edit log

Wedge Document Edit Log PhL 9.24.15 This log covers the time period Sept 24, 2015 to the first release on May 19, 2016, a period of 8 calendar months. See in retrospect note elsewhere. Sept 24, 2015 I have been working on this stuff for several weeks now, and I think an "thought log" would be helpful, so that is starting right now at 3 PM on 9.24.15 Issue #1. In my section 2.1 I suddenly start talking about metric tensors and differential distance and dot products etc. I write (ds)2 = Σij ijdxidxj. But the reader can ask, "what is dxi" ? I have been talking about vector space V, not vector space Rn. What is the meaning of "distance" in generic vector space V? It has no distance, so why am I bring it up? The reason is that I want to talk about whether or not the basis vectors ei are orthogonal or not. I keep thinking I need the notion of an a b dot product in order to talk about ei ej = δij. Where do "components" come into the picture? I say V = Σi viei so I can refer to the coefficient there as a "component". So "component" seems like a well-defined concept. Then can I say ek = Σi (ek)iei ? Why not, since ek is like any other vector. Then I have components of the vector ek. Once I have such components, I can then at least talk about this quantity fij ≡ Σn (ei)n(ej)n I can then write this same equation in matrix/vector notation as fij ≡ ( (ej)1. (ej)2....(ej)n') One could then speak of cases where fij ≡ Σn (ei)n(ej)n = δi,jgi // orthogonal fij ≡ Σn (ei)n(ej)n = δi,j // orthonormal Key point: One can do all this stuff for a pure vector space with no metric and no dot product! Nothing I have done above talks about norms, metrics, or inner products. But one could certainly DEFINE an inner product a b = Σi aibi. And this could in turn be used to define a norm and a metric, and then you have a Hilbert space. So any vector space becomes a Hilbert space by this definition. But this then opens to door to consider other metric tensors. I could have a vector a ϵ V and talk about a small change in that vector da, and I could then say (da)2 = Σij ijdaidaj is the square "length" of this differential vector. What has one done in introducing this last equation? I have opened many cans of worms be talking about differential vectors. Suppose you just define a dot product by a b = Σijijaibj and say nothing about distances or metrics. Think of this as () (a,b) = Σijijaibj, then the function () is a bilinear function since then all the usual rules would apply! No reason for ij to be symmetric. Plan A: Suppose I just stop talking after getting to these lines Σn (ei)n(ej)n = δi,jgi // orthogonal Σn (ei)n(ej)n = δi,j // orthonormal Don't mention metric tensors, or norms, or dot products, or non-Euclidean spaces, or Cartesian spaces. Just leave it at the above point. Don't say anything about "covariant notation" or up and down indices. I could say that en is just "some other basis" which is true. OK, let's now apply this plan to section 2.1 and just remove stuff that is not relevant. OK, I have redone section 2.1 to avoid all mention of dot products. I guess I have to go back and remove these from earlier sections. Have to do that soon. // It is done, and I think thereby improved. As of 7 PM 9/24/15 I have finally woven the fabric between my opening Tensor Sections, and the Wedge Sections I wrote earlier. It took yet another full day. I had a lot of trouble combining these two documents into one, had to decide what order to put things, etc etc. Can't repeat things, can't have things out of order, on and on. 9.25.15. I have now battled my way through Section 2.2 on the V*W* space. There were many changes. I am finally ready to move into the first wedge section which is now 2.3. 9.26.15. I am not going through Section 8, things look good. Maybe I should replace my multilinear rule with the statement everyone else uses, rather than insist on my two liner? Not sure of this. The two liner makes things clearer, but one might miss the combined idea. Question: Does it make sense to talk about V^V as the wedge product of two vector spaces? None of my authors talks about this, but I did find an exchange comment which refers to a book of Fulton and Harris which has the following, [ right at this point I had a word crash with this log and the 5,6,7,8 doc open. A strange crash, no CPU time, and I did it with a bad control key combination. Saved -1 versions of both files, but think all OK and will continue working in the originals. Save frequently! ] The right side above seems to mean ( V^V^V...) which is what I am looking for. And then the left side is the space V*^V^* which I will soon be looking for. Can I see into this book? Yes, I have a no-OCR PDF file. Wedge is not in index, but "exterior algebra" is on page 475. Right up my alley, I will read a little in this section right now. Mentions a word "functorial". Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.[6] This book also writes which I think shows the meaning of the symbol |→ which I have been wondering about. Does this symbol have a name, does anyone mention it? Here from wiki list of math symbols, So there you are! However, I cannot find the quoted equation in this book!!! I don't see Λ^Λ type construction anywhere. I do see π : Vk → Λk as a projection and maybe that will help me later. So keep searching elsewhere. He says directly that Λ ^ Λ ≠ Λ2 but does not explain why. Here are some other references: Bourbaki is just too heavy duty for me. Edelen has no pdf I can find. Idea: What would the basis be for the vector space V^V ? Try ei ^ ej ? I comment on this as follows: One could rearrange the n2 basis vectors of VV into these two groups: (ei^ ej) = [eiej - ejei] n(n-1)/2 independent elements in this set (ei * ej) ≡ [eiej+ ejei] n(n)/2 independent elements in this set I found someone who says it! It is in projmec.pdf which I converted from ps on download. Here is what they say, Then later this author says, He seems competent, looked him up here, PhD win.ua.ac.be/~penne If he does it, I guess I can do it. I will just comment on this. New proofing of Section 2.3 (a) OK, but may throw out red stuff (b) OK, and I just found an easier counting method, include both. (c) OK, one red caveat New proofing of Section 2.4 -- OK Somewhere I need to throw in the quotient thing for wedge, but not now! ***** Section 2.5 OK! Section 2.6 OK, and it is very long! Section 2.7 OK, very short Section 2.8 I just learned the significance of evaluating the functions like T(v,w) at (ei, e'j)! Back and fill into the earlier sections. Sept 27, 2015. Added a new section (d) to 2.3 on "components". This then has an exact parallel in Section 2.4. Trying to get full parallelism between these various sections. OK, yet another read through, I will add equation numbers and continue with endless edits. 1. The Tensor Product opening text OK 1.1 The Tensor Product as a Quotient Space entire section OK 1.2 The Tensor Product in Category Theory entire section OK STOP, bug detected. Things are not clear with the idea of [ab]ij = viwj . Is this meaningful only in the simple basis, or for all bases ei and e'j ?? [ answer: it is valid on all bases, as long as you treat both sides the same.] For example, suppose I write T = ab = Σijaibj(eie'j) Since a = Σiaiei in ANY basis ei, the above equation must be correct in ANY basis, since ab = Σiaiei Σjbie'j = Σijaibj(eie'j) Now what happens when you take components? Trs = Σijaibj (eie'j)rs Not too fast now! Here is a little paradox. We can write (eie'j) = Σrs (eie'j)rs (ere's) where we expand (eie'j) on itself, so to speak. We don't yet have an ab component rule, so what can one do next? Staring at the above equation, it seems to me that one is forced to say (eie'j)rs = δi,r δj,s for ANY basis ei This would suggest that (ei)r = δi,r But my notes show that in fact' (ei)r = Rir So how do I resolve this paradox? Well go back to this simpler situation which should show the same problem, v = Σiviei v = Σi vi(e) ei Now use this equation to "expand er on itself" er = Σi(er)iei er = Σi (er)i(e) ei Now the paradox is more simply stated, This equation seems to imply that (er)i = δi,r (er)i(e) = δi,r But I know that (ei)r = Rri BUT, I am sure this is a Cartesian space component! It really says (ei)r(u) = Rri Take a component to get vr = Σivi(ei)r It seems that you are then forced to have (ei)r = δi,r in ANY basis! But I thought this was valid only for the simple basis! This is a very major issue and needs to be understood and repaired! Maybe you have to start over and say v = Σi vi(e) ei Then the contradiction goes away. For some other basis fi you would have v = Σi vi(f) fi When you take a component on the left side, you have to say which basis you are dealing with. This really gets me back to the dot product, which I keep trying to get rid of!!! Ouch! Suppose I go ahead and define Status 9.28.15. I have made several attempts to restart the Outer Products section, mainly how to deal with "components", and each attempt seems wrong, too complicated and so on. So this is my holding point at the moment. To talk components with general basis ei, you really have to deal with the metric tensor gij . I tried talking un and en and a transformation between them, but that seems so repetitive, it is really all there in tensor doc for the using. I am now planning to get rid of u and have only e. Question: In tensor doc, when you see Vi as a vector component, what basis is this associated with? V = V1 u1 + V2 u2 +... = ΣnVn un where un V = Vn un = gni ui V = V1 u1 + V2 u2 +... = ΣnVn un where un V = Vn V = V'1e1 + V'2e2 +... = Σn V'n en where en V = V'n en = g'ni ei V = V'1e1 + V'2 e2 +... = Σn V'n en where en V = V'n (7.13.10) So the answer must be: Vi are the expansion coefficients onto the ui . The coefficients onto the ei are V'i. Status 9.28.15. Today I finished a very long section on tensors and covariant notation, so my document is now going to be in full covariant notation! That decision has now been made, finally. I cut off the first two sections on tensor products which required no component and put them first as Chapter 1. Then Chapter 2 is this new thing on the tensor algebra. I will now do a proofing pass of Chapter 2. The eq nums are at least placed, but references are all wrong and will require lots of work to get right. cap section titles ? I forget what my standard is. 2. A Brief Review of Tensors in Covariant Notation opening text OK 2.1 R, S and how tensors transform I am not going to check tensor doc primed references on this pass ****** text and eq nums are OK. 2.2 The metric tensors OK. 2.3 The basis vectors en and en OK 2.4 The basis vectors un and un OK 2.5 Expansion of vectors onto the various bases in x-space OK 2.6 A change in notation long section, all OK 2.7 Tensor Expansions OK 2.8 The Outer Product of Tensors a long section, all OK 2.9 The Inner Product (Contraction) of Tensors OK All done proofing Chapter 2. I did this in a single sitting for coherence. The next task is figuring out how to follow this with the old outer product section which will be rewritten or eliminated. I do want the Kronecker delta section . 10.1.15. A new month, not much stability. Unhappy with the V(e)notation, will try again today. Also how to write the tensor expansion -- you have to explain when you write it! Just reread 2.1, it is OK as is. And 2.2 is also OK. Added comments declare x and x' dependence, some thing I had omitted. And 2.3 is OK. Idea that en(x) can be considered arbitrary. And Sec 2.4 is OK, the reminders at the end on x dependence work well. Sec 2.5 OK. So far things "read" smoothly with no catches at least for me. (um(u))n = δmn (um)(u)n = g(u)mn (em)(u)n = Rmn (em)(u)n = Rmn (um(u))n = g(u)mn (um)(u)n = δmn (em)(u)n = Rmn (em)(u)n = Rmn (um(e))n = Rnm (um)(e)n = Rnm (em)(e)n = δmn (em)(e)n = g(e)mn (um(e))n = Rnm (um)(e)n = Rnm (em)(e)n = g(e)mn (em)(e)n = δmn I have just revamped Sec 2.6 with the paren changes noted above (left column). I now think the label like (e) applies to the vector itself, specifying its space, and is NOT so much associated with the index itself. OK now thru Section 2.7 with new notation. I need to talk about outer products BEFORE tensor expansions, so shuffle sections! At 1:30PM I have a temp stable wedge doc through Section 2.10 on tensor expansions. 10.2.15. I just altered the Kronecker section to be all-covariant, so I am now going to proof it in temp3 and then install it as Section 3.2. // Proofing done, I like it, I will save out the old version and then install this version. // Done with Kronecker! I have already proofed Section 3.1 several times, so lets move on! Section 4.1 editing k = 2 tensor product Section 4.1 : edit this inside old tensor wedge doc. Keep out of the new version until that editing is done, then install it. Do this section by section. opening text OK I am doing major hacking here, since now Chapter 2 exists! Things are all clearer and I can make this section shorter. 