k or [hi-1hkΓ kji + hi hk-1 Γijk] = 2 hk hi-2 (∂jhi)δi,k or hi-1hkΓ kji = - hi hk-1 Γijk + 2 hk hi-2 (∂jhi)δi,k (J.7.18) Installing this into (J.7.17) gives (ΓLai)ijk = [Γ()]kji = hi-1hk Γkji – hi-1(∂jhi) δk,i = - hi hk-1 Γijk + 2 hk hi-2 (∂jhi)δi,k - hi-1(∂jhi) δk,i = - hi hk-1 Γijk + hi-1(∂jhi)δi,k [ 2 hk/hi - 1] = - hi hk-1 Γijk + hk-1(∂jhi)δi,k = hk-1 [ - hi Γijk + (∂jhi)δi,k ] so that (ΓLai)ijk = [Γ()]kji = hk-1 [ hi Γijk + (∂jhi)δi,k ] (J.7.19) This is an alternate form of (J.7.17) ********** I spend the rest of this day trying to grok the Lai presentation around page 503. There must be some way to tie that into my notation, but after 6 hours I was unable to nail it. I think it needs a rest for a while. So this leaves my little Lai version of Γ in limbo, located in Appendix J.7. So once again, I am unable to finish "proofing" Appendix J, but it is OK up to that point. Update 4/25/15 I added a whole new section J.9 on how Lai computes the big tables on pages 504 and 505. This renders obsolete certain items at the end of Section J.7 which I now park here: The Lai method of computing (T)'ijk is different from ours and is presented on page 503. Lai et. al. use Mijk = (T)'ijk with the expansion T = Σijk Mijk ijj (p 501) which provides an example of the polyadic notation (E.3.2) (ijj= ijk). They use the notation en to represent our n which adds an extra level of confusion. We shall not attempt The Alternate form of Γ used by Lai et. al. First of all, our Γ definition for Picture C1 of (F.1.4) says (den)i = Γkjn dxj eki // see (F.1.5) (J.7.12) where the en are the tangent base vectors. How can this be written in terms of the unit vectors n ? (den)i = (d[hnn])i = (dhn)ni + hn(dn)i = (∂jhn)dxj ni + hn(dn)i (J.7.13) so (J.7.12) becomes (∂jhn)dxj ni + hn(dn)i = Γkjndxj hk ki so (dn)i = (hk/hn) Γkjndxjki – hn-1(∂jhn)dxj ni = hn-1 [hk Γkjn – (∂jhn) δk,n] dxj ki ≡ [Γ()]kjn dxj ki (J.7.14) Thus, dn = [Γ()]kjndxj k where [Γ()]kjn ≡ hn-1 [hk Γkjn – (∂jhn) δk,n] (J.7.15) Taking n→ i gives di = [Γ()]kjidxj k where [Γ()]kji ≡ hi-1 [hk Γkji – (∂jhi) δk,i] (J.7.16) Lai et.al. p 502 (8A.12) define their version of Γ in this manner (developmental notation), dei = (ΓLai)ijk dxj ek where their ek are our k. Thus we can make this connection, (ΓLai)ijk = [Γ()]kji = hi-1 [hk Γkji – (∂jhi) δk,i] . (J.7.17) Example: For spherical coordinates we know that, Γ r = Γθ = Γφ = (I.7.3) hr = 1 hθ = r hφ = rsinθ (5.13.15) We can then compute some sample ΓLai elements : (ΓLai)rθθ = [Γ()]θθr = hr-1 [hθ Γθθr – (∂θhr) δθ,r] = hr-1hθ Γθθr = 1 r (1/r) = 1 (ΓLai)rφφ = [Γ()]φjr = hr-1 [hφ Γφφr – (∂φhr) δφ,r] = hr-1 hφ Γφφr = 1 rsinθ (1/r) = sinθ (ΓLai)φφr = [Γ()]rφφ = hφ-1 [hr Γrφφ – (∂φhφ) δr,φ] = hφ-1 hr Γrφφ = (rsinθ)-11 (-rsin2θ) = - sinθ (ΓLai)φrφ = [Γ()]φrφ = hφ-1 [hφ Γφrφ – (∂rhφ) δφ,φ] = (rsinθ)-1 [ rsinθ (1/r) – ∂r(rsinθ)] = (rsinθ)-1 [ sinθ – sinθ] = 0 The first three agree with the first three of Lai p 503 (8A.14), while the last is one of the many that vanish. Reader Exercise: Recall the identity (F.3.1) stated for Picture C1 of (F.1.4), (∂jgik) = – [gin Γ kjn + gkn Γijn] . (F.3.1) Apply this identity in orthogonal coordinates to obtain this alternate form of (J.7.17), (ΓLai)ijk = hk-1 [ – hi Γijk + (∂jhi)δi,k ] (J.7.19) This is perhaps a nicer form since the gamma i,j,k indices are in the same order on both sides. Recall now (T)'ijk = (h'i h'j h'k)-1 (T)'ijk (J.7.8) where (T)'ijk = h'i h'j(∂'kT'ij) + ∂'k(h'i h'j) T'ij – Γ'nik(h'n h'jT'nj) – Γ'njk(h'i h'nT'in) which we rewrite as (T)'ijk = h'i h'j(∂'kT'ij) + ∂'k(h'i h'j) T'ij – Γ'nik(h'n h'jT'nj) – Γ'njk(h'i h'nT'in) So here is our status proofing App J where we left off above J.1 Total time derivative as prototype equation 350 OK J.2 Computation of components (T)'ijk 351 OK J.3 Tensor expansions of T on the un and en base vectors 352 OK J.4 Tensor expansions of T on the e^n base vectors 353 OK J.5 Total time derivative equation written in unit-base-vector curvilinear components354 OK J.6 Shorthand notations and a continuum mechanics application 356 OK J.7 Maple computation of the (T)'ijk components for spherical coordinates 358 OK J.8 Maple computation of the (T)'ijk components for cylindrical coordinates 362 OK J.9 The Lai Method of computing (T)'ijk for orthogonal coordinates 363 OK full eq num check OK FINALLY after many days I have finished proofing Appendix J. AND, this means that all appendices have now been proofed and here is our current status Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7. So next on tap will be Chapter 8 which I know will be painful. Update 4/26/15 Proofing of Chapter 8 8.1 Overview 135 OK 8.2 The differential N-piped mapping 136 OK 8.3 Properties of the finite N-piped spanned by the en in x-space 138 OK 8.4 Back to the differential N-piped mapping: how edges, areas and volume transform140 (a) The Setup. 140 OK (b) Edge Transformation. 141 OK (c) Area Transformation. 141 OK (d) Volume Transformation. 143 OK (e) Covariant Magnitudes. 144 OK (f) Two Theorems : g' / h'n2 = g'nn g' = cof(g'nn) and |(Πxi≠nei)| = cof(g'nn) 145 OK (g) Cartesian-View Magnitude Ratios. 146 OK (h) Nested Cofactor Formulas. 147 OK (i) Transformation of arbitrary differential vectors, areas and volume. 147 OK (j) Concatenation of Transformations. 150 OK (k) Examples of area magnitude transformation for N = 2,3,4 150 OK Example 2: Spherical Coordinates: area patches 151 OK 8.5 Transformation of Differential Volume applied to Integration 152 OK 8.6 Interpretations of the Jacobian 155 OK 8.7 Volume integration of a tensor field under linear transformations 155 OK verify all eq num. OK! Task: Bold en in components, so (en)i. Do this globally ******* Storing a tail end of Section 7.18 here. ************************* Without proof, we claim that one can in general construct a transformation x' = Fb(x) for which some generic complete set of vectors bn(x) are the tangent base vectors en(x). For this transformation Fb, one would have (bn)i = (Sb(x))in from (3.2.6), so matrix Sb(x) would be known. Then Rb = (Sb)-1 is also known, and one would then integrate (2.1.6) dx' = Rb(x) dx to obtain a viable Fb(x). As discussed in Section 6.2, for this transformation Fb, we can interpret wnm as the contravariant metric tensor g'nm and so on. Here then are the translations of our en results to the generic bn of Section 6.2 : (2.1.6) e'n = R en b'n = R bn // by definition, Section 2.9 (e'n)i = Rij (en)j => (b'n)i = Rij (bn)j (7.18.1) (7.13.1) (en)i → (en)i => (bn)i → (bn)i (7.13.1) (En)i → (en)i => (Bn)i → (bn)i (7.18.2) (7.4.1) g'ij → g'ij => wij → wij // contra metric tensor for Fb (7.4.1) 'ij → g'ij => Wij → wij // cov metric tensor for Fb (7.18.3) (7.13.6) en em = g'nm bn bm = wnm (7.13.6) en em = δnm bn bm = δnm // orthogonality (7.13.6) en em = g'nm bn bm = wnm (7.18.4) (7.13.4) en = g'nm em bn = wnm bm // raise label (7.13.4) en = g'nm em bn = wnm bm // lower label (7.18.5) (7.13.1) Σn(n)i(en)j = δi,j Σn (bn)i(bn)j = δi,j // completeness (7.18.6) (7.13.1) Σn n enT = 1 Σn bn bnT = 1 // completeness (matrix form) (7.18.7) V = Σn V'n en V = Σn (Vb')n bn // expansion (7.13.10) V'n = En V (Vb')n = bn V (7.18.8) Status as of 6:30 PM: Chapter 8 is done and has no red showing. I thought it would be worse than it was. I did rewrite it in the past at least once, earlier it was a total mess. So Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8. Proofing of Chapter 9 opening material OK 9.1 Geometric Derivation of the Curvilinear Divergence Formula 157 OK 9.2 Various expressions for div B 160 OK 9.3 Translation from Picture B to Picture M&S 162 OK 9.4 Comparison of various authors' notations 163 OK eq num check full OK It is done. Update 4/27/15 Resume on Ch 9 above. Done! Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9. Proofing of Chapter 10 10.1 Expressions for grad f 165 OK 10.2 Expressions for grad f • B 167 OK eq num check OK Yikes!! I think I have completely messed up the gradient stuff! I am claiming that ∂nf = f,n is a vector when in fact it is f;n that is a vector! However, in (F.9.1) I do say B;α = ∂α B so why is that?? Look at the definition, Babc..x;α ≡ ∂α Babc..x – ΓnaαBnbc..x – ΓnbαBanc..x – .... – ΓnxαBabc..n + W Γκκα Babc..x There is one correction term for each of the indices abc,,x plu the W term. So if there are no indices a...x then there are no correction terms at all if W = 0. Thus B;α = B,α = ∂αB. So I guess yes, ∂αf really is a vector if f really is a scalar. Show this here: ∂'α f'(x') = Rαβ∂α f(x) Tempest in a teapot. Chapter 10 is now proofed, it was an easy one. Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9,10. Proofing of Chapter 11 Only one section. OK eq num check OK Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9,10,11 The next Chapter 12 on curl will not be so easy, so I defer it to post-lunch. 12.1 Definition of curl B 171 OK 12.2 Computation of the line integral 172 OK 12.3 Solving for the curl 174 OK 12.4 Various forms of the curl 175 OK 12.5 The curl in orthogonal coordinate systems 177 OK 12.6 The curl in N > 3 dimensions 177 OK eq num check OK This comment is misleading below 12.3.4, archive it here Comment 1: In the first line above one can replace [curl B](x) by [ x B](x) with the understanding that the LHS is the curl in Cartesian x-space and the RHS is expressing this LHS in terms of x'-space coordinates and objects. The RHS is certainly not equal to ' x B' = ε'nab ∂'aB'b(x') ' . It is to avoid this possible confusion that the curl is written out as the word curl, and the same comment applies to the other differential operators. (B'a;α – B'α;a) = ∂'α B'a + g'αβΓ'aβn B'n – ∂'a B'α - g'aβΓ'αβn B'n = ( ∂'α B'a - ∂'a B'α ) + ( g'αβΓ'aβn B'n - g'aβΓ'αβn B'n) = ( ∂'α B'a - ∂'a B'α ) + ( g'αβΓ'aβn - g'aβΓ'αβn)B'n Now maybe an identity to the rescue? They all have plus signs. So this is it (B'a;b – B'b;a) = ( ∂'aB'b - ∂'bB'a ) + ( g'acΓ'bcn - g'bcΓ'acn)B'n To avoid confusion, we shall assume a right-handed coordinate system as shown in (12.1.3) and as discussed in Section 6.9. This means that det(S) > 0. But from (5.12.6) J = det(S) > 0. Furthermore, from (5.12.14) with g = 1 we have J = g'1/2 . Thus we write An = Jen. To this we add the differential scaling factors (Πi≠ndx'i) to move from the finite N-piped having edges en to the differential N-piped having edges endx'n as shown in (12.1.3). Thus, the area of the far face n is given by, dAn = g'1/2 en (Πi≠ndx'i) . (12.1.4) Comment: This factor of J-1 won't appear in the final results, as will be seen, but we must include it in the analysis in order to claim that C'k are the components of vector C in x'-space. At 8:30 PM I have finished proofing Chapter 12! It consumed most of the day and it is much improved. Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9,10,11,12 Update 4/28/15 Proofing of Chapter 13 13.1 Derivation of the Vector Laplacian in general curvilinear coordinates179 OK tent 13.2 The Vector Laplacian in orthogonal curvilinear coordinates 181 13.3 The Vector Laplacian in Cartesian coordinates 183 Bug found! In (E.2.20) I claim that V = JW ΣnV'nen and similarly for more general tensor expansions. However, let's look back at the original derivation as shown in (6.6.9). We start with V = Σnαnen where αn are unknown. Dot this into en to get V en = αn and therefore V = Σn (V en) en Now write we know that (V en) has weight -1 of V does, so therefore, (V' e'n) = J-1(V en) But (V' e'n) = V'i (e'n)i = V'iδni = V'n (V en) = αn Therefore V = Σnαnen = Σn (V en) en = Σn J1(V' e'n) en = J1 V'nen OK, I need to add this in Appendix D (if that is the right one). Another bug: Using the normal dot idea, I found that V = ΣnVn un with un V = Vn Here it is: V un = ΣnVn un But direct evaluation gives. un V = (un)i Vi = gni Vi ≠ Vn OK, I have added a new section in tensor density appendix about what expansions look like! I then refer to this in the (E.2.20) area, and also in the curl Chapter where the subject comes up. I then use it in (13,1,11) and this leads to another bug: Equation (13.1.13) is wrong! It should say V'n = ε'ncd ∂'c{ J-1C'd} = . J ε'ncd ∂'c{ J-1C'd} (13.1.13) So here is then a corrected section from Chapter 13, (fixes in red) Since equations of (13.1.11) can be rewritten as C'n = ε'nab ∂'a{ B'b} (13.1.12) V'n = J ε'ncd ∂'c{ J-1C'd} . // J = (13.1.13) Now lower the index on C'n in (13.1.12) to get C'd = g'deC'e = g'de ε'eab (∂'aB'b) , (13.1.14) and insert this into (13.1.13) to get V'n = J ε'ncd ∂'c{ (1/) g'de ε'eab (∂'aB'b) } = J ε'ncd ε'eab ∂'c { (1/) g'de(∂'aB'b) } . (13.1.15) Then from (13.1.11) we have this expression for V, V = curl C = J-2V'n en = [ (1/)ε'ncd ε'eab ∂'c { (1/) g'de(∂'aB'b) }] en . (13.1.16) and it comes out the same!!! We can then summarize our results: B = G – V (13.1.7) G = ∂'n{ (1/) ∂'i ( B'i)} en (13.1.10) B = B'nen V = [ (1/) ε'ncd ε'eab ∂'c { (1/) g'de(∂'aB'b) }] en (13.1.16) B = B'nen = [ (1/) ε'ncd ε'eab ∂'c { (1/) g'de(∂'a[g'bfB'f]) }] en . B = B'nen (13.1.17) All is well again, Section 13.1 is now officially OK and I move to 13.2. Good changes were made! Proofing of Chapter 13 13.1 Derivation of the Vector Laplacian in general curvilinear coordinates179 OK 13.2 The Vector Laplacian in orthogonal curvilinear coordinates 181 OK 13.3 The Vector Laplacian in Cartesian coordinates 183 OK full eq num check OK Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9,10,11,12,13 