1:21PM Start new proofing of Section 4.1 with eq nums in place. done This section is a very concise summary of much that went before, I am have various references into earlier sections. Here are the topics generalities about VW bilinearity of the dot product as a tie in with Chapter 1. special case of dot product for orthonormal basis etc outer product in matrix notation, same for inner product restate the rules two ways state wrong rules which are really for show tensor expansion for rank 2 tensor T So let's leave 4.1 be fore now, eq nums are in. Section 4.2 editing k = 2 dual tensor product Major changes here! Since I replace the qk with ek and am forced to write λi(v) = ei v so the index on λ is now up instead of down, and this is everywhere! // OK, I have battled my way through Section 4.2 in the dual world. I will now copy 4.1 and 4.2 into the main document. Just renamed from v2 to no v, so main doc is now called tensor and wedge, while the remaining two sections are in wedge sections doc. Section 4.3 editing k = 2 wedge product (a) done, has eq nums now. (b) done as well. (c) done as well. (d) done as well, lots of work getting things updated to my current way of doing things I will now install this Section. Done. Tomorrow continue on with Section 4.4. 10.3.15. Continuing Section 4.4 editing made lots of changes, added matching EQ nums, it seems OK and I will now copy it into the main doc. Section 4.5 editing Done, was pretty easy, very few components mentioned. I will install it now. Section 4.6 editing STOP I have digressed to write the quotient derivation of the wedge product. It turns out there are two ways to write this, I have them both. This was a big missing piece of my puzzle. These will be Chapter 5 Sections 5.1 and 5.2, I have it roughed out. I will now resume editing on Section 4.6. 10.4.15 Resuming at 9 AM on Section 4.6 Section 4.6 is now done. It was brutal, there are 49 equations to worry about! I guess there is just a lot to say. I have edited earlier sections along the way. I will now move the edited Section 4.6 into position in the main doc. Section 4.7 editing Did this fairly quickly. Question: How does a thing like T(v1,v2....vk) in (4.7.19) "transform" under a transformation??? I have never addressed this question! This will take a little pondering. I think I have seen this somewhere, dim recollection. Of course I should address this in earlier dual sections! 10.4.15. Continuing. I am wondering if I have perhaps a theory problem on this question. In section 4.2 I note that basically λi = ei as an operator and then λi(v) = ei v (allowing for a metric tensor there). Now to the extent that λi = ei , I should know how this "transforms" say in going from a no-prime ei world to a primed one: (e'i)a = Rab(ei)b Subquestion: I would also write (e'i)a = Rab(ei)b . Does this force (e'i)a = δia as it appears in tensor doc? I think I could go more generally from {ei} world to some completely arbitrary {e'i} world, and not the specific transformation of tensor doc. Well if {ei} are the tangent base vectors of that transformation, I think it would force (e'i)a = δia, but I could have some other transformation I suppose. In any event, look at (e'i)a = Rab(ei)b . It seems hard to deny that I would then have (λ'i)a = Rab(λi)b for how the λi transform. That in turn implies at once that the coefficients like Tij transform the way any other tensor like Tij transforms. So then Tij really are the "components", so why am I calling T(v,w) the components? Look at (4.2.24). I could add to that list (and should! V = Σi Vi λi This is a vector in the dual space. Then one has V(v) = Σi Vi λi(v) V(ej) = Σi Vi λi(ej) = Vj I am just closing the functionals with their argument vectors. Think now of λi(v) as a vector field. Then [λ'i(v')]a = Rab[λi(v)]b which is completely standard for tensor doc. So I would say V(v) = Σi Vi λi(v) in x-space V'(v') = Σi V'i λ'i(v') in x'-space Is V(v) a scalar field? Well, consider V = Σi Viei V = Σi V'ie'i Here in some sense the vector V is a scalar. Well this is the passive view. I am expanding the vector in two different coordinate systems, but same vector. So with the same idea I would write V = Σi Vi λi V = Σi V'i λ'i This is the same dual vector expressed in two coordinate systems. But then I guess you would have V(v) = Σi Vi λi(v) V(v') = Σi V'i λ'i(v') Are these equal? Well, the argument is just a dummy argument so you could write V(V) = Σi Vi λi(V) = Σi Vi λi( Σj Vjej) = Σij Vi λi Vj λi(ej) = Σij Vi Vj δij = Σi Vi Vi = V V V(V) = Σi V'i λ'i(V) = Σi V'i λ'i(Σj V'je'j) = Σij V'i λ'i V'j λ'i(e'j) Σij V'i V'j δij = Σi V'i V'i = V' V' I like these equations. Since V(V) is a linear combination, the functional V works right! So this seems to say that V(V) is a scalar! The functional is a vector, but when closed with its argument you get a scalar. So V(V) = V V = V' V' = V'(V') Let's try the next case up the ladder T = Σij Tij λiλ'j T(v1,v2) = Σij Tij λiλ'j(v1,v2) = Σij Tij λi(v1)λj(v2) = Σij Tij λi(Σn v1nen)λj(Σk v2kek) = Σijnk Tijv1nv2kλi(en)λj(ek) = Σijnk Tijv1nv2kδin δjk = Σij Tijv1iv2j This is a scalar! So I could write T(v1,v2) = Σij Tijv1iv2j = Σij T'ijv'1iv'2j = T'(v'1,v'2) So ALL these "function things" are scalar functions. Wow, how could I miss such a major fact? Let's reread (4.2) right now and see where this would fit in. This is my first discussion of dual anything. Realization: I need to add a section about "dual space of V" somewhere! ! Could it be Section 2.11 at the end of my "review" ? That might be an excellent place. This would not be V* W*, it would just be V* I have started into a new Section 2.11, but now on Monday 10/5/15 it is time for Cape Cod prep operations. 9:30 AM signing off. // Back later Section 2.11 first cut is done, lots of basic facts, I think I will install it and they see how this affects later stuff. I will first review 4.1 which is non-dual. // I removed orthonormal definition and put in the earlier section on the en where it belongs. Change equation numbers then in 4.1 which is a very short section. I have reviewed 4.1 several times now, it is short and I think say only the basics. I have now rewritten 4.2, taking into account the new section 2.11. It is shorter now. It needs to be read a lot more times, but I am tired. A lot happened today in tensor/wedge world. Things got shuffled around, some interpretations changed. I am a bit light on tying into other peoples work however! I have reviewed 4.1 and 4.2 a few times today, with some of 2.11, and think it is OK. I am trying very hard to have a parallel between rank-2 in V2 and rank-2 in V*2. Only in the second case does one deal with functions. So let's now take a look at Section 4.3 and see how it reads after changes made yesterday. Section 4.3 review (a) is very clean (b) is just fine, counting elements of L2 versus V2. (c) just fine, the geometry connection (d) practicing with components, all just fine Section 4.4 review (a) fast and clean (b) short and clean (c) short and sweet, no geometry stated (d) functions OK This whole section is OK. Section 4.5 review (after changes made elsewhere earlier) short and simple, I saw no problems on this pass Section 4.6 review. This is longer, chock-a-block full of good things IMHO. This is Lk non dual. At this point I have not yet processed Section 4.7 and 4.8. I just checked eq num seq of all sections 4.1 on and they seem OK. Section 4.7 edit: first reading OK, added the key Fact at the end. Let's read it again. STOP Just added more data at the end of Section 4.5 regarding TS. How to multiply tensors of rank k and rank k' This was just plain missing and is pretty important! Problem: can I invert this, Aii...i = ΣP (-1)S(P) TP(i)P(i)...P(i) i1 < i2 < ..... < ik Imagine a permutation where i1 = Q(j1) and so on. then write the above as AQ(j)Q(j)...Q(j) = ΣP (-1)S(P) TPQ(j)PQ(j)...PQ(j) AQ(j)Q(j)...Q(j) = (-1)S(Q) ΣP (-1)S(PQ) TPQ(j)PQ(j)...PQ(j) (-1)S(Q) AQ(j)Q(j)...Q(j) = ΣP (-1)S(PQ) TPQ(j)PQ(j)...PQ(j) Now apply ΣQ to both sides to get ΣQ (-1)S(Q) AQ(j)Q(j)...Q(j) = ΣP ΣQ (-1)S(PQ) TPQ(j)PQ(j)...PQ(j) Use the rearrangement theorem on the right on either sum, say the P sum, to get ΣQ (-1)S(Q) AQ(j)Q(j)...Q(j) = ΣP ΣQ (-1)S(Q) TQ(j)Q(j)...PQ(j) I don't think you CAN solve for T in terms of A. Idea: Suppose I redefine my object A so that by scaling it down by a factor k! T = k! Σi<i<....<i Aii...i (ei ^ ei ^ .... ^ ei) (4.6.22) Then (4.2.30) would say k! Aii...i = ΣP (-1)S(P) TP(i)P(i)...P(i) Aab = (Tab - Tba)/2 k = 2 a < b Aabc = (Tabc - Tacb + Tcab - Tcba + Tbca - Tbac)/3! k = 3 a < b < c (4.6.31) [Alt(T)]ii...i ≡ ΣP (-1)S(P) TP(i)P(i)...P(i) = Aii...i (4.6.32) and then a simpler result A = Alt(T). OK, what happens next in my product deal? I will copy all the stuff to here: T = Σii....i Tii....i (ei^ ei^ .....^ ei) . rank k, T ϵ Lk S = Σjj....j Sjj....j (ej^ ej .....^ ej) . rank k', S ϵ Lk' (4.6.47) T^S = Σii....i[Tii....i Sii....i] (ei^ ei ......^ ei) (4.6.48) (T^S)symii....i = Tii....i Sii....i (4.5.24) Clearly the wedge product T^S is an element of Lk+k'with the following tensor components, (T^S)symii....i = Tii....i Sii....i = (TS)ii....i (4.6.49) So I have this symmetric form: T^S = Σii....i(TS)ii....i (ei^ ei ......^ ei) Now we want to cast this into an ordered sum like (4.6.22) above, T = k! Σi<i<....<i Aii...i (ei ^ ei ^ .... ^ ei) (4.6.22) so that would say, (T^S ) = (k+k')! Σi<i<....<i(TS)Aii....i (ei^ ei ......^ ei) ≡ (k+k')! Σi<i<....<i(T^S)ii....i (ei^ ei ......^ ei) So it seems then that (T^S)ii....i = (TS)Aii....i = ΣP (-1)S(P) (TS)P(i)P(i)...P(i) = (k+k')! Alt(TS)ii....i But then I get this wrong result T^S = (k+k')!Alt(TS) just as I got without redefining A. I cannot find where Spivak explains his constant. He said it would be clear later. His equations have no numbers, there is no way for him to back reference anything. Spivak says this but maybe his ω and η are not really my T and S. Maybe they are somehow the A-form. Do other authors have a different normalization more in line with me? Spivak never uses ordered indices, for example. Scanning: Korman no Benn He has these results: which seems the same as Spivak's Alt function (note the factorial). But he claims, with no coefficient at all. I think then Benn would say a^b = (1/2!) (ab - ba) so he is then normalized differently from me. Here is his footnote, Benn Tucker DO use ordered indices. I think they are more on the ball than Spivak (1965)maybe. I see I have a huge mess to detangle now. I will sign off pre Cod right here. Maybe switch to the nice Benn scaling so that Alt then stands exactly for my expansion in terms of the P operator, with no constant needed. That expansion has no name, and Alt provides one! Resuming Friday 10/16/15. I see that Suter puts the 1/2 factor as well into the wedge definition, so I will do it. I will now edit all of my document starting at 4.3 where the wedge first appears. Am working now in tensor and wedge v2 just in case I decide to undo all these changes! (a) done (b) done (c) done (d) lots of work needed here on factors of 2, but I think it is all OK now. I changed the definition of Ars to include 1/2. 4.4 The wedge product of 2 dual vectors (a) done (b) done (c) done (d) done, some shuffling around here. 4.5 Involves only tensor products so no changes. 4.6 The wedge product of k vectors This is where major changes are going to happen. // Things have slowed down. Question: What can be said about the "normalization" of the wedge product when written in terms of the tensor product. Spivak page 79. Applying his general rule, you would say ω ^ η = 2! Alt(ω η) = 2 * (1/2)(ωη - ηω) = ωη - ηω but he never writes this explicitly! Spivak is pretty hazy on normalization IMHO. But Spivak very clearly puts a 1/k! factor in his definition of the Alt function. Benn. page 22 is very explicit: and this is exactly what I have done in normalizing a wedge of vectors. He also has these statements Both of these exactly agree with what I am now doing. The Benn Alt also has the 1/k! factor. Wiki. Very hazy, they say physics people do like Benn, whereas Spivak does it differently. They don't say either way is wrong. Ablamo is too fancy. Denker has same as Ben for the many vectors wedge normalization, except his products are geo products and not tensor products, so not really useful. Conrad is different, and does this and later says this so he does not use the leading factor at all. But later he says, so I guess he would support me. He makes a real argument for why the k! should be there. I should reference his discussion of this factor, that will get me off the hook. Hitchen does not show any symbols at all. Korman is more on Clifford, no connection. Cooper has nothing, a very short piece. So I will stick by my guns and add a pre-emptive comment. OK, I have added a comment on this (1/k!) factor at I think the right place, early on in the k-wedge section where I state the candidate form. I think the other k! factor is a separate convention. Now, how about my OTHER k! factor which appears in (4.6.22). I think this is independent, but maybe not, so check it out please. OK, I did some cleanup and added headings in 4.5 on tensor product of k, then added similar new headings in 4.6 on wedge of k. Added to the titles of these two sections. I think today had good forward motion. I want to resume post-Hawaii at the end of Section 4.6 where I have marked things. Want to know how to multiply elements of L(V) and somehow that means you are doing wedge multiplication. Again, I am stopping right here at 6:15 Sat 10.17.15 and will resume post-Hawaii, assuming there is a post-Hawaii. Well, I am getting in a few extra AM hours. Realized something I overlooked. Go back to this point where I compute the A coefficient Aii...i = (1/k!) ΣP (-1)S(P) TP(i)P(i)...P(i) = (1/k!) ΣP (-1)S(P) (v1)P(i)(v2)P(i)...(vk)P(i) = (1/k!) ΣP (-1)S(P) M1P(i)M2P(i)...MkP(i) = (1/k!) [ M1iM2i...Mki + signed permutations ] = (1/k!) determinant of a certain minor of a matrix with k rows and n columns Keep these facts in mind: (1) Aii...i is totally antisymmetric and so vanishes if two indices are the same (2) the index set has n values where n is generally larger than k, so the ir is not a permutation of the full index set. (3) since n > k in general, be very careful trying to use the ε symbol to express anything involving components of vectors. Here I have defined a matrix M which is this: Mab = (va)b so that, M = (v1)i (v1)i ..... (v1)i (v2)i (v2)i ..... (v2)i (v3)i (v3)i ..... (v3)i ... (vk)i (vk)i ..... (vk)i This is a square matrix, but it is k x k and not n x n. The ir are some subset of a partition of the full set of n components. So you cannot claim that M = [v1 v2 .... vk ]T where bolded vectors are the full vectors with all components. The matrix M is a minor of a matrix with k rows and n columns. How might I annotate this matrix M which is a minor of the larger matrix. k! Aii...i = det I need to transpose the above matrix to get, Then T = v1^ v2^ .....^ vk = Σi<i<....<i k! Aii...i (ei ^ ei ^ .... ^ ei) = Σi<i<....<i det (ei ^ ei ^ .... ^ ei) = Σi<i<....<i { k! [Alt (v1v2...vk)]ii...i } (ei ^ ei ^ .... ^ ei) Now consider some special cases for this new determinant form. For two vectors we get T = v1^ v2 = Σi<i det (ei ^ ei) det and I think this agrees with earlier results, need to check. Resume Weds 10/28/15 after Hawaii Trip. Clarifying limits on sums in Section 4.6, Consider [ Σi<i<...<i + Σi<i<...<i + many similar reorderings ] = ΣP ΣP(i)<P(i)<...<P(i) = ΣP Σ i< i<...i I think the second form is what I really mean. I am not permuting the ir, I am permuting the r. So how does this affect all subsequent equations? I copy paste and edit: We then partition the summation space as follows (1 ≤ ir ≤ n) Σi≠i≠...≠i = [ Σi<i<...<i + Σi<i<...<i + many similar reorderings ] The total sum can be written in this manner, using the permutation sum notation, Σi≠i≠...≠i = ΣP Σ i< i<...i (4.6.25) where P are the k! permutations of the k integers {1,2,...k}. Thus, we can write T = ΣP Σ i< i<...i Tii...i (ei ^ ei ^ .... ^ ei) . (4.6.26) At the end of this section we shall prove the following Lemma, which in the meantime we hope seems at least plausible to the reader, ΣP [Σi<i<...i] fii...i = Σi<i<...<i [ΣP fii...i] . (4.6.47) Within the ΣP permutation sum, the permutation operators have moved from the summation index subscripts to the summand index subscripts. Accepting this Lemma, we then have T = Σi<i<...<i ΣP [ Tii...i (ei ^ ei ^ .... ^ ei) ] . (4.6.27) But we know that (ei ^ ei ^ .... ^ ei) = (-1)S(P) (ei ^ ei ^ .... ^ ei) (4.6.28) where S(P) is the number of swaps associated with permutation P. Thus. T = Σi<i<...<i [ΣP (-1)S(P)Tii...i] (ei ^ ei ^ .... ^ ei) (4.6.29) which we can compare with the ordered sum (4.6.22), T = Σi<i<....<i k! Aii...i (ei ^ ei ^ .... ^ ei) . (4.6.22) Thus, the relation between the A and T coefficients is given by Aii...i = (1/k!) ΣP (-1)S(P) Tii...i i1 < i2 < ..... < ik = (1/k!) [ Tii...i + all signed permutations ] // k! terms (4.6.30) The A coefficients are just the T coefficients fully anti-symmetrized. The motivation for adding the k! in (4.6.22) is to have the (1/k!) factor in (4.6.30). Then if it happens that Tii...i is already totally antisymmetric, one ends up with all expansion terms being the same and then Aii...i = Tii...i . I don't really know where I am going with this paper, but I can see that it is stifled by its organization. So today I have moved to version 3 where things are broken out in a way that allows a lot more elbow room. I now have ALL the k = 2 stuff in Chapter 4. Chapter 5 is the k = k tensor product, and Chapter 6 is the very long k = k wedge product. I will probably need follow ups for both dual cases. Now sure yet where to put products of tensors other than vectors. I need now to add subsections to the new Chap 5. Thurs 10.29.15 I have sectioned my new Chapter 5 on tensor product of k vectors, and am now renumbering all equations as I do a proofing of this Ch 5. I am now happy with Chapter 5, did not check pre-5 refs however. Now reviewing the new Chapter 6. 6.1 OK 6.2 OK [ IMHO this is a good paper, getting all these facts clearly stated! ] 6.3 OK 6.4 OK 6.5 OK 6.6 OK 6.7 OK 6.8 OK 6.9 OK Finally I am ready to start again on tackling the wedge product of two arbitrary tensors. Now for the 10th time, why can't I get ε into this thing, Aii...i = (1/k!) ΣP (-1)S(P) TP(i)P(i)...P(i) = (1/k!) Σjj....j εjj....j ?? Because the LHS has free ir indices, this just does not work the way it does in (6.1.2). Trouble: I have argued in (6.5.11) that the elements of Lk are totally antisymmetric tensors, but then in the very next equations I seem to claim that Tii...i = ai bi ... qi is an element of Lk even though this Tii...i is obviously NOT a totally antisymmetric tensor!! OK, fixed that up. As I look at section (6.10), I think I have only done a special case of the general wedge product. Here are the ordered expansions, T = Σi<i<....<i k! Aii....i (ei^ ei .....^ ei) . rank k, T ϵ Lk S = Σj<j<....<j k'! Bjj....j (ej^ ej .....^ ej) . rank k', S ϵ Lk' (6.10.1) and here are the symmetric ones,. T = Σii....i Tii....i (ei^ ei .....^ ei) . rank k, T ϵ Lk S = Σjj....j Sjj....j (ej^ ej .....^ ej) . rank k', S ϵ Lk' . (6.10.4) I can go ahead and show that T^S = Σii....i Σjj....jTii....i Sjj....j(ei^ ei .....^ ei) ^(ej^ ej .....^ ej) = Σii....ijj....jTii....i Sjj....j(ei^ ei .....^ ei^ej^ ej .....^ ej) = Σii....iii....i[Tii....i Sii....i] (ei^ ei ......^ ei) = Σii....i[Tii....i Sii....i] (ei^ ei ......^ ei) = Σii....i[TS]ii....iii....i(ei^ ei ......^ ei) (6.10.5) where, similar to (5.6.7), we use in the last line our standard outer product notation, [TS]ii...iii...i = Tii...i Sii...i . (6.10.6) But then how do I get this into the ordered expansion form? Let us momentarily replace TS by the symbol T, so that (6.10.5) reads, T^S = Σii....iTii...i(ei^ ei ......^ ei) (6.10.8) According to (6.4.1), this symmetric sum can be replaced by the ordered sum (6.4.1), T^S = Σi<i<....<i (k+k')! Aii...i(ei^ ei ......^ ei) (6.10.9) where, according to (6.4.12) and then (6.5.2), Aii...i = (1/(k+k')!) ΣP (-1)S(P) TP(i)P(i)...P(i) = [Alt(T)]ii...i // A = Alt(T) (6.10.10) Replacing T = TS we get this final result for the wedge product of tensors T and S, T^S = Σi<i<....<i { (k+k')! [Alt(TS)]ii...i } (ei^ ei ......^ ei) (6.10.11) OK, I have now repaired Section 6.10, I think it is OK and I recover an earlier result. Tomorrow consider: 1) doing S ^ T and obtaining the usual commutation rule 2) maybe do some fancier examples of T ^ S. 3) extend result to wedge of three tensors. Fri Oct 30, 2015 I did all three items above and made major improvements. I then got rid of the k! annoying factors by redefining A. First I did this with A, but decided to go all the way with A. Then I realized that both alt and Alt have their roles to play, and they are both there. I then did much more work on the Sym and Alt business. Overall, things are much improved and more efficient. Loose end: in an earlier chapter, there is some possible confusion with symbol A, go check that out. This was the reason I put the k! into the ordered expansion in the first place. The next chapters of course will be the "dual space" versions of things! Sat Oct 31, 2015 Went back and cleaned up my A normalization in earlier wedge sections, seems OK now. The earlier sections are now so vague, I think it is time to write summaries for them before I go on. Chapter 2 Reproofing. Section 2.1. This is a great review of tensor doc notation and the concluding comments are also very good. My motivation for putting out this paper increases. Overview of Chapter 2: Written after reading each today section! In a sense, this Chapter 2 is a 27 page summary of my tensor doc which is 391 pages long! Section 2.1 is a review of covariant-notaton tensor analysis using the conventions of Ref **. It is emphasized that the notion of "tensor" is based on an underlying transformation between x-space and x'-space. A general non-linear transformation x' = F(x) is characterized by its differential matrix R. Section 2.2 continues this review, bringing in the metric tensors g and g' and covariant dot product . Section 2.3 describes the tangent base vectors en and dual base vectors en and shows how these base vectors are connected to the metric tensor g'. Section 2.4 describes the "axis-aligned" base vectors un and their dual base vectors un and shows how these base vectors are connected to the metric tensor g. Section 2.5 shows various ways of expanding a vector V onto basis vectors, and demonstrates that the R matrix is the basis-change matrix relating the e and u bases. Section 2.6 changes notation from "Picture A" to "Picture E" and computes all components of all basis vectors. More precise notation cleans up a vague paradox presented at the start of Section 2.1. Section 2.7 shows how a Picture E system can be built up from a set of constant base vectors en. Expressions are given for R, g and en, and these are then computed by Maple for a specific example. Section 2.8 describes the outer product method of creating higher rank tensors from lower rank ones. It then introduces a "third approach" to the meaning of the symbol, by having this symbol provide a name for the outer product tensor. The first two approaches to were those of Chapter 1. Section 2.9 derives the "tilt reversal rule" and then discusses the notion of an inner product (contraction) of two tensors which produces a tensor of lower rank than the total rank of the contracted tensors. In a sense the inner product is the reverse of the outer product described in the previous section. The notion of a dot product is extended to tensor product spaces like VW and VV. Section 2.10 shows how to expand a rank-2 tensor on an arbitrary basis, and how to project out from such an expansion the coefficients. Use is made of the dot products introduced in the previous section. The results are generalized for a rank-k tensor and the multiindex notation appears for the first time. Section 2.11 introduces the notion of the dual space V* to V. The basis "vectors" (functionals) are called λi. After considering an expansion for the most general vector in V*, the discussion moves to rank-2 tensors in the space V*V* and finally to rank-k tensors in V*k = V*V*..... The section ends with a restatement of various equations in multiindex notation. [ the last two section summaries above updated on Nov 1 ] I am having trouble now with Section 2.11. I want λi to be both a linear functional AND to be the vector ei. Is there a reasonable way to do this? Lots of people write ei(ej) = δij λi(ej) = δij without needing any dot product. I would say ei(ej) = ei ej = δij λi(ej) = ei ej = δij using the dot product as an intermediary. I just don't think you can say λi = ei as a standalone statement if by ei you refer to a certain basis vector in x-space. The notation ei(v) then makes no sense. So that means I am going to have to unwind a lot of stuff!!! λi(v) = ei v clearly defines a linear function of v, no problem λi = ei = an operator ?? I just don't know how to do this. OK, I have now done a full rewrite of Section 2.11 removing my various false statements such as λi = ei and components thereof I just saved off the old section 2.11 and installed the new into tensor wedge v4. Nov 1, 2015 I did more updating of Section 2.11, then fixed up the end of Section 2.10 to use the same systematic notation and then the multiindex notation. What then comes next? Added the two summary sections above. Ready now for Chapter 3. Section 3.1 shows that the componentization implied by the Section 2.8 outer product with is consistent with the no-components theory of the tensor product presented in Chapter 1. More examples of outer products are given. The notion of a direct sum is defined, and then used to define the tensor algebra T which is a graded algebra. Two distinct notions of "mixed tensor" are delineated. Section 3.2 defines the Kronecker product ST of two linear operators (in covariant notation) and uses Maple to compute this product for some sample cases. This section is included only because the symbol is involved in this product. The Kronecker product is not used elsewhere in our document and so this is a good section for the uninterested reader to skip. It is definitely a bit tedious. Chapter 3 Section 3.1. OK Section 3.2 OK We are now on to Chapter 4. I know there have to be downstream changes because I have altered and added much to earlier sections. Section 4.1 is a brief review of earlier material (including Chapter 1) relating to the tensor product of two vectors, and more generally relating to the arbitrary rank-2 tensor. Chapter 4 Section 4.1 OK. Section 4.2. I expect trouble here since on the dual subject! There was trouble, but not of the expected type. The trouble was overlap with Section 2.11 and I think I have cleaned that up, so this section is now OK. It needs an overview. But I am done for this day at 8 PM, dental tomorrow. Nov 2, 2015 I just did a quick reading of 2.11 and 4.1 and 4.2, constantly making small changes. I now resume starting with section 4.3. Section 4.3 (a) OK (b) OK (c) OK (d) OK Section 4.4 This does seem a tedious repetition of Section 4.3, but I guess it is necessary to keep the parallel presentation going. There are no gross errors in this combined wedge + dual section, so I will say it is OK. Chapter 5. 