Proofing of Chapter 14 14.1 Summary Conventions 185 OK 14.2 divergence 186 OK 14.3 gradient and gradient dot vector 186 OK 14.4 Laplacian 187 OK 14.5 curl 187 OK 14.6 vector Laplacian 188 OK 14.7 Example 1: Polar coordinates: a practical curvilinear notation189 OK eq num check Proofing status is now: Have done A,B,C,D,E,F,G,H,I,K and have done 1,2,3,4,5,6,7,8,9,10,11,12,13,14 Update 4/29/15 Maybe somewhere add a section on "separation", but review what already exists on this subject inside tensor doc. [ I don't think PDE's and separation of variables really fits into tensor doc, that is M&S stuff.] Make clear curl is circulation divided by area, not sure I got that in. Done Review again B'n;j;j difficulty. Done, whole new Section 15.8 ! Proofing of Chapter 15 15.1 Review of Chapters 9 through 13 191 OK 15.2 The Covariant Method 192 OK 15.3 divergence (Chapter 9) 194 OK tent 15.4 gradient and gradient dot vector (Chapter 10)194 OK tent 15.5 Laplacian (Chapter 11) 194 OK tent 15.6 curl (Chapter 12) 195 OK tent 15.7 vector Laplacian (Chapter 13) 195 OK tent comment in the intro on repetition (already there). I just did a major rewrite of Section F.1, we now have Γcab = qc (∂jqb) and there should be no mention anywhere in tensor doc of the former Γcab = ec (∂jeb) !! Also, I moved the proof that Γ = 0 for Cartesian space to the right place which is after I show the identity that proves it, see (F.4.16) which used to be (F.1.3), so I updated all those references. Now I can finally get back to Section 15.2!! Done! Bug:: I have to redo below (G.2.12) since en→ qn. I see there are many problems now that I have changed Section F.1 to involve those qn vectors!! Now is the time to get this all cleaned up. OK through end of Appendix F. I continue to scan down looking for Γ problems. Big Problem just below (G.2.12). I now see that the Γ used here is the Lai one, not mine. I will just remove this entire footnote and park it below. I could fix this all up with qn vectors, but just not worth it. ___________________________________________-- Footnote 2: We know that [(v)(e)]ij = [∂'jv'i + Γ ' ijc v'c] from the general covariant derivative formalism of Appendix F leading to the example (F.9.3) which says B'a;α = ∂'α B'a + Γ'aαn B'n applied in x'-space. However, the fact that [(v)(e)]ij = [∂'jv'i + Γ ' ijc v'c] can be reached by the following alternate path which avoids that formalism, and which is more typical of calculations done in the Lai et. al. book (e.g, p 54-66, sections 2.33,2.34 and 2.35). We shall use Picture B, (G.2.12) Comparing this to Picture C1' of (F.1.4), one sees that the x'-space affine connection is Γ'cin = ec (∂'ien) with the corresponding (∂'jen) = Γ'kjn ek. These are the Picture B versions of (F.1.6) and (F.1.5). Here then is our alternate proof that [(v)(e)]ij = [∂'jv'i + Γ ' ijc v'c] : dvi = (∂jvi)dxj (chain rule) => dv = (v)dx . Expand: dx = dx'i ei and v = v'n en => dv = dv'n en + v'n den . But den = (∂'jen)dx'j = Γ 'ijn ei dx'j // see just above that ∂'jen = Γ 'ijnei But this is the Lai affine connection, not my affine connection! so dv = dv'n en + v'n Γ 'ijn dx'j ei = [dv'i + v'a Γ 'ija dx'j ]ei . But dv'i = (∂v'i/∂x'j)dx'j = (∂'jv'i) dx'j so dv = [(∂'jv'i) dx'j + v'a Γ 'ija dx'j ] ei = [∂'jv'i + v'a Γ 'ija ] dx'j ei . Then dv = (v)dx => [∂'jv'i + v'a Γ 'ija ] dx'j ei = (v) dx'j ej or [∂'jv'i + v'a Γ 'ija ] ei = (v) ej => ei (v) ej = ei [∂'jv'i + v'a Γ 'ija ] ei = (∂'jv'i + v'a Γ 'ija ) => [(v)(e)]ij = ei (v) ej = [∂'jv'i + Γ 'ijc v'c ] QED (G.2.13) ________________________________________________________________ I am OK through Appendix G on this Γ stuff, keep going. OK through H. OK through (I.8.11) then some trouble. But I added (F.1.11) and then bailed out (I.8.11) stuff. More trouble with the Lai stuff at J.9.4, all fixed up!!! I think things are stable again. Back once again to 15.3. Proofing of Chapter 15 15.1 Review of Chapters 9 through 13 191 OK 15.2 The Covariant Method 192 OK 15.3 divergence (Chapter 9) 194 OK 15.4 gradient and gradient dot vector (Chapter 10)194 OK 15.5 Laplacian (Chapter 11) 194 OK 15.6 curl (Chapter 12) 195 OK 15.7 vector Laplacian (Chapter 13) 195 OK tent Wrapping the day with that last section still to go. I think I can shorten it based on extra work done in 15.6. Update 4/29/15 I started into 15.7, but then got bothered about the "two forms" issue. I looked again at my attempt to prove it and this led to a lot more stuff which I am now working on. I am not ready yet to continue with Section 15.7. Time now 12:30. I added a new rule (F.9.23) that was missing, pushed other numbers down with no xref costs. Bug Found: I found these results in Appendix D: det(Mij) = (1/g) (1/N!) εab..x εAB...X MAa MBb ...MXx (D.12.15) det(Mij) = g det(Mij) // scalar density of weight -2 g = det(gij) (D.12.18) Suppose I apply this to Mij = gij, the metric tensor. Then det(gij) = (1/g) (1/N!) εab..x εAB...X gAa gBb ...gXx = (1/g) (1/N!) εab..x εab...x Now use εABC... = g εABC... to get det(gij) = (1/g) (1/N!) g εab..x εab..x = 1 Then det(gij) = g det(gij) = g no bug yet. I suspect this is also true det(Mij) = (1/N!) εab..x εAB...X MAa MBb ...MXx . and this would say g = det(gij) = (1/N!) εab..x εAB...X gAa gBb ...gXx Let's check on this: det(gij) = (1/N!) εab..x εAB...X gAa gBb ...gXx = (1/N!) εab..x εAB...X gAa gBb ...gXx = (1/N!) εab..x εAB...X δAa δBb ...δXx = (1/N!) εab..x εab...x = (1/N!) εab..x g εab...x = g, correct Now consider g;j = (1/N!) [ εab..x εAB...X gAa gBb ...gXx ];j Expand this with Liebnitz with this idea (ABC);α = (AB);α C + (AB)Cα = [A;αB + AB;α]C + (AB)Cα = A;αBC + AB;αC + (AB)Cα so we then get [ εab..x εAB...X gAa gBb ...gXx ];j = [ εab..x εAB...X];α gAa gBb ...gXx + [ εab..x εAB...X] gAa;α gBb ...gXx + etc = [ εab..x εAB...X];α gAa gBb ...gXx So we end up with g;j = (1/N!) [ εab..x εAB...X];α gAa gBb ...gXx But direct calculation shows that g;j = 0, so how does that come out of the above? I think I have this theorem not yet documented, εabc...x;α = (1/) ∂α()εabc...x If that is correct, then [ εab..x εAB...X];j = εab..x [ εAB...X];j + εAB...X[ εab..x ];j = εab..x (1/) ∂j()εABC...X + εAB...X (1/) ∂j()εab..x = 2 (1/)∂j() εab..x εABC...X and then we get g;j = (1/N!)2 (1/)∂j() εab..x εABC...X gAa gBb ...gXx = 2 (1/)∂j() g = 2 ∂j() ≠ 0 but this does not seem to vanish!!! I have a screw loose somewhere! We have a Paradox, so the show grinds to a halt. Update 5/1/15 Will prove this theorem: Theorem 1: If rank-4 tensor Mabcd is totally antisymmetric on the first three indices, then Mabcd can be written as Mabcd = εabcVd (*) where Vd is a vector density of weight +1. Proof: You are given Mabcd and so you surely compute Xc = εijkMijkc Then εabcXd = εabc εijkMijkd = δabc;ijk Mijkd = Mabcd + all signed permutations of abc But, because -Mbacd= +Mabcd and so on, all the terms in this sum are the same. Thus εabcXd = 3! Mabcd or Mabcd = εabc [ (1/3!)Xd ] Therefore Mabcd can in fact be written in the form εabcVd where Vd = (1/3!)Xd = (1/3!)εijkMijkc QED. I guess it really is true. Theorem 2: If rank-4 tensor Mabcd is totally antisymmetric on all four indices, then Mabcd can be written as Mabcd = εabcd V where V is a scalar density of weight +1. Proof: You are given Mabcd and so you surely compute X = εijknMijkn Then εabcdX = εabcd εijnkMijkn = δabcd;ijkn Mijkn = Mabcd + all signed permutations of abc But, because -Mbacd= +Mabcd and so on, all the terms in this sum are the same. Thus εabcdX = 4! Mabcd or Mabcd = εabcd [ (1/4!)X ] Therefore Mabcd can in fact be written in the form εabcdV where V = (1/4!)X = (1/4!)εijknMijkn QED. So this is an alternate proof to what I show in tensor doc. I am now off in separate doc on Paradox of the Day 5/1/15 related to my claim that tensorization is unique. // My proof survived, and the Paradox has gone away. I did make a logic error. When AB=0 assuming that A=0 was wrong. Now I want to deal with the issue of g versus |g| and try to resolve that mess. In Chapter 5 I do introduce s = sign(g) and I state that |J| = / = | det(S) | = with the idea that g and g' always have the same sign. So I allow that you can have J < 0 as well as g < 0. Where does |g| first appear in tensor doc? It is in Appendix D below (D.5.4) so that is a good place to start probing. There I say |g| only is used in the "added sign s convention". I repeat this a few times, but now go down to section D.9. BUT go back to Example 2 below (D.2.3) where I consider g'-W1/2 = J-W1 g-W1/2 . // no R factors since scalar and I say that (gW/2 Bij) transforms under F as a regular tensor. This would certainly be hard to interpret if we had g < 0! I guess I would have to break it down into two statements. g'-W1/2 = J-W1 g-W1/2 . s = sign(g) = sign(g') > 0 (-g')-W1/2 = J-W1 (-g)-W1/2 . s = sign(g) = sign(g') < 0 You never have both cases at once. So it does seem you could combine this to say |g'|-W1/2 = J-W1 |g|-W1/2 Now go back to Section D.9. In (D.9.2) I use |g|-1/2 for the same reason above, we don't want anything imaginary going on. I could write this as (sg)-1/2 of course. So I like (D.9.2) as is, and use |g| is required here. // I agree with (D.9.7). OK through (D\.9.9) and starting section 2. This is entirely on "added sign s" convention, I skip it for now since I am all-Weinberg. Now on to Section D.10. I follow through and I guess I have to agree with this |g|-1 εcabεca'b' = gaa'gbb' – gab'gba' = EcabEca'b' (D.11.10) OK, now I see more trouble. Anyplace there is I have trouble! So let's now search for where those occur. Lots of them in (5.16), but I think g is always Cartesian for cont mech so OK. But g' might not be? So maybe some repairs are needed in Section 5.16 *****. More occurrences in Section 8.4. I think if g is Cartesian, then g and g' are always both positive, so not to worry in these sections. It is det(S) that changes sign if you change your 1,2,3 ordering. Fact: Since det(') / det() = [det(S)]2, the sign σ of det(S) has no effect on the sign s of g. If you change ordering of r,θ,φ you change sign of S maybe, but not the sign of g. That is good. The next real occurrence is just below (15.2.2) so I can just start repairing things there! OK, I think we are there!! One could, for example, write the last item as (see next section) (1/|g|) εabcde..x εabcd'e'..x' = 3! δd,e...x;c,d',e'...x' = 3! δ ... (D.10.40 and then this Kronecker delta symbol can be interpreted as a true tensor of rank 2(N-3), but we shall not pursue this notation further. OK I think I am all happy with the |g| question now, as well as with my additions to Appendix D. Finally I think I can return to Section 15 where we are trying to get proofing finished! Now I want to wedge in a few theorems involving the ; technology. Oops. Another appearance is in Corollary of (F.4.2). It is fixed. Added my little extra theorems to the end of Section F.10, all done. Now I am ready to go back to Chapter 15, I think all pieces are in place to continue. I will resume on this tomorrow, hopefully no more snags. Update 5/2/15 Chapter 15 again. // AT 10:15 I have finished proofing Section 15.7. But I want to add Section 15.8 to show why the two tensorizations are the same. Could make this App L I suppose, but it is short and I can just get it done. // OK, it is done. So: 15.1 Review of Chapters 9 through 13 191 OK 15.2 The Covariant Method 192 OK 15.3 divergence (Chapter 9) 194 OK 15.4 gradient and gradient dot vector (Chapter 10)194 OK 15.5 Laplacian (Chapter 11) 194 OK 15.6 curl (Chapter 12) 195 OK 15.7 vector Laplacian (Chapter 13) 195 OK 15.8 Verification that two tensorizations are the same 203 OK eq num check: OK Now I will go through all of tensor doc and deal with all references still in red, some of which are to Chapter 14 which is finally stabilized. ok through: start of Section J.1 The following was already done in (13.3.4), so archive off here Comment: In Cartesian coordinates the above can be written, using εnab = εbna, (B)n = ∂n(∂jBj) - εnab∂a (εbde∂dBe) = ∂n(∂jBj) - εbnaεbde∂a (∂dBe) = ∂n(∂jBj) - (δn,dδa,e - δn,eδa,d) ∂a (∂dBe) // (D.10.22) = ∂n(∂jBj) - ∂e (∂nBe) + ∂d (∂dBn) = ∂n(∂jBj) - ∂j(∂nBj) + ∂d (∂dBn) = ∂d (∂dBn) = 2(Bn) verifying (13.1.3) and the Cartesian-coordinates vector identity [(B) - x ( x B)]n = 2(Bn) . At 7 PM I finished my "get out the red references" pass. There are still some things that are purposely highlighted in red, so don't ever change the entire doc to black! Proofing of Overview and Notations. Done. search for word "section" where I may have missed doing xrefs ******* Done. I decided to capitalize Section when it refers to one of the actual Sections which are parts of Chapters, otherwise lower case. Make sure the circles look good in PDF, and find out why they are off in Word! ******** Well, the alignment varies with magnification, so I don't really know how it comes out in a PDF. I will check on this closer to production time. Task: Bold en in components, so (en)i. Do this globally ******* Did a search on (en) whole word and fixed quite a few that were not bolded. Then did (ei), all were OK. That is enough on this! Treatment of name Lai ? ***** OK, I will leave this as is. Don't cap all standard notations, looks weird. Just some of them. Reproofings of rewritten sections. 