5.1 OK 5.2 OK 5.3 OK 5.4 OK 5.5 OK 5.6 OK, I added a new notation generalized result to this section. Chapter 6 6.1 OK 6.2 OK 6.3 OK 6.4 OK 6.5 OK 6.6 OK 6.7 OK 6.8 OK 6.9 OK 6.10 OK 6.11 OK but need to add equation numbers. Nov 3, 2015 Another full day push. I found a bug which required me to change from Tii....i to Tii....i as the coefficient of the symmetric sum. Otherwise you get horrible inconsistent nonsensical results. This was a good find, and it is good to have it all fixed. It of course had implications through the entire document, as do all my "finds". As part of this fix I was able to entirely dispense with Sym and all that T = A + S + X stuff. I am now working on the v5 version. I also added good statements about the components of wedged objects and a good theorem about same which was just missing in v4. I also cleaned up and reduced the sizes of those two "components" sections. I also stated the default notation so the (e) need not be written. A lot happened today! Nov 4, 2015 . Still wondering about the SAX decomposition. Suppose T = S + A + X T = Σii...i Tii...i (ei ^ ei ^ .... ^ ei) where S = Sym(T) A = Alt(T) = A = k! T We can see that the S piece gives nothing, the A piece gives something, but what about X? TX = Σii...i Xii...i (ei ^ ei ^ .... ^ ei) = Σii...i [Tii...i - Aii...i - Sii...i ](ei ^ ei ^ .... ^ ei) = Σii...i [Tii...i - Aii...i] (ei ^ ei ^ .... ^ ei) = T - Σii...i Aii...i (ei ^ ei ^ .... ^ ei) 7. Theorem: The tensor Yii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) is totally antisymmetric. Proof: We shall show that Yii...i = - Yii...i and then the same argument applies to any pair of indices. Start with, Yii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) where as usual P(ir) ≡ iP(r) so that P acts on the i subscript. The sum ΣP is over all permutations of the integers {1,2...k} and S(P) is the number of swaps involved in permutation P . Define the specific permutation P' by P'{1,2...k} = {2,1...k} which has S(P') = 1. Since in this case i2 = P'(ii), we can write i1 = P(i2) = PP'(i1). Similarly, i2 = P(i1) = PP'(i2). For any other subscript like i3 we have i3 = P(i3) = PP'(i3). The above sum can then be written Yii...i = ΣP (-1)S(P) RPP'(i)PP'(i)....PP'(i) Since S(PP') = S(P) + S(P') for any P and P', and since S(P') = 1, we know S(P) = S(PP') - 1, so then Yii...i = ΣP (-1)S(PP')-1 RPP'(i)PP'(i)....PP'(i) = (-1) ΣP (-1)S(PP') RPP'(i)PP'(i)....PP'(i) . Finally we can use the "rearrangement theorem" of group theory applied to the permutation group (see PL ***). This theorem says that ΣP f(PQ) = ΣP f(PQ) = ΣP f(P) where Q is any fixed permutation. The first two sums are just reorderings of the third sum and so equal the second sum. Selecting Q = P' we have ΣP f(PP') = ΣP f(P) so then Yii...i = (-1) ΣP (-1)S(P) RP(i)P(i)....P(i) = - Yii...i QED Corollary: The tensor Y12...k = ΣP (-1)S(P) RP(1)P(2)....P(k) is totally antisymmetric. Proof: Repeat the previous proof replacing ir → r. Then the above sum may be written this way since, for example, i = P'(i) and then P(2) = P(P'(1)) = PP'(1), Qii...i = ΣP (-1)S(P) RP(i)P(i)....P(i) ********** OK, I have now made these changes: Appendix A created which has various supporting proofs Added References to this appendix as needed, removing junk from main line Cleared up the S + A + E mystery for tensor decomposition. I have rewritten Section 6.5 and think it is now good. I am perhaps ready now to make another attempt at Section 7. OK at 7:30 PM I have a first cut at Section 7! This is a big milestone. Appendix A grew a bit, has more good theorems. Nov 5, 2015 I want to stabilize Chapter 7, and then add a new section about "functions" including the alt function for functions in wedge products. This might go at the end of Section 7.10, not sure yet what it looks like, so do it here. For two tensor functionals I get this Disaster at 4:30 PM. I am trying to get the function alt thing figured out. I updated Appendix A with more stuff, new Theorems A.6 and A.7, but now I have run aground on the old εjj...j has no meaning if the indices run 1 to n. I think my (A.7.1) theorem is correct, but my proof is wrong. I know the case k = 2 works explicitly. I have a lot more work to do now, perhaps other parts of Appendix A are now going to wobble. I am used to this situation of non-stability! I have resolved the disaster and obtained the main desired result. Tomorrow try to firm this up with better threads and better notation. Nov 6, 2105 I started into yet another "new Appendix A" which more concentrates on particular "forms". I then rewrote Sections 6.1 and 6.2, removing technical details which are now in Appendix A. It is all much better now. Equation (6.1.2) maintains its same number. I think updated relevant equation number references in later sections. I now continue with new appendix A. I renamed it just now to be v3, so I will use it from now on in my ongoing edits. I am now grinding through Section 7 again, in light of all I have done recently. I did it all and I am up to the ADDER section where I dealt with the functions in a first cut only. But I did not do 7.11 yet. Here is the order for tomorrow. 1. clean up 6.11 which is not very good right now (it is a complete mess really) 2. create a dual section 7.11 to match 6.11. 3. add new section 7.12 to do all the function ADDER stuff. Now 8 PM, done for this day. Nov 7, 2105 Reading through Section 6 again, lots of good changes due to new App A and other things. Renumber Section 6.1 globally: each number is bumped up 1. Done, including Section 7 and App A v3. Am now at the start of 6.5. Made sure the adjective "totally" is always used with sym or antisym, done. Resume post lunch with 6.6. Resuming... Section 6.7 is just a relax interlude for the reader, it had to go somewhere. Section 6.8 I have now enhance with side-by-side multiindex notations! Section 6.9 done and I made a few additions there. Now finally we come to Section 6.10 and it is already 3 PM!! OK through start of Comm section. T^S = Σii....i(T^S)ii..i(ei^ ei ......^ ei) I think this is wrong when you take components of both sides, and this is the right way (T^S)jj....j = Σii....i(T^S)ii..i(ei^ ei ......^ ei)jj....j = Σii....i(T^S)ii..i(1/(k+k')!) ΣP (-1)S(P) δjP(i) δjP(i)...δjP(i) = (1/(k+k')!) ΣP (-1)S(P) Σii....i(T^S)ii..i δjP(i) δjP(i)...δjP(i) = (1/(k+k')!) ΣP (-1)S(P)(T^S)P(j)P(j)..P(j) = alt(T^S)jj....j so the coefficients must be written (T^S)ii..i OK, I have done the rehash maintaining the words so the reader can keep track of what is going on. That was the problem with my original presentation. Status 8:30 Sat 11/7: I have immensely improved the tensor product sections of Chapter 6. * I need to retitle sections 6 and 6 or break out more chapters. It is more than "k vectors" unsure how to do this, don't want too many tiny chapters. * Go back and clean up Chapter 5 in light of today's changes DONE * move on to Chapter 7. * Show that ^ is associative. I have never done this!! An Appendix A item. [ Well, it is DONE, but it ended up as an axiom ] * Eventually search web to see if many of my results are not already there. Nov 8, 2015 Reread Section 6 before doing Section 5. 6.1 ok 6.2.ok 6.3 ok, in fact good after yesterdays improvements. 6.4 ok and good 6.5 ok, and I like all my simple examples strewn around 6.6 ok 6.7 ok, checked all and replaced plot with Maple one showing code 6.8 ok, very good examples of multiindex notation 6.9 ok and good 6.10 ok, and an excellent intro to multiindex notation, could not do this better! 6.11 ok, including derivation of the sign rule. I'll bet that would be hard to find! Done with Chapter 6. I just added an "associativity of " section. to Chapter 2. I then quote that result in the later Section 5.6 where I am multiplying two tensors! I am now bogged down on how to justify the ^ rules. I claimed the follow from the rules, but I think that is wrong! This needs lots of work. BUT, since I just reviewed Chapter 6, I want to do Chapter 5 right how while Chapter 6 is "fresh in my mind" which it does not stay for long! Then I can return to the rules problem. Reread Chapter 5 5.1 ok 5.2 ok 5.3 ok 5.4 ok with one red TBD item 5.5 ok 5.6 ok, had to clean up a few things here. All done! Now we can get on to the messy subject of the ^ Rules! Are they derivable from the rules, or are they new axioms? I have basically rephrased things saying both and ^ associativities are axioms, as I do recall reading somewhere for the case of ^. I added this into the Exterior Algebra section, right before it is needed for the first time. Next bug: We know this is true since L = Σk=0∞ Lk and we showed in (6.3.2) that each Lk is a vector space. // Oops, not a bug. Question 1: Why is Vk a vector space? Ooo, remember that vector spaces does not involve multiplication, it is algebra that involves that. Vk is not closed under but it is still a vector space. I forgot that detail, back on track now. Question 2: Why are the "rules" needed to make Vk be a vector space? And later Lk? OK. Working on getting and ^ tensor algebra sections more parallel, always refer to L as L(V) and T as T(V). Fixed confusion between vector space and algebra in both sections. I will reproof these two sections tomorrow. Nov 9, 2015 Read through Section 3.1, it dupes all the tensor algebra graded algebra stuff of Section 5.6, The purpose of repeating the outer product idea here is to show that it is compatible with Chapter 1 where there were no components. The reason for the tensor algebra part is less clear. Suppose I just remove it? Then I will have to move the direct sum section to Section 5.6. Let's see what equations in that tensor algebra section are referenced/ (3.1.15) on direct sums: one rules ref above 4.1.6, then similar above 6.9.7 moved (3.1.16) quoted near 5.4.1, no other moved (3.1.17) none (3.1.18) none (3.1.19) none Not much affected if I move this stuff and remove duplication. This problem arose just due to the history. So I now do a series of edits: 1. Remove that tensor algebra section from 3.1 and save it in a file 2. Move the direct sum section to the start of Section 5.4. 3. Globally adjust all (5.4.* equation numbers: all get bumped up one. DONE 4. Adjust references to (3.1.15) to be (5.4.1) 5. Adjust section headings for Chapter 3 and Section 3.1 All done. This was a good cleanup, remove stupid duplication. Kronecker Products. Why do I have that section in this document? Answer: paper is on tensor product, and this is the tensor product of two operators. I advise readers to skip this section now. Look at Section 2.11. I read through it. Unbolded all λi references since this is NOT a vector, whereas ei is a vector. DONE. I will remove my small product dot now in Section 6.8, done, now consistent. So I have reviewed Chapter 3 a bit just now. Let's do some Chapter 4 right now. 4.1 ok and I like it. 4.2 ok, links heavily back to Section 2.11 4.3 (a) OK (b) OK (c) OK geometry connection (d) OK and now I like the idea that T = A/2 since this appears later with k! (e) OK, dot products 4.4 (a),(b),(c),(d), OK, tedious regurgitation of the previous section, but I want to have it. That concludes reading 2.11, 3 and 4, always making small edits. To be consistent I really need to have these chapters., 5. The Tensor Product of k vectors : the vector spaces Vk and T(V) 6. The Tensor Product of k dual vectors : the vector spaces V*k and T(V*) 7. The Wedge Product of k vectors : the vector spaces V*k and L*(V) 8. The Wedge Product of k vectors in the dual space : the vector spaces Λk and Λ(V) I just spaced this somehow. Chapter 5 already exists. Today I created Chapter 6 by a very massive copy, paste and edit on Chapter 5. I first tried to do this with a translation table, but I just could not get happy with that approach. Equations involving components don't translate, etc etc. I now have to renumber the existing Chapter 6 to be Chapter 7. I am being consistent by doing the dual right after the non-dual section. OK, chapter is installed and things are renumbered. Done for today, * Add new section to Chapter 6 which shows tensor multiplication in the language of functions rather than functionals. Eventually I will have a similar section in Chapter 8. Now 8 PM. Tues Nov 10, 2015 Will add functions section to new Ch 6 first thing. But first proof the just-written Chapter 6. 