5.16 done 5.12 done 6.8 done 7.18 done fixed an error in the b table 15.8 done fixed a text error 7.9 done D.12 done F.1 done D.11 done There is just too much for me to reproof the entire document. I just did some repairs and improvements at the end of Section D.10 and save this off, all xrefs done. Example: According to the second entry in (D.10.37) we know that (implied sum on c) εcabεca'b' = = εabεa'b' of (D.11.2) Thus we can rewrite (D.11.2) as εcabεca'b' = δa,a' δb,b' – δa,b' δb,a' . We now put this into properly tensorized form as described in Section 15.2, |g|-1 εcabεca'b' = δaa'δbb' – δab'δba' = gaa'gbb' – gab'gba' = EcabEca'b' Raising indices a,b then gives this useful identity relating the ε or E tensors and the metric tensor: |g|-1 εcabεca'b' = gaa'gbb' – gab'gba' = EcabEca'b' (D.11.14) Both sides of this equation are true rank-4 tensors (weight W = 0), with contraction on index c. Pausing now at 5 PM Sunday 5/3. Time to build a Visio and Maple index. I have never done this for tensor doc. // Got it done but will do the Maple items manana. Update 5/4/15 Question: Is it possible that Bn;j;j gives a much simpler formula for vector Laplacian that I have overlooked? In my Section 13 I use the curl curl form as the starting point of course and end up with a result that has the nasty εε structure. The whole answer after all is this simple form, so Bn;j;j = (Bn;j);j = ∂j Bn;j + Γnjs Bs;j + Γjjs Bn;s = ∂j [ ∂n Bj + gnβΓjβs Bs] + Γnjs[∂s Bj + gsβΓjβr Br] + Γjjs [ ∂n Bs + gnβΓsβr Br] This is NOT very simple! It is a sum of 2nd and 1st and 0th derivatives of Br, including some ∂jΓjβs stuff, and some Γ Γ factors, very nasty indeed. On the other hand, in Section 15.8 I do take this path, B'n;j;j = g'ne g'adB'e;d;a but this really does nothing in terms of simplification. Added a complexity comment at the end of Section 13.1 commenting on the above. Let's now fill out the Maple part of the index. // Done at 10 AM, just cancelled Brent by phone. Called Lorraine to ask who to see on tennis elbow, just in case. Spell Check. Use print layout so you can pick a specific batch of pages for each check. In the scratch doc, each time it craps out, just delete the doc up to that point and then continue. Spell Check. p 1-82 batch: it crapped out at 6.5.1 which is the first 83 pages, but I found 9 typos in that batch. Web still gives no fix for this problem. So I just do batches. Spell check p 80-125, but only got to 7.14.4 on page 125, so maybe only good for 50 pages at a shot. Found 2 fixes. Got to 10.2.3. spell check done through: all done! total typos found: 24 Just proofed the References since I forgot to do that, done! So the spell check took maybe 2 hours at most. Here is how to do it. How to Spell Check a Large Document 1. Select the first 100 pages or so, paste into a scratch document, make sure spell check is running. 2. Page down and look for red underlined spelling errors. For each, repair in the original document. This is where human pattern recognition sparkles if there are many math expressions etc that get underlined which are not spelling problems. 3. If Word shuts down the checking system due to too many errors, delete the scratch document from that point back to the start, you can then resume spell checking where you left off. 4. Repeat the above steps for each 100 pages or so of the document. It helps to keep track of how far you have gone in the check at each step. Pagination. TOC OK but needs final edits remember! Overview OK 2:35PM, made some additions here Chap 1 OK Chap 2 OK Chap 3 OK Chap 4 OK Chap 5 OK now 3:35PM Chap 6 OK Chap 7 OK Chap 8 OK Chap 9 OK Chap 10 OK Chap 11 OK Chap 12 OK Chap 13 OK Chap 14 OK Chap 15 OK App A OK App B OK App C OK App D OK App E OK App F OK App G OK App H OK App I OK App J OK App K OK Refs OK Reviewed (7.17) is OK. Reviewed (7.16) is OK. Reviewed (5.16) is OK. Update 5/5/15 Continuing on the above pagination starting with App E. I want to replace => with everywhere. Word has trouble doing that even if I paste in the symbol.  => => My charcode macro says for that it is this in TNR Well I see that this thing is in the Symbol font, not TNR. So in Replace, just specify that font and then it works. Do this globally in tensor doc! It did 178 replacements more or less! DONE. Idea: Present the messy Trθφ stuff in a Maple Table instead of the mess I have. 1) the Maple binder where I did an example of this is missing, not in basement anywhere. 2) I have no search handle for the concept. Table and Array is something else. 3) my example mws file showing this I cannot find among the 1600 Maple files, don't know the name 4) I cannot find this in my Maple User Guide, don't know what to search for I think I made a table of the Legendre polynomials for a few low ordrs. there are 48 MWS which contain the word Legendre Ding!!! The magic word is "spreadsheet". File is "spreadsheet example.mws" in physics/E&M/Maple area. So my code of interest lies in file Now, where is my just-made Maple index? desktop tensor doc folder (where it should be) YES Datum of interest is (J.7.12). Index says (J.7.9-12) (T) sph Maple: gradTsph App K.mws, in Physics/Lai CM area OK, I now remember the problem: You cannot turn off the ~ characters so get things like ∂/∂θ~ in equations and all is ugly! OK, I give up on making it look better. Print allows no alignment. Lucky it is as good as it is! User would have to manually create a table. Completed pagination at 11:30 on 5/5/15. Let's now do a final check of EQ sequencing AND left-right positioning. Eq (6.3.5) is missing, go fix! There are refs to 6.3.10 only which I will have to change to 6.3.11 BUT page view cannot be trusted I see. I will have to do this in the PDF. First PDF Cycle 5/5/15 Making first PDF on Alta, says 238 bookmarks, then sits a long time with no apparent action. Then the pile driver starts up. Then all message screens go away and it has failed. I start it again without closing the doc. 2nd pass: gets the 238 bookmarks. Sits but CPU is 98% word. I have it not working off the flash drive. After a minute or so, pile driver starts. Then gets right to "creating pages" and we are on track. Done. Need to directly edit the TOC. Check to see if any bookmarks are missing. 7.9 is missing! Think is fixed most of the 8.4 subsections are missing (need to be on single line!) Fixed E.7 has extra space at the start. Fixed Should I be displaying this low level a,b,c stuff? If so, why not in other places? Pagination check Page 43: orphan bottom line. on page with (5.2.10), add a page break! Done Page 47 maybe rethink where that context pic should go Done Page 61 add page brak where 5.14 starts Done page 78 bottom add page break Done This is very annoying, there are lots of differences between the PDF page ends and my Word page ends. Start Over. I restored Black tensor doc to as it was before fiddling just now, it now matches the PDF and the file on Alta desktop. On Alta in page view, do side by side compare of page view in Word versus PDF. Page at a time. Page 14 is first with a difference. Word is failing to display the last line at all! So page at a time page view is no good in Word!! So change both to 125% display and continue the comparison. OK thru: page 42 Page 43 is first issue. At page view 125% Alta and Black Word do not agree!!!! Rule: You must paginate on the Alta where the PDF Generation takes place!!! I resynced the two files, and it is true: Page 43 displays differently at same 125% page view on Alta versus Black. It might be some OS difference! OS, fonts, or printer -- those are the choices. So lets see if I can find some printer setup difference. But they both see the same printer really. Aha!! Although it is exactly the same *.doc file on both machines, the Margins in page setup are different!!! I am amazed, I thought that was part of the doc. Black has a 1" bottom margin whereas Alta has a .75" bottom margin!!! When I create a new doc on Alta, it has a 1" bottom margin, so what is going on here? Experiment #1. Change to 1" bottom margin on the Alta version. So then both have 1" bottom margin, Alta and Black. But Page 43 still looks different! Headers 0.5" in both, so now I see no difference. Experiment #2. After changing bot marg to 1" I save as **1.doc on Alta. The margin setting goes with the doc for sure! Compat options are the same! I tried to make a macro that does ActiveDocument.repaginate, but it seems to do nothing at all. How to force repagination. I think if you actually print to the spooler with a selected doc, it will have to repaginate for that printer. Then you can go back. Do this with printers off then delete the spooled files. But when I did this, spoolers kept feeding the printer long after deleted jobs! So don't do it that way! Question: Why does the printer type say 4MP on Alta, but says 4 Plus on Black? Was that just a name I entered, or does it maybe affect the print driver? I had to reboot both PC to stop printing. I reinstalled the driver on Black, I gave it the 4MP name, but it still installs the LaserJet Plus driver, fine. Page 43 looks good on Black. It still looks different on Alta. I guess changing the bottom margin made no difference at all! Theory: It is the OS difference somehow. Plan: Do pagination always directly on Alta because THAT is what the PDF is going to match! I will now once again make a fresh copy of the Black tensor doc, move it to Alta, and redo my pagination. I am doing a complete repagination now on Alta. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 A,B,C,D,E,F,G,H,I,J,K,Refs OK, this Alta repaginate went pretty fast because distinctions between Alta and Black are small, but just enough to make a mess of a few alignments. I fixed everything noted above. The current tensor doc is now on Alta desktop! Did TOC edits and will do another PDF cycle on Alta right now, the second. Second PDF Cycle 5/5/15. This time there are 240 bookmarks, only up by two. Sits then pile driving starts. It then fails with no message box, so I just start it over, same thing as in the first cycle. Second time works, as before. Done and I will now examine the PDF: bookmarks? the Section 7 problem is fixed it only sees (a), (f) , (k) and Example 2 in Section 8, same as last time I guess I have to re-enter them all? Maybe to a smaller test doc!! All other bookmarks are OK! orphans? looked at every page bottom, PDF has done exactly what Word told it! line integrals They look perfect in the PDF. I take just Section 8.4 put into test1.doc. Made a PDF of test 1, it shows the problem exactly. b,c,d,e,g,h,i,j, Re-entered them all, tested in test2, Third PDF Cycle 5/5/15 I guess I have to let it fail each time. So I start it up as usual. Bookmarks = 248, up from 240 which is exactly right for the edits I just did. It works first time probably because I ran those little temps in betweeen. Examine the PDF bookmarks are fixed, maybe it can be shipped!! Copied the DOC and PDF back from Alta to tensor area on Black. OK, I am now doing some proofing in the actual PDF at high mag, and am recording errata. Some former errata are in the list as well, I did not look at them. This is actually "enjoyable proofing" when the main work is done. I will continue tomorrow around section 5.2 in Xchange at 125% mag. Update 5/6/15 Today I read some of "the heart" which is Chapter 5 on metric tensors. I recorded errata, and I see that the most common problem is eq num's being wrapped around for reasons I don't yet know. I will have to study this directly on Alta because I fear OS issues. So I will now just scan from Sec 5.16 to the end looking for thrown eq num and record them in the errata log. DONE! They are rare, but I did find some. If there are 5 eq per page, then about 2000 of them, but only about 10 were out of plce, maybe 1/2 of 1%. I am glazed over after proofing early sections today, so I think I will do another PDF cycle to fix the 30 or so problems I already found. Edit only on Alta !!! Fixed the 30 items and now have going on Alta: (this time a tried a tiny conversion first with a test file, and IT failed on the first try! ) Bug: I cannot tell whether or not Acro 8 is going to fail until about 4 minutes passes on Alta after it does the post-bookmarks pile driver sequence. Hypothesis #1: If will always succeed after succeeding on a tiny file conversion. Fourth PDF Cycle Did this, worked first time, checked all my fixes in the errata log, found 2 that still needed fixes. Did about 5 in a row of the little test pdf conversions, then Fifth PDF Cycle Worked first time, checked final 2 items. At this point, I know of no errors in tensor doc on Alta. Bug: It has blue entries in the TOC, even though they are not blue in the Word doc. Perhaps you cannot edit anything or it remakes the TOC during the export. Then you would have to re-edit after each edit to tensor doc. Just for fun, I will make new TOC, edit the TOC, then do another cycle to see whether or not this fixes things. Sixth PDF Cycle Conclusion: If you make a TOC after all edits are done, then edit the TOC, the problem entries are NOT blue in the PDF, and all is OK! I guess it does not make a new TOC if it is up to date. The bookmarks fonts of course are still not OK, but no problem with that. Update 5/8/15 The Aatrix Font Mystery. Now that things are done, I might look at bit into the difference between Alta and Black's Word in terms of text