6.1 ok and good 6.2 ok and excellent 6.3 ok and just ok STOP. I now realize that the direct sum concept has to be restated somehow in the dual world! Discussion: In my original version, I defined direct sum in terms of my vector visualization picture, and I had trouble extending that silly picture to functionals. The solution is Appendix B now fully written and it depends much less on the tall vector idea and completely solves my problem. Now I just refer to this Appendix B from Section 5 and Section 6. Along the way I realized I need to put signs in the fruit salad typical elements. All this is edited in now. It is 4:10 PM. I have to now adjust equation numbers in Chapters 5 and 6 because there is no longer a number (5.4.1) and no longer a (6.4.1). First: It still seems to me that you ought to be able to write t = Σij Tij eiej Σijk Tijk eiejek ϵ T(V) ϵ V2V3 and t' = Σijk Tijk eiejek Σij Tij eiej ϵ T(V) ϵ V3V2 Well, I added an isomorphism comment on this to Appendix B. Repair the Section 5.4 equation numbers globally. My change is that all numbers downtick by 1. So I can just make that global change (manually of course). Done. Repair the Section 6.4 equation numbers globally. My change is that all numbers downtick by 1. So I can just make that global change (manually of course). Done. Now I was doing a read through of Section 6 before I was so rudely interrupted. Start again. 6.1 ok and good 6.2 ok and excellent 6.3 ok and just ok 6.4 ok 6.5 ok 6.6 ok Now I want to add a "function" section to Chapter 6. I will write it up in a scratch doc. DONE and it is installed as section 6.7, it went very fine. We now have this for Chapter 6 I am trying harder to stick with Spivak's notation. The next step is to review Chapter 7 on the wedge situation and add function section(s) to it similar to section 6.7 above. Once I get that done, then we need a bunbuster Chapter 8 which is a dual version of Chapter 7. I need to go off and look at what other authors are calling things. Spivak says k-tensor, but I think others might just use rank-k tensor. Other notations for spaces? Due diligence! And then I have to go back and get the Tk(V) notation into earlier sections if that is what I decide to go with. I do need a notation guide at the beginning which lists off all the many spaces used in my doc! Weds Nov 11, 2015. Will take an editing break, now that I see the lay of the land, and will review some PDF's to see what names people use for things. Benn Tucker. They have a chapter on tensor algebra, but they use the strange symmetric approach that I don't like. They do use V and V*. For them, x = my v, and X = my α. My α(v) is their X(x). But then they get right into x(X) which would for me be v(α). It is not very nice. Field is F, not K. Unclear what a tensor of degree k really is. Multilinearity mentioned. Tr(V) ≡ Vr is a space. Basis is ei. But then at top of page 3 they get right to the function idea. Degree is "formerly called rank". So B&T get right into the "function business" with page 3 top, like Spivak. T(V) = sum Tρ(V). They are very clear that Tr(V) = Vr so I think we are close on that. Z-graded algebra. Word automorphism is used, unclearly to me. η operator mults a tensor by (-1)k. Antisymmetric tensors in Tp(V) are called Λp(V), but I think that is my Lp. These antisymmetric tensors are called exterior p-forms. The (n,p) count is given. Their ALT operator definition is the same as my Alt shown in (1.2.1). But it acts on a function in (1.2.1). This is the section I have not yet written but I expect it to come out right. They have not yet used multiindex. I guess I better write my function section and then come back to this review of PDF's. Overview should stress my frequency use of examples! Chapter 7 review. 7.1 ok, I like it, but needs a few red support items 7.2 ok and an excellent summary of the properties of a wedge of k vectors. 7.3 ok, leaving no stone unturned, I like all of this 7.4 ok, connection between A and T, I like it still 7.5 ok, gets into decomposition of a general tensor which I like 7.6 ok, three forms of the wedge of k vectors 7.7 ok, ratio of sizes of the two spaces, very short, one graph 7.8 ok, this is a great section with multiindex equations examples 7.9 ok, the blade table, this is a real masterpiece IMHO 7.10 (a) OK, gets rule for T^S with tensors k and k' (b) OK, excellent in fact, the graded comm sign rule (c) OK, the multiindex rehash of the above two sections. 7.11 ok, including the swap rule (original research?) The time has now come to write the very important last section 7.12 on functions. I will mimic section 6.7 where I do the same thing for the product, whereas in 7.12 we have the ^ product. I will do this in a separate doc for now. STOP, wrong!! This function section will go in Chapter 8, not chapter 7, because you have to be in the dual world to even talk about functions! So I now have to "face up" to Chapter 8. I am going to have to do the parallel approach again. It is 25 pages long, what else can I do? At least for myself I really should do it that way. Then when I am done, I might somehow abbreviate Chapter 8. This is going to be slow going, so I will mark progress here. comment ok 8.1 ok 8.2 ok 8.3 ok 8.4 ok first cut 8.5 ok 8.6 ok 8.7 ok 8.8 ok multiindex worth repeating all stop. problem with numbering in section 7.9, fix globally right now: continuing 8.9 ok, this was pretty major. 8.10 ok, that was major too break Sprouts and walk and fee made 8.11 ok, this took a ton of editing. Now I have to create a new functions section. I know this will require creating new equations in earlier sections of Chapter 8 (hopefully always at subsection end), because this is what happened going from Chapter 5 to Chapter 6. OK, I am writing 8.12, got confused, thing I have done enough tedious hours for today at 6 PM. Nov 12 2015 Something is fishy with my wedge product in terms of associativity. Consider for example: (ej ^ ej ^ .... ^ ej)ii...i = (1/k!) ΣP (-1)S(P) δP(j)i δP(j)i... δP(j)i (ej ^ ej)ii = (1/2!) ΣP (-1)S(P) δP(j)i δP(j)i (ej ^ ej)ii = (1/2!) ΣP (-1)S(P) δP(j)i δP(j)i (ej ^ ej^ ej ^ ej)iiii = (1/2!) ΣP (-1)S(P) δP(j)i δP(j)i δP(j)i δP(j)i Is there some way to TEST this claim? (ej ^ ej^ ej ^ ej) = (ej ^ ej) ^ (ej ^ ej) The problem is this: I know nothing a priori about (ej ^ ej) ^ (ej ^ ej) . I have no expansion I can write for this that does not already assume assoc is true. I have to assume the assoc property in order to get any tensor product formulas. So maybe my "definition" claim is correct. My Chapter 8 2-tensor product rule is this: T^S = Σii....i (T^S )ii....i(λi^ λi ......^ λi) (T^S )ii....i = Tii....i Sii...i This is the product of tensors in Λk and Λk' : T = Σii....i Tii....i λi^ λi .....^ λi. rank k, T ϵ Λk S = Σjj....j Sjj....j λj^ λj .....^ λj . rank k', S ϵ Λk' . (8.10.3) There is no ΣP(-1)P in these two expansions. Question: In my Section 8.12, I first show this, T^S = Σii...ijj...jTii....i Sjj....j (λi ^ λi ...^ λi ^ λj ^ λj ...^ λj) . (8.10.4) which seems a no-brainer. I then evaluate at (v1,v2....vk, vk+1....vk+k'). But first go back to T = Σii....i Tii....i λi^ λi .....^ λi. rank k, T ϵ Λk Evaluate this at (v1,v2....vk) to get T(v1,v2....vk) = Σii....i Tii....i (λi^ λi .....^ λi)(v1,v2....vk) = Σii....i Tii....i (1/k!) ΣP (-1)S(P)(vP(1))j(vP(2))j ... (vP(k))j = (1/k!) ΣP (-1)S(P) Σii....i Tii....i(vP(1))j(vP(2))j ... (vP(k))j This IS a totally antisymmetric function. I think you can write this as T(v1,v2....vk) = (1/k!) ΣP (-1)S(P)T(vP(1),vP(2)....vP(k)) which shows that the function T in Λk is an action on T in V*k. . Suppose we define T = [Alt(T)] so that T(v1,v2....vk) = [Alt(T)](v1,v2....vk) ≡ (1/k!) ΣP (-1)S(P)T(vP(1),vP(2)....vP(k)) where for the first time EVER I have my "function Alt" concept. The definition is [Alt(X)](v1,v2....vk) ≡ (1/k!) ΣP (-1)S(P)X(vP(1),vP(2)....vP(k)) which compare to [Alt(X)]ii...i ≡ (1/k!) ΣP (-1)S(P) XP(i)P(i)...P(i) (7.5.3) Note that a "function Alt" only has meaning in a dual + wedge world, and that is Chapter 8. Maybe I need to add something new to Section 8.5 on the Alt stuff. I have done this, at least a first draft. I then got sidetracked on a Theorem which took me about 3 hours to prove and I think this will end up in Appendix C. That theorem is the last line here: (T^S)(v1,v2....vk, vk+1....vk+k') = (1/k!) ΣP (-1)S(P) T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) = (1/k!) ΣP (-1)S(P) T^(vP(1), vP(2) ....vP(k))S^(vP(k+1), vP(k+2) ....vP(k+k')) first hat! I now want to state this using an Alt operator. Well, I have launched into an Appendix C and I am confused by the very basic claim (C.1.2) about the ordering of P and Q. ________________________________ C.1 Notation to describe the permutation of function arguments. Let f(v1,v2,...vk) be a function of k arguments and let Q be a permutation of {1,2..k}. Then define the permuted function Qf as, Q[f(v1,v2,...vk)] = f(vQ(1),vQ(2),...vQ(k)) . (C.1.1) Example: If Q{1,2,3} = {2,1,3} then we think of Q(1) = 2, Q(2) = 1 and Q(3) = 3 so that Q[f(v1,v2,v3)] = f(vQ(1),vQ(2),vQ(3)) = f(v2,v1,v3) . Suppose in (C.1.1) Q is the product of two permutations Q = PR. Then the above says (PR)[f(v1,v2,...vk)] = f(vPR(1),vPR(2),...vPR(k)) On the other hand (PR)[f(v1,v2,...vk)] = P[ R[f(v1,v2,...vk)] ] = P[ f(vR(1),vR(2),...vR(k)) ] Therefore P[ f(vR(1),vR(2),...vR(k)) ] = f(vPR(1),vPR(2),...vPR(k)) (C.1.2) Example: Suppose P{1,2,3} = {2,1,3} and R{1,2,3} = {3,2,1}. Then P(1) = 2 P(2) = 1 P(3) = 3 R(1) = 3 R(2) = 2 R(3) = 1 PR(1) = P(3) = 3 PR(2) = P(2) = 1 PR(3) = P(1) = 2 so then (C.1.2) says P f(v3,v2,v1) = f(v3,v1, v2) which is WRONG!!! I really want this to swap the first two arguments of the function f. Suppose P is some other permutation of {1,2..k}. Then P{Q[f(v1,v2,...vk)]} = P[f(vQ(1),vQ(2),...vQ(k))] = f(vP(Q(1)),vP(Q(2)),...vP(Q(3))) Example: If P{1,2,3} = {2,1,3} then P{Q[f(v1,v2,v3)]} = P[f(vQ(1),vQ(2),vQ(3))] = f(vQ(P(1)),vQ(P(2)),...vQ(P(3))) = f(vQ(2),vQ(1),...vQ(3))) Notice therefore that P[f(vQ(1),vQ(2),vQ(3))] = f(vQ(2),vQ(1),...vQ(3))) merely swaps the first two arguments of the function f. In abbreviated notation, PQ[f(v1,v2,...vk)]} = P[f(vQ(1),vQ(2),...vQ(k))] = f(vQP(1)),vQP(2)),...vQP(3))) (C.1.2) Ouch ouch ouch, I am flailing . Consider f (vP(1),vP(2),...vP(k)) = ΣQ(-1)S(Q) F( vQ(P(1)),vQ(P(2)),...vQ(P(k))) . (C.1.3) = F( vQ(1),vQ(2),...vQ(k)) – F( vQ(2),vQ(1),...vQ(k)) + etc I think this is what I WANT when I antisymmetrize something. What would this be F( vQP(1),vQP(2)),...vQP(k)) f (vP(1),vP(2),...vP(k)) = ΣQ(-1)S(Q) F( vQP(1),vQP(2)),...vQP(k)) Here maybe you argue that P(1) = (2) and then it does the right thing. My notation has many weaknesses and here they are coming to light! My most basic idea is this: v1^ v2^ .....^ vk = (1/k!) ΣP (-1)S(P) ( vP(1) vP(2) ..... vP(k)) (*) Recall that 1,2,..k are labels on the left. But suppose va^ vb^ .....^ vq = Q (v1^ v2^ .....^ vk ) = vQ(1) ^ vQ(2)..... ^ vQ(k) Would you then say, va^ vb^ .....^ vq = (1/k!) ΣP (-1)S(P) ( vP(a) vP(b) ..... vP(q)) ?? This does not seem clear whereas (*) is totally clear. I think this is also pretty clear vj^ vj^ .....