alignment. First I need a doc that shows differences. Bug: For some reason, Alta cannot open a Word file it sees on Black! It just hangs. But if I copy the file to Alta then open, it works fine. So I try the bu tensor doc of 5/5 opened on both PC's. I look at (5.16.8). The eq num alignment is correct on Black, but it is wrapped on Alta. So there is our prototype example to study. I display tabs and there is some kind of difference! Both Words are 2003 8405 SP3, so no service pack is missing. Very strange. here is the difference on line (5.16.8) On Black (show hidden), there is tiny a character dec=hex=9 just after the R. If I blow up to 500%, it appears to be a tiny tab arrow! I verify that other large tabs have this char code. So here is what we have on Black: S-1 = Rt, as t is the tab On Alta we instead have S-1 = R t, as so the tab is much larger in size. And this causes the eq num (5.16.8) to be wrapped around to the next line on Alta but not on Black. Is this a tabs difference? [ no] Is it a font difference? [yes, read on] Fact: On Black, the insertion point after the R at 500% is just to the left of the 4.5" tab. but on Alta this insertion point is exactly AT this tab. That is the cause of the problem. There is then room for a tiny-tab on Black but not on Alta. So what causes this tiny left/right positional difference? Start at the left end of the line on both PC's and go left to right a bit at a time to see where it gets "off". Insertion point just after the r of Curvilinear: it is more to the right of the tab mark on Alta! So by this time we have the problem. It is the ' field code where the difference lies! The left edge of this g is aligned on both PC's but the right edge is not! Also, the over bar is heavier on Alta viewed at this 500% But in both cases the overbar is the same font character. The overbar is WIDER on Alta than on Black! This is true for an overbar typed all by itself! This as measured versus the screen top ruler. On Black it is about 55% of a tickmark. ON Alta maybe 70% of a tick mark. If you instead select the p of up on both machines, the p width is the same on both. For some reason, that font AAtrix OCR A Extended is different for this character on the two machines!!!! I think I am zeroing in on this problem. Alta: The wide overbar is in Aatrix OCR A Extended font, it says, size 11. This font exists in the font list on the Alta Word font list. But Black does not have this font!! It is not in the drop down list as it is on Alta. So Black is substituting something for this font I suspect. Experiments on Black 1. Create a new symbol. The overbar here can be bolded or not (looks different) but is from Times New Roman! My Macro program says to add ChrW(175) for this overbar. Where do I get these character codes and what does ChrW mean? Answers: ChrW means a Wide character, that is, a 2-byte Unocode character. and this is a VBA usage. The argument is decimal. But oddly 175 seems like a normal non-unicode code. Hex is AF, and this is the usual code for an overbar. OK, so on Alta I specified this to be in that Aatrix font, but I failed to load that font onto Black, so Black subs in the TNR (or perhaps something else) and that explains the mystery. So what do I want to do? I want Black to do what Alta has always done. Alta has my legacy of documents. so I want to alter Black, I don't want to alter Alta! What OTHER fonts might Black be missing that I commonly use? I can go into the PDF of released tensor doc and here is the font list: I use 18 fonts! Here they are from the PDF So here is a more compact listing: Alta Black TNR stuff: 7 yes yes Wingdings 1 yes yes Symbol 1 yes yes Script MT Bold 1 yes yes Palatino linotype 1 yes yes Lucida 2 yes yes Courier 4 yes yes AAtrrix 1 yes NO So in tensor doc, only this one aatrix font is missing! The Fix: Add this font to Black. This will make past-entered overbars look the same on Black as they do on Alta. Newly created overbars on Black will be TNR which is fine since Alta also has TNR. How do I add this aatrix font family to Black? I have done this before. Here from my setup doc Fonts. I surely added fonts to Word. An example is Script MT bold which is on Alta and not on Black. There is a control panel fonts item. Fonts are in C / windows / fonts. I look at the fonts on Alta from Black. You have to add "title" column to see what is in the "font name" column of Alta. Web says to just copy and paste, I will try it. This method worked for MSBold. So what other fonts did I add myself and do I use? Nothing comes to mind, but when I run into the problem I will come here. So I do a copy and paste using two trees in Cube Explorer and all done! Aatrix 9 fonts are now all on Black. The fonts appear at once in the Word drop down list. Effect of adding this font on Black: now that bu tensor doc looks the same on both machines. The equation number wraps on Black now just as it always did on Alta. Problem solved, missiong accomplished (I hope). May 6, 2014 Release ____________________________________________________________________________________ Tensor doc has rested now for almost two years, but now more activity: Update 3/16/16 While trying to write my differential forms chapter of wedge doc, I noted some errors in tensor doc. I put these all into the errata log and then implimented all the simple ones (changing to red as usual). But there are some more difficult items remaining. I did the simple stuff one or two days ago. Today I am attacking one of the harder errata, concerning the notion of "covariant transpose". I think I have it all figured out, but there will be some equation number changes in Section 7.9 only. In fact, in that section all equations above (7.9.3) will be changed, and this will require action soon: update tensor doc globally for all equation numbers after (7.9.3) ******** DONE I just pasted in new stuff for the Section 7.9 changes and am ready to do the above item tomorrow first thing. Update 3/17/16 Did the global update noted above as **** but need to check again after further edits to come. I don't think tensor doc has a Maple or Visio index, check into this!!! // I do have a file called Visio-Maple Index.doc it is down one level from top level. It seems up to date. Add dchange Maple code for (E.9.19). Want to see this code which uses dchange, not obvious to reader what is going on. But the index does not know where this code is! Just search mws for dchange! But I decided to redo this from scratch without the "dchange" call, code is in E_9_19.mws. Done and gives the same result. Need to increase size of ∞ symbols in summations as I did elsewhere, a font change ? If I search on ∞, I see only 4 occurrences. Is that possible?? Yes, word confirms. So only fix ones in sums. This is in (K.4.1). Where did I fix this before? Answer: charged bowl, put ∞ in Verdana 12 superscript. Just paste them out of bowl, only two to do, done! ****** full repagination will be needed DONE. I did final proofing of a new tensor doc section 7.19 on basis vectors and R and S matrix elements and installed it today around 2 PM. Done. Update 3/18/16 Major realization on the covariant transpose, so rewriting the end of Section 7.9. Equations 7.9.14,15,16 are all going to get bumped up by 5. Update all refs to these right now: 14 none, 15 none, 16 none DONE. So I have done some rewriting on Section 7.9, it is much better I think. I now have to go "fix up" places where I used the covariant transpose more tentatively. Fixed up (E..7.4): done Fixed up Section which starts with (E.7.14): done Fix up (E.9.19): done I think I am done now with this second round of transpose fixes to tensor doc (3/18/16). But there is one other place to maybe improve. Fix up near (7.19.19) : this is still in progress, leaving for the Bernie rally. // Rally over, DONE. So ends another round of covariant fix-up to tensor doc. Now I can pop the stack and return to the issue of the pullback operator and how to deal with it. Update 5/15/16 I reviewed some undone items above and did them. Next is to install my new section 8.4 (h) which I have been storing in "volume element v2.doc". I save the previous 8.4 (h) right here since it is so short ____________________________________________ (h) Nested Cofactor Formulas The object dAn is the "area" of a face on an N-piped. This face, which is itself an (N-1)-piped, in turn has its own "areas" which are (N-2)-pipeds, and so on, so there is a hierarchy of "areas" of dimensions N-1 all the way down. The area ratios of corresponding areas under transformation F are determined by equations similar to that above. For example, the mth face of face n of an N-piped has area ratio . This at least makes some sense since the matrix cof(g'nn) has dimension N-1, so its cofactor matrix has dimension N-2, and so on. For N=3, these faces would be line segments and one would have cof(g'33) =  cof[cof(g'33)]22 = g'11 = h'12  = h'1 (8.4.h.1) and h'1 is in fact the edge length ratio given above in (8.4.g.2). __________________________________________________ I have installed the new section and will proof it right now one more time. // Proofing done, fixed a few things! It is installed. Is there anything else? Look at the errata list please. Yes, there are more items to worry about! Here is the old start of section __________________________________ E.1 Direct Product Notation This entire Section uses the general Picture A context where x-space need not be Cartesian, (E.1.1) The Standard Notation of Chapter 7 is used throughout. Useful forms can be found in Section 7.18. The key tool required for the expression of tensor expansions is the notion of a direct product (tensor product, outer product) of n tensorial vectors defined in this simple way, (ABC ...)abc... ≡ AaBbCc..... (ABC ...)abc... ≡ AaBbCc..... etc . (E.1.2) The tensor ABC ... is nothing more than the outer product of vectors A,B,C as in (7.1.1) for contravariant vectors, but later extended to any mixture of vector types. One can define a direct or outer product of two rank-2 tensors in this way, (MN)ab,AB ≡ MaANbB // rank(M) = rank(N) = n = 2; number of tensors = I = 2 (MN)ab,AB ≡ MaANbB etc (E.1.3) Notice how the indices are arranged on the left side of each equation. The same idea can be applied to form a direct product of tensors of any rank, for example (MN)ab,AB,αβ = MaAαNbBβ etc // rank(M) = rank(N) = n = 3; number of tensors = I = 2 (E.1.4) On the left side the number of groups of indices equals the tensor rank n of the tensors on the right, and the number of indices within each group matches the number I of tensors being direct-product-multiplied. In what follows, only the (E.1.2) direct product of vectors shall be considered. One can define the dot product of two direct-product-space vectors in this obvious manner, (ABC ...) (A'B'C' ...) ≡ (ABC ...)abc... (A'B'C' ...)abc = AaBbCc..... A'aB'bC'c..... = AA' BB' CC' ... (E.1.5) where of course the indices abc can be "tilted" in any way desired according to (7.11.3). ___________________________________ So as of tdoay 5/15/16 8:30 PM I have no other errata known for tensor doc. It needs complete repagination! I don't really want to go hunting up problems. I did add Wedge as a new reference and I refer to it in two places. May 19, 2016 ***** Remove blue from TOC **** Verify title and author Starting pagination of tensor doc! TOC OK except for blue Overview OK Ch 1 OK Ch 2 OK Ch 3 OK Ch 4 OK Ch 5 OK Ch 6 OK 1 adjustment Ch 7 OK and perhaps 6 adjustments. Back up and rescan: OK, VERY long chapter Ch 8 OK , made a few changes, I added stuff here! Ch 9 OK Ch 10 OK Ch 11 OK Ch 12 OK Ch 13 OK Ch 14 OK Ch 15 OK App A OK App B OK App C OK App D OK changes made App E OK many changes a brutal section App F OK App G OK no change App H OK App I OK App J OK App K OK So the pagination is done. It is not perfect, but it is good enough and better than it was. Starting a PDF cycle now on Alta. Bookmarks 249, OK first try. On review I found a bug! The header for App B is lost! The problem existed in the last release and I never noticed it, I was just checking them now. Several appendix headers were a mess, lots of section marks missing, I think I may have knocked them out by raising first lines of sections!!! Fixed TOC both underlines, colors, and alignment, after making a new TOC. Stop! In word doc App H is missing from level 2TOC, is in level 3! Fixed this Second PDF cycle in progress, still 249 total bookmarks. I am working directly off the USB drive in these exports. // Done, brought back to Black, all headings are now in place. Forgot to push an orphan in the TOC, does not matter. Well, I fixed that and will do third PDF cycle, might as well get it right, Alta does not mind another cycle. I think about 5 min per cycle. Got it back, looks good. Will review both tensor and wedge pdf's tomorrow, then upload. May 20, 2016 Forgot to spell check my new STS section 8.4 (h), will do right now: done and OK. How about verifying some erratas to make sure they are done. Noticed that my release area has tensor doc in cap letters, so I will update that right now in the work area. That is the way all other docs are. At one time I thought this might be a search problem. May 19, 2016 Release ____________________________________________________________________________________