^ vj = (1/k!) ΣP (-1)S(P) ( vP(j) vP(j) ..... vP(j)) But then if I select j1 = a, j2 = b... I get the above. OK, if ?? is OK, then we have vQ(1) ^ vQ(2)..... ^ vQ(k) = (1/k!) ΣP (-1)S(P) ( vP(Q(1)) vP(Q(2)) ..... vP(Q(k))) Now, what is the meaning of: vP(Q(1)) if P = P12. Meaning #1: vP(Q(1)) = vQ(2) probably what I want it to mean Meaning #2: vP(Q(1)) = depends on what integer Q(1) equals. Meaning #3: vP(Q(1)) = v(PQ)(1) = vPQ(1) = same as meaning #2 In order to get Meaning #1 clearly, you would want to write vQ(1) ^ vQ(2)..... ^ vQ(k) = (1/k!) ΣP (-1)S(P) ( vQ(P(1)) vQ(P(2)) ..... vQ(P(k))) where now the P's are on the right. Now the sum seems completely clear. Fact: I want the summation index to be on the right. Tiny Theorem. If f(v1,v2,...vk) = ΣQ(-1)S(Q) F(vQ(1),vQ(2),...vQ(k)) Q = permutations of {1,2..k} then if P is some particular permutation, f (vP(1),vP(2),...vP(k)) = ΣQ(-1)S(Q) F( vP(Q(1)),vP(Q(2)),...vP(Q(k))) . (C.1.3) OK as of 5 PM I have Appendix C going again, and the first significant theorem is therein proved. ΣP (-1)S(P) T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) = ΣP (-1)S(P) T^(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) (C.2.1) OK now 8:20PM, have made good progress with Appendix C and then more editing in Chapter 8 in two places. The next act is to get the Alt stuff going and hopefully to verify Spivak's little theorem and to state probably many other theorems. It's been a long time coming! Nov 13 2015 Continuing on Section 8.12 on functions. I have [Alt(X)](v1,v2....vk) ≡ (1/k!) ΣP (-1)S(P)X(vP(1),vP(2)....vP(k)) (8.5.11) Alt(Alt(Y) = Alt(Y) T^ = Alt(T) = Alt(T^) S^ = Alt(S) = Alt(S^) (T^^S^)(v1,v2....vk, vk+1....vk+k') = (1/k!) ΣP (-1)S(P) T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) . (8.10.5) (TS)(v1,v2....vk, vk+1....vk+k') = T(v1,v2....vk) S(vk+1,vk+2....vk+k') (6.7.1) (T^^S^)(v1,v2....vk, vk+1....vk+k') = (1/k!) ΣP (-1)S(P)(TS)(v1,v2....vk, vk+1....vk+k') (T^^S^) = (1/k!) ΣP (-1)S(P)(TS) (T^^S^) = Alt(TS) Since T^ and S^ are both in Lk as well as Λk, we can write (T^S^)(v1,v2....vk, vk+1....vk+k') = T^(v1,v2....vk) S^(vk+1,vk+2....vk+k') (6.7.1) and then (T^^S^)(v1,v2....vk, vk+1....vk+k') // T^ ϵ Λk, S^ ϵ Λk' (T^^S^) ϵ Λk+k' = (1/k!) ΣP (-1)S(P) T^(vP(1), vP(2) ....vP(k))S^(vP(k+1), vP(k+2) ....vP(k+k')) (8.10.6) reads (T^^S^)(v1,v2....vk, vk+1....vk+k') = (1/k!) ΣP (-1)S(P)(T^S^)(v1,v2....vk, vk+1....vk+k') (T^^S^) = (1/k!) ΣP (-1)S(P)(T^S^) (T^^S^) = Alt(T^S^) Now what combinations can I come up with: Alt(T^^S^) = Alt[Alt(T)Alt(S)] = Alt[ TAlt(S)] = Alt[ TS^] = Alt[ Alt(T)S ] = Alt[ T^S] = Alt(TS) Suppose T^ = A^^B^ and S^ = C^. Then since ^ is associative, we have Alt[( A^^B^)^C^] = Alt[ A^^B^^C^] = Alt[( A^^(B^^C^)] Can you replace any or all of these Question: can you way T^ = A^^B^ T = A^B ? There are still lots of missing pieces. Well here is a proof: A^B = (TS)(v1,v2....vk, vk+1....vk+k') = T(v1,v2....vk) S(vk+1,vk+2....vk+k') (6.7.1) = But how do I get to triple products? I conjecture that Alt(T^^S^^R^) = Alt[Alt(T)Alt(S)Alt(R)] = Alt[TAlt(S)Alt(R)] = Alt[Alt(T)SAlt(R)] = Alt[Alt(T)Alt(S)R] = Alt[TSAlt(R)] = Alt[TAlt(S)R] = Alt[Alt(T)SR] = Alt[TSR] Here is the one from Spivak, Can I make Appendix C proofs more concise so I can do more of them. C.2 Theorem ΣP (-1)S(P) T(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) = ΣP (-1)S(P) T^(vP(1), vP(2) ....vP(k))S(vP(k+1), vP(k+2) ....vP(k+k')) (C.2.1) or Alt(T[k]S[k']) = Alt(T^[k]S[k']) RHS = Alt(T[k]S[k']) = Alt(T^[1..k]S[k+1...k+k']) = Alt Rewrite with k' → k" as Alt(T[1..k]S[k+1...k+k"]) = Alt(T^[1..k]S[k+1...k+k"]) Now replace S[k+1...j+k"] → S[k+1...j+k'] R[k+k'1...k+k'+k"] to get Alt(T[1..k]S[k+1...k+k'] R[k+k'1...k+k'+k"]) = Alt(T^[1..k]S[k+1...k+k'] R[k+k'1...k+k'+k"]) = Alt (T^{k} S{k'} R{k"}) Question: How about Alt(kT) = kAlt(T) k ϵ K S^T = (-1)kk'T^S Alt(S^T) = (-1)kk'Alt(T^S) I then went off and started rewriting Appendix C because I overlooked some trivial facts. I have finished section C.1 and now I have to see how my various Theorem proofs are affected by these changes in Section C.1. As of 7 PM, I am happy with Sections C.1 through C.4. Continued in Section 8.12, but I still have missing stuff which I can jam into Appendix C. OK, today went OK, now 8:20 PM. Sat Nov 14. Putting else on hold while I deal with a stone so far unturned. Back in Chapter 7 which is non-dual, what is the meaning of objects like (T^S^)^ ? I did a first cut of this as doc Section 7.12, BUT I already have to make changes in the meaning of my notation. It is inconsistent!! Also, I have stuff in App C that relates, but is in dual space notation. I need this all first in regular space notation. I should have done App C in regular space, not dual space, oy! Had no sleep last night, it is 7 PM, I am tired. Sun Nov 15. I am rewriting my draft Section 7.12. In doing so I keep adding things in earlier sections. One of those things is this installing the rules for special cases of ST and S^T when one or both is a scalar. It is done, and the new equations are (7.10.16) and (5.6.15). I may have to alter later dual sections due to these additions. Also added examples showing this is consistent with the graded commutivity rule. OK, I am done with my new v2 of Section 7.12, it is much better, time is 4:30 PM. Next step is to develop some Alt theorems separately from Appendix C. These theorems will have nothing to do with "functions", just tensors. Not sure yet where these should go. Got this started, and did show that Alt(Alt(T)S) = Alt(TS) as my new Theorem One. The proof is very similar to my App C for dual things. There must be some simpler way to do all this stuff. At least if we set T = AB this says Alt(Alt(AB)S) = Alt((AB)S) = Alt(ABS) which I think is the infamous Spivak theorem. Why is HIS proof so short? Nov 16, 2015 I spent the AM session trying to abstract away the essence of Theorem One of Appendix C. This attempt was fairly successful, but when I then started using Q and Q' to distinguish Q from its extension Q', things did not work out right. So this is where I stand at noon. I think there is something fishy in the notation I am inventing and using at the same time. // Somehow by 8 PM I got this cleaned up (I think) and have finished App C v4 sections on the three Theorems. Nov 17, 2015 Will now add application sections to App C v4. Started around 5 AM, it is now 11:30 AM and Appendix C (version v4) is looking very very good! Beyond my expectations in terms of the basis change idea. The appendix is relatively self-contained and by itself makes a great addition to this wedge document. I have perhaps changed notation a bit, so I now have to go back and review Sections 7.12 and Chapter 8 which are not yet installed and are sort of in development. Review of Section 7.12. Nov 18, 2015 Should I start using T^ notation earlier in my doc, as in "everywhere there is a wedge product? For vectors there is no difference, but for any expansion there is. By having no T^ I was forced to use coefficients T. I tried this on Sections 4.3 and 4.4 in scratch doc, it alleviates the need for italic use of T for example. And T^ is a very concise notation! I like it. Let's try some other scratch changes. Chapter 7 7.1 no changes 7.2 no changes 7.3 no changes 7.4 major changes STOP. AT this point I decided on going to App A v 4 where I have now gathered up a new selection of pieces into I hope something that is coherent. It has the rearrangement theorem from scratch, and then I show all the "permutation theorems" in generic notation. The in Section A.5 I "apply" these results to tensors Tii...i . This means I have to do all those theorems only once, not twice. I remove material from Section 7, so I really have to go now to v6 of the document just in case. So what is the current "appendix status" ? App A v4 was just violently altered today. App B is on direct sums is unchanged by what has gone on, though I will have to check notation. App C v4 is the most recent version there, but it will have to have its opening section removed since that info is now in App A. I will hold off a little on App C mods because I want to return to a rewrite of Chapter 7, and for that we go to v6 of the main doc!! Status 8:40 PM. I have started my rewrite of Section 7 in place inside tensor and wedge v6.doc. I now have available all the horsepower of the new Appendix A v4 to use as needed. Thurs 11.19.15 Will now edit Chapter 7 in v6 doc: I will put working issues above the check list Item 7 below. I want to show that P can move up or down and this requires an old appendix A result. How can I get this into the new appendix A? 7.1 OK now 7.2 1. OK 2. OK 3. OK after I added missing item to App A v4. Removed template need. 4. OK 5. OK 6. OK 7. OK after a LOT of work and Appendix A v4 support work added 8. OK it is now 12:30 just getting this far!!! 7.3 OK, did some cleanup and improvement. 7.4 Here I have to pause. I did some practice already using T^ notation, but I think there is more involved. For example, I can now say T^ = Σii...i Tii...i (ei ^ ei ^ .... ^ ei) = Σii...i Tii...i Alt(ei ei .... ei) = Alt Σii...i Tii...i(ei ei .... ei) ] = Alt(T) Then what about this: T^ = Σ1≤i<i<....<i≤n Aii...i (ei ^ ei ^ .... ^ ei) . (7.4.1) = Σ1≤i<i<....<i≤n Aii...i Alt(ei ei .... ei) That one is a little tricky! But it must be allowed, just a direct sub. Then go on to get = Alt[Σ1≤i<i<....<i≤n Aii...i (ei ei .... ei) ] = don't know what to do with this. Not too useful I think I still need my fancy argument connecting T and A OK, I have done lots of scratch work on Section 7.4 in my scratch doc. Section 7.3 and earlier were edit in the full v6 document, be careful. Idea: Consider vj^ vj^ .....^ vj = Alt(vj vj ..... vj) Can I derive the wedge "rules" from this??? I say that you cannot below (7.2.3). But let's try again v1^ v2^ .....^ vk = Alt(v1 v2 ..... vk) v1^ (v2+ v'2) ^ .....^ vk = Alt(v1 (v2+ v'2) ..... vk) = (1/k!) ΣP (-1)S(P) ( vP(1) [vP(2)+ v'P(2)] ..... vP(k)) = (1/k!) ΣP (-1)S(P) { ( vP(1) vP(2) ..... vP(k)) + ( vP(1) v'P(2) ..... vP(k)) } // (5.3.1) = Alt(v1 v2 ..... vk) + Alt(v1 v'2 ..... vk) = vj^ vj^ .....^ vj + vj^ v'j^ .....^ vj Now you cannot derive it. They are by fiat, and then I just show that my "candidate form" satisfies them. I have now finished editing 7.4 and I will install it into the main doc. I have to keep cleaning up my old permutation notation to now show P acts directly on subscripts, here are some aids for doing that = (1/k!) ΣP (-1)S(P) (vj)i(vj)i ... (vj)i // outer product form = (1/k!) ΣP (-1)S(P) (vj)i(vj)i ... (vj)i // (A.1.19) Mab = (vj)i Moving onto section 7.5. The new Section 7.5 is now done inside v6. This is taking a LONG time! Section 7.6 is now trivially done. Section 7.7 is also done, with some labor to get eq num refs good. Pausing for the 4PM late walk Done and then did Section 7.8, it is OK. So here is where we stand on this edit pass: 7.1 OK now 7.2 1. OK 2. OK 3. OK after I added missing item to App A v4. Removed template need. 4. OK 5. OK 6. OK 7. OK after a LOT of work and Appendix A v4 support work added 8. OK it is now 12:30 just getting this far!!! 7.3 OK, did some cleanup and improvement. 7.4 OK 7.5 OK 7.6 OK 7.7 OK 7.8 OK Now comes the Main Act which is the next Section. I will do this scratch temp5 in a separate file. Here is my edit log before I change Ch 10 to Ch 9 7.10 (a) OK (b) OK Status as of 8 PM. I have the new Section 7.9 sitting in temp5 being edited. I found that T^^ S^ = Alt(TS) and this is going to cause some rethinking of how to present this stuff. The status is that I have shifted all equation numbers from 7.10 to 7.9, I have added all the hats ^ and I have undone the italics on T and S. But somehow I don't feel it is the way it should be! T^ = Alt(T) S^ = Alt(S) T^^ S^ = Alt(TS) Is this somehow obvious? Alt(T) ^ Alt(S) = Alt(TS) ? [Alt(T)]I = (1/k!) ΣP(-1)S(P) TP(I) I think you have to show it! I am now much closer to this theorem of Spivak. But where are his factors coming from? He never says, though he promises to say. I just defines them into his wedge product definition! But Benn Tucker are going to agree with me! Fri 11.20.15 I just rewrote Section 5.6 on tensor product of N tensors and it is MUCH better. I did this in temp6 and will right now install it into v6. After installing I edited it some more! Now back to Section 7.9. I am removing to scraps the method of using the ordered sum to do a tensor product. In retrospect, there seems to be no payoff in this information, but I can reinstate it later if need be. AT 10:15 AM I am finally happy with Section 7.9 developed in temp5. I had to go to (7.9.a.1) type equations numbers, there was not way out. So save the old in scraps and install the new. I reduced the section count again since no separate section for 2 and N. So Chapter 7 is now much better and shorter I think. But still it is 109-84 = 25 pages long. I just HAVE to come up with a better way to do the dual versions of sections! Chapter 5 is 75-66 = 9 pages long. Maybe I provide translation rules, and THEN a new section for functions. Note that Chapters 6 and 8 are now completely out of date and cannot be used. Meanwhile, what happened to my little Spivak theorems? Where is my stuff about "combinations of and ^ ?? I have lost track of this, do a search. Having trouble finding it. Need clean search handle. // Found it, file is "Section 7.12 rough draft v2", you have to put phrases in quotes for Ransack! Lets start be rereading this little piece. Just made a v3 of it since lots of editing needed. OK, I did some editing, and I get the idea. I am studying combination wedge/tensor products, a good topic. Now what about those Spivak theorems? I state one of them. Where did I do a proof? The proof is in Tensor Alt Theorems, and I can see a revamp is needed. Maybe this needs to be proved at the generic level!! But I now see that this is all addressed in App C v4 where I show that Alt[(TS)] = Alt[(T^S)] Now that I have the Alt connection, this says Alt[(TS)] = Alt[(Alt(T)S)] and this is the Spivak theorem, boom, all done! But wait, look again at which theorem we are talking about, My proof of this is now Alt(ωη) = (ω ^ η ) = (ωη)^ So want to show Alt((ωη)^ θ) got i!~ I know from App C v4 that Alt[(TS)] = Alt[(T^S)] = Alt[(TS^)] = Alt[(T^S^)] = T^ ^ S^ Set T = (AB) so we then have Alt[( (AB)S)] = Alt[( (AB)^S)] = Alt[( (AB)S^)] = Alt[( (AB)^S^)] = (AB)^ ^ S^ Thus for example Alt[(ABS) = Alt[( Alt(AB)S)] and that is it! Maybe I need to do App C all in the generic world??? Maybe pick up at this point. Here are the three theorems collected from above : 1. ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f^[P(z)] F[P(Z)] (C.2.10) 2. ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f[P(z)] F^[P(Z)] (C.3.10) 3. ΣP (-1)σS(P) f[P(z)] F[P(Z)] = ΣP (-1)σS(P) f^[P(z)] F^[P(Z)] (C.4.1) Maybe write f[P(z)] F[P(Z)] = (fF)[P(1,2...k+k')] Then my theorems say 1. ΣP (-1)σS(P) (fF)[P(1,2...k+k')] = ΣP (-1)σS(P) (f^F)[P(1,2...k+k')] (C.2.10) 2. ΣP (-1)σS(P) (fF)[P(1,2...k+k')] = ΣP (-1)σS(P) (fF^)[P(1,2...k+k')] (C.3.10) 3. ΣP (-1)σS(P) (fF)[P(1,2...k+k')] = ΣP (-1)σS(P) (f^F^)[P(1,2...k+k')] (C.4.1) But recall now that [Alt(f)](1,2...k) ≡ (1/k!) ΣP (-1)S(P)f( P(1),P(2)...P(k) ) (A.2.1) so then [Alt(fF)](1,2...k+k') ≡ (1/[k+k']!) ΣP (-1)S(P)(fF)( P(1),P(2)...P(k+k'') ) (A.2.1) Throwing out the factorial common factor, the three theorems say (with σ = 1), 1. Alt(fF) = Alt(f^F) (C.2.10) 2. Alt(fF) = Alt(fF^) (C.3.10) 3. Alt(fF) = Alt(f^F^) (C.4.1) and these are now GENERIC results. So I need to revamp Section C.5. OK, time to revisit Appendix C, remove first section etc etc. This is a stack pushdown, good time to do it, shut all else down for a while. As of 5 PM I have finished on a long pass through Appendix C v5, it is now compatible with App A v4. Back at 6 PM, right arm hurts after walk. I think that my candidate Section 7.12, now 7.10, adds no value to my document so I am going to leave it out for now. So I am back to the problem of how to deal with the "dual" sections. Do I start again with a huge copy and paste to make some kind of Chapter 8? OK, to a fresh copy of 7 and edit all over again. Log the edits 7.1 done 7.2 done 7.3 done 7.4 as best I could 7.5 running aground Back to the idea of some translation rules. I need to get the rule for functions stated Tii...i = (v1)i (v2)i ... (vk)i = (v1 v2 ..... vk)ii...i T(vi,vi....vi) = (α1)(vi) (α2)(vi)... = (α1 α2 ..... αk)ii...i I started a new doc just to do Ch 8 translations. BUT I now see that a piece of Appendix A is missing! I need a section that says Application to Tensor Functions. I think that is where I go next. No, it should be Section A.8. Another copy paste job from (A.5) I guess. I started to create Section A.8 and left an "ok to here" mark. Just continue on this tomorrow. Have to decide on a name for the function-tensor space. When A.8 is done, install it, and go back tot he Chapter 8 translation stuff starting I guess Section 7.4. Sat Nov 21, 2015 I have finished A.8 and installed it. So back to Chapter 8 translations. // It is in progress. I have a funny feeling about my coefficients in the dual world. Consider, T = Σii....i Tii....i (λi λi ..... λi) T^ = Σii....i Tii....i (λi^ λi .....^ λi) T^ = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) . I show then that Aii...i = k! [Alt(T)]ii...i and then I say A = k! Alt(T) This is confusing because HERE in Alt(T) I refer to coefficients Tii....i, whereas earlier I refer to Alt(T) as a functional in V*k, so the notation is AMBIGUOUS! What if I do this instead T = Σii....i Tii....i (λi λi ..... λi) T^ = Σii....i Tii....i (λi^ λi .....^ λi) T^ = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) . so the coefficients are non-italic. Then I will get A = k! Alt(T) and that is identical to what happens in the non dual world. I like it! Let's do it. DONE, need to do this in earlier chapters too, put on the stack. After much viewing and editing. here is my Ch 8 Nov 20 status intro not there yet ****** 8.1 updated 8.2 updated 8.3 updated 8.4 updated 8.5 updated 8.6 updated 8.7 updated 8.8 updated Next is the most important section 8.9 which is also very long: products of tensors. I will pause right here on my edit of Ch 8 Nov 20 in order to back-update other areas. Pause to make a change in Chapter 6: In dual world, replace Tii....i with Tii....i Edit pass on Chapter 6 6.1,2,3,4,5 OK, changes noted are now made, but now need to do some work. I want Sections 6.6 and 6.7 to be replaced by a new Section 6.6 that looks like Section 7.9 Let's do a FULL EDIT on Chapter 6, because it has other important loose ends. Chapter 6 intro OK 6.1 OK greatly improved 6.2 OK finally I have clear definitions of "tensor","tensor function", and V*k vs V*kf 6.3 OK added a clarity last sentence. 6.4 OK stressing distinction and new notation V*kf . 6.5 is this a place to stick in tensor function transformations? STOP!!! What about this idea T'(vi, vj) = RiaRjb T(va, vb) That is a lot different from my claim that T(va, vb) is a scalar! What's the deal here!! Where do I make this scalar claim? Go back to section 2.11 where dual first appears. The key stuff is right below (2.11.3) where I say α(v) = α v = α' v' = α'(v') . (2.11.13) i MUST have this wrong, so now is a good time for cleanup/ I have a feeling that v does not transform, it is a dummy argument. Maybe C.5 can shed some light on this subject. * Tab = < ea,eb | T > T'ij = RiaRjbTab = RiaRjb< ea,eb | T > = < e'i,e'j | T > ?? * T(va,vb) = < va,vb | T > T'(vi,vj) = RiaRjbT(va,vb) = RiaRjb < va,vb | T > = < v'i,v'j | T > = T(v'i,v'j) * α(va) = <va|α> α'(va) = Ria α(vi) // not a vector field!!! If I want to get some kind of Ria form I have to NOT transform arguments to the new space. I think I have seen something like this in a reference somewhere. I don't see this offhand in Benn. I don't see it in Spivak. I have looked through all my local PDF's and I just cannot find the equation I remember. Might be best to add something to Section C.5 doing it right, rather than arm-waving in Chapter 6. Stack push: back to Appendix C.5 now. From ???? : This looks like my equation! T with no prime so to speak on both sides. Spivak's norm factor. I finally have a hint of why he does this from my new multilinear mappings paper. My result is this T^^ S^ = Alt(TS) (T^^ S^)(vi,vi....vi) = [Alt(TS)](vi,vi....vi) = ΣP(-1)S(P) (TS)(vi,vi....vi) // (A.8.3) = ΣP(-1)S(P)T(vi,vi....vi) S(vi,vi....vi) // (A.8.22) Notice that there is that big fact factorial sitting in the last equation. My 15Chap11 paper instead says this First I note his interesting notation, I think that is somehow my Σ' . I think it has to do with the situation when T and S are already antisymmetric, so you are doing a counting thing. Well, OK, that is another subject I will have to deal with. Here is this guy on multiindex notation: Maybe this is a symbol I have been missing. So there is his Alt operator, just a script A as I originally intended. Is this a theorem I know about? I know that A^^B^ = Alt(AB) Then replace A by Alt(A) = A^ to get A^^^B^ = Alt(A^B) = Alt(Alt(A)B) = Alt(AB) = A^^B^ So I would say A^^^B^ = A^^B^ = A^^B^^ or (Alt(S^))^T^ = S^^T^ = S^^(Alt(T^)^ so there is his little theorem, very good. But look at his crazy equation with that silly factor in it! This 15Chap really is closed to my world, stuff I have in other papers is here too. What is this book? It is somehow from here http://cseweb.ucsd.edu/~gill/CILASite/ This is a 1989 book and it is on line for free at the above site, which is where I got it from. So he has a Chapter on my topic, it just happens! k-linear New source: This guy uses yet another normalization. Again those factorials! The above σ as my T is very close to my transformation rule. STATUS at 9 PM. I have added a "transformation section" to Appendix C.5 and I think it is good. Not many people talk about this, as the above search shows. But my entire document is about tensors, and tensors are related to transformations in my book. Along the way I picked up some ideas above. I see other people's notations and normalizations. I get a hint on that Spivak norm factor. Sunday 11.22.15 Realized the transformation rule is trivial due to multilinearity. But then realized that both arguments are wrong! Back to square zero on transformations at 5:30 AM. Side Question: Is T(v1,v2 ...) (nk)-multilinear in the components of the vectors? I don't think so. Consider the case k = 1. Linear means f(αx+βy) = αf(x)+βf(y) . Define F(x,y,z) ≡ f(x) Calculate: F(αx,y,z) = f(x') x' = (αx,y,z) F(αx,y,z) = f(x') αF(x,y,z) = α f(x) Would need to show that f(x') = α f(x). But we DON'T know that. We only know that of the entire vector is scaled, so x' = αx. Answer to side question: NO. ψ(r1, r2...rk) = <r1, r2...rk| ψ> (C.5.4) where ri is the position of particle i. Status 1:30 PM 11//22/15 Sunday: I finally have Appendix D stabilized and I removed that material from Appendix C which no longer has a Section C.5. This took a LOT of fiddling around. Status 7:30 PM. I have lost track of where I was editing before Appendix D had to be written to clear up two questions (a) how does a tensor function transform? (b) how can α(v) be both a scalar and a vector? Basically today was spent resolving those matters, and Appendix D is done (for now). I know I was pondering Section 6 and here is where I was (copied from above) Chapter 6 intro OK 6.1 OK greatly improved 6.2 OK finally I have clear definitions of "tensor","tensor function", and V*k vs V*kf 6.3 OK added a clarity last sentence. 6.4 OK stressing distinction and new notation V*kf . 6.5 is this a place to stick in tensor function transformations? STOP!!! What about this idea T'(vi, vj) = RiaRjb T(va, vb) Here is I think the batting order now, starting tomorrow Monday: 1. Clean up all of Section 2.11 which concerns dual things. 2. Review all of Chapter 4 which has several dual sections in it. 3. Then resume at Section 6.5 and finish off Chapter 6. 4. Then resume the long-stalled edit in Chapter 8 Nov which has an ok to here marker in it. It is very hard to maintain coherence with such a massive amount of stuff, but I am trying. Note that the appendices are not installed and so far I have A,B,C and D. It is useful to have them kept separate for now during development. Mon Nov 23. Starting at 8 AM on the above list. I will do all of Chapter 2 because very many things have happened since my last review of this underpinning chapter. Chapter 2 opening material OK, no mention of tensor functions 2.1 OK, a few edits for clarity 2.2 OK, more small edits 2.3 OK and all this stuff is very good IMHO I am removing this comment at the end of 2.4 Reminder: In general, we have en(x) and un(x) and gnm(x). Then (un(x))i = δni is valid at every point x in x-space. I don't think un is ever thought of as a function of x, since it is axis-aligned basis vectors. I may change this later if I see something. 2.4 OK 2.5 OK 2.6 the Picture E change, OK if tedious. All is laid out. 2.7 fine, the Maple example, keep it. 2.8 the symbol, outer products, the rabbit is in the hat 2.9 ends with those dot products, all is well so far. 2.10 (a) OK k = 2 (b) OK k = k. Pause: Section 2.10 (b) is almost identical to Section 5.2. I added comments to smooth this over. Found error in number sequence, there are two (2.10.15)'s! Warning: I am now altering Section 5.2 and that will have implications for Section 6.2 *(**** New plan: Don't discuss the k = k case in 2.10 since I have a whole Chapter 5 dedicated later to that subject./ I have a huge amount of replication!!! Get rid of it! But on the other hand it fits right in! I replaced Section 5.2 with an exact duplicate of Section 2.10 (b), I don't think it makes sense to get rid of either copy, each is a special context and I will keep both. It is not very long. I continue now with the above. it is now feeding time, so it took all morning just to get UP TO section 2.11 on the dual stuff. 2.4 OK 2.5 OK 2.6 the Picture E change, OK if tedious. All is laid out. 2.7 fine, the Maple example, keep it. 2.8 the symbol, outer products, the rabbit is in the hat 2.9 ends with those dot products, all is well so far. 2.10 (a) OK k = 2 (b) OK k = k. 2.11 opening material OK (a) finally done see next paragraph Heavy editing of Section 2.11 (a) is now underway! I think this has taken maybe 2 hours! It was a complete rewrite and reorg job. The eq nums got changed a bit, but I managed to maintain (2.11.14) as the first equation in section (b), to which I now turn. I added a proof that λi form a basis. Added the scalar vector stuff back at the λi level. Continuing 2.11 opening material OK (a) finally done see next paragraph (b) Why at this early stage am I getting all involved in the dual space general k case? Or even the k = 2 case? That is the subject of a whole chapter section later on in Chapter 4, here I am only in Chapter 2. It is true that this is where I first learned things. More reorg needed, I fear things are in a bad state. But in fact Chapter 4 is about the wedge product, and I only want to quote the k=2 stuff then from dual Chapter 2. I will ponder after a break. Back. Major discovery that I never realized before right now: (λiλj)(v1,v2) = λi(v1)λj(v2) = (v1)i(v2)j = (v1v2)ij This relates a certain tensor function to a certain tensor!! (λi λi ..... λi)(vj, vj ....vj) = (vj vj.... vj)ii...i What does this mean for an expansion? T = Σii....i Tii....i (λi λi ..... λi) . (6.2.1) T(vj, vj ....vj) = Σii....i Tii....i(λi λi ..... λi)(vj, vj ....vj) = Σii....i Tii....i (vj)i(vj)i ....(vj)i = Σii....i Tii....i (vj vj.... vj)ii...i Restate this: T(vj, vj ....vj) = Σii....i Tii....i (vj vj.... vj)ii...i Implications for Appendix D? There we found in Appendix D that (vj vj.... vj)ii...i = (vj vj.... vj)(ei,ei, .... ei) So this is a third way to write it. Summary: (λi λi ..... λi)(vj, vj ....vj) = (vj vj.... vj)ii...i = (vj vj.... vj)(ei,ei, .... ei) This "new discovery" is only useful for the λi λi ..... λi) object. I will now keep it in mind. Then we get T(vj, vj ....vj) = (vj vj.... vj)(ei,ei, .... ei) which is strange looking. Continue on Paradox: Consider my new equation, (λiλj)(v1,v2) = λi(v1)λj(v2) = (v1)i(v2)j = (v1v2)ij = (ei v1)(ej v2) . (2.11.16) 1 2 3 4 Looking at 1 and 4, I see that we have a "scalar" for fixed i and j. This is well hidden in forms 2 and 3. For example, we have (v1)i = (v1)(e)i whereas in x'-space it would be (v1')(e')i so that is the mystery Paradox. Consider this simple equation Vi = (V ei) Since the dot product is a scalar under a component transformation, Vi is a scalar and not a vector!! V'i = (V' e'i) = (V ei) = Vi Well this is true: If you rotate the apparatus and you rotate the coordinate system, you get V'i = Vi. This subject is discussed in tensor doc STOP, I have reread that tensor doc section, and I am now going to have to completely rethink what IO have said everywhere about vector vs scalar! This includes Appendix D. OK, I see the error of my ways. I have to stop now and redo Appendix D.3, and then there will be no need for a D.4 section! But more trouble. Consider V'a = RabVb As a vector field I write V'a(x') = Rab(x)Vb(x) where x' and x I think of as position vectors, but they are just vectors in x-space, whatever that means. Now suppose in x'-space we have some velocities of particles. Can you write V'a(v') = Rab(x)Vb(v) ? where v'a = Rabvb x' = F(x) Does that even mean anything? I ask because I am going to have stuff like this in dealing with the transformation of tensor functions. Maybe v is the velocity of a point particle, and Vb(v) is the vector potential that particle creates. But that makes no sense because you would need to have Vb(x, v) . So I don't have an example yet. My case of interest is this T '(vi,vi, .... vi) = T(v'i,v'i, .... v'i) = RijRij .... Rij T(vj,vj, .... vj) I know that T is multilinear and THAT is why the right equation is valid. The left equation is my interpretation of things. I guess I could interpret, T(vj,vj, .... vj) = Fjj ...j = a tensor Then the above says, F'ii ...i = RijRij .... RijFjj ...j T '(vi,vi, .... vi) = RijRij .... Rij T(vj,vj, .... vj) But the object T(v'i,v'i, .... v'i) does not fit into this picture! End of Day Essay. I have spend probably years of my life dealing with tensor transformations, such as tensor doc and such as Chapter 2 of this paper. I always want to know what things look like in another frame of reference so to speak. So I naturally feel the need to "transform" the "tensor function equations". THAT is where I am having a huge problem. I don't even know what it means, much less how to do it. I do know that this is true for any tensor function, (just from multilinearity). T(v'i,v'i, .... v'i) = RijRij .... Rij T(vj,vj, .... vj) . v'i = Rijvj I urgently want to understand this in terms of a transformation. In some frame S I have some velocities v, and in some other frame S' I have velocities v' . So there IS a transformation going on here. But I am having trouble connecting this with a function T' which "lives in frame S'. The transformation here really is "that different one", which I call a "vector transformation". It is different from all my Chapter 2 stuff, even after I repair it to be talking Experiment #2 instead of #1. I did some whiteboard work and think I have a plan for tomorrow. Tues Nov 24. I descended into confusion, but think I saw the way out by day's end. I was thinking that there were two kinds of component indices, such as (V)(u)n and (V)(e)n , when in fact those (u) and (e) labels belong on the vector, and there is only one kind of component index, the one that enumerates an expansion! Weds Nov 25. Rewrote Section 2.6 with a new Picture E and default means no superscript. Add the V' expansions as well although they are ugly in the new notation. I had things very balled up before in terms of this new Picture E notation. Section structure was not changed, but equations renumbered so will need to fix all references at some future time. While doing this, I did edits in Section 2.1 and I added new material such as (2.1.6). Moved the inversion rule to its rightful place, and corrected it. I left end of Section 2.1 in a mess starting with the tensor fields because I am not sure yet which tensor field examples I want to use there. Sections 2.2, 2.3,2.4 were unchanged. Section 2.5 has a new right column in (2.5.4), and that was the only change to that section. I will now save out the old Section 2.6 to scraps and insert the new Section 2.6. No later Sections of Chapter 2 were changed, but that may happen soon. Trying to restabilize Chapter 2. I next want to go through yesterday's "barrage of questions" document and possible other docs to make sure confusions of yesterday are all resolved. I started into this barrage document called "transformations and experiments", and indeed there are some "good questions" there that were not easy for me to answer, I worked on some of those questions in another doc called "an experiment question". Certain things got cleared up, especially how you get from Chapter 2 to the general notation of Chapter 5 which I use elsewhere. But the question of what those "experiments" are and how you describe them in bra-ket notation is unstable and fuzzy. Luckily I faced this exact same confusion 7 years ago in "confusion about rotations" in my phys/QM/angmom area and I think I should review this carefully before going on with my work here. No need to reinvent the wheel if I already have a wheel ready to go. I will have to annotate that document in a new doc because I see it is a bit messed up. Weds Jan 13. Well, wedge doc has been dormant now for 7 weeks while I updated bowl doc. I guess I will just start completely over with Chapter 1, I have completely lost all context. This will happen again when I do the Cod trip coming up in 2 weeks, but maybe I can get Ch 1 and Ch 2 stabilized before that trip. So I begin by reading Chapter 1 right now. The most recent version of wedge doc is tensor and wedge v6, I presume, and it shows last mod date of 11/25 which is when I stopped as shown above. Chapter 1 opening section and "nomenclature". 1.1 long and quite good I think, it all made sense reading after a 7 week gap. No claim to precision. 1.2 the category diagram approach,. Comment: I think both these sections are good for a reader like me who has wondered about these arcane structures but who has no real interest in all the details. My reader is a practical engineer/physicist who is not interested in math niceties, BUT who appreciates having a little survey/overview of such matters. ****** I have to clean up the α,β versus s1,s2 business for scalars. Chapter 2 opening material, very good 2.1 OK up to red markings, then needs clean up . Issues came up here which I will have to review. 2.2 OK 2.3 OK 2.4 OK, very short on u basis 2.5 OK 2.6 OK, paradox resolved 2.7 OK, example of building the tensor framework given a set of constant en vectors 2.8 OK, outer products and a third meaning for 2.9 OK, inner products and contraction, mixing in the notation 2.10 (a) OK on tensor expansion rank-2 (b) OK on general rank-k, multiindex has appeared 2.11 OK opening material (a) Thurs Jan 14. I slide a few files into wedge tensor obs folder to try to clean up the mess, but there are still a lot of doc files in the tensor wedge folder. Right now I am confused by the appendix status. The main doc ends right now with Chapter 7 which is "wedge k". Chapter 8 is supposed to be "dual wedge k", but that seems to be resting in separate file Ch 8 Nov 20 for the moment. No appendices are installed. So what was I planning for appendices? Possibilities are A,B,C,D. Appendix A Appendix A v4: Permutation Support. Let's review this right now, should be no problem. This is a very long appendix, I only dimly remember writing it! I got tired of not having all these facts "on hand". The appendix needs an overview at the start!!!! Appendix A v4: Permutation Support A.1 excellent! Made a few fixes. This was rearrangement theorems. A.2 OK, there is a LOT of stuff in here, I totally forgot. This was Alt. A.3 OK, a copy paste edit of Alt for Sym, same EQ nums. A.4 OK, very good, combinations of Alt and Sum and Else. A.5 Application to tensors (a) Alt equations written for tensors, OK (b) Sym equations for tensors, OK (c) involving both Alt and Sym and Else A.6 About ε. Again, lots of stuff here. No mention of Levi Civita. A.7 OK. Wedge product Alt equation A.8 Application to tensor functions (a) OK (b) OK (c) OK Yes, definitely needs an overview section. // Wrote an Overview and installed it! Comments: Appendix A is very good and contains critical information that will be used throughout the main doc. Good for it to be the first appendix, it is definitely "a keeper". It is quite long at 24 pages and could be shortened, but I hesitate to cloud things by doing that. Earlier much of this information was mixed in with the main doc. The Big Idea is to do it once generically, then apply those generic results to tensors and tensor functions. I wonder if anyone else has ever done that? Appendix B This is everything one needs to know about direct sums. I first avoided this topic, but in later drafts realized it was important, and here we have it all in a small 6 pages. Read it all, added front fake TOC. Appendix C Some confusion here and I have to clean up right now! This appendix contains a certain set of "three theorems". There are 6 versions of Appendix C! At an earlier time I had "unified view of tensors and tensor functions" as the last section of Appendix C, but then I moved that to be Appendix D. I think the current version of Appendix C is called Appendix C v5 . Here is a summary: Summary of the Three Theorems Theorems One, Two and Three have shown that, in our generic function space, Alt[TS] = Alt[T^S] = Alt[TS^] = Alt[T^S^] where T^ = Alt(T) S^ = Alt(S) (C.4.1) Sym[TS] = Sym[TsS] = Sym[TSs] = Sym[TsSs] where Ts = Sym(T) Ss = Sym(S) (C.4.2) After proving these theorems, I generalize. So OK, I now remember what Appendix C is all about. I will move all the previous versions out to storage in a folder. I then went off and reviewed some docs in the wedge folder, trying to prune down so only active docs are in there, not just retired or installed old versions of things. Continue on this next time! Example: in wedge doc below (7.4.12) I refer to my reordering theorem in App A, but that theorem does not presently exist in Appendix A. I am not sure where it would fit. Maybe have to add new sections to Appendix A for this and other permutation sum related items. Fri Jan 15, 2016. Things are just a real mess. The v6 main doc I think is only partially edited with changes I intended to make, such as putting wedge coefficients in italics. I do this right now in (4.3.5) for the k=2 wedge section, but it is not installed yet in Section 7 on k wedge. So I really need to start proofing again starting with Chapter 3, since I did 1 and 2 in the last few days. I need to see what conventions I am using for the k=2 stuff and then make sure those same conventions get used for the k = k world. But first I just have to declutter the tensor/wedge directory, it is filled with docs that are no longer at issue, and I therefore can't find things. I am right now very confused about my "hat notation". In "section 4_3 with hat notation. doc" (created 11.18) I write things like this, T^ = Σij Tij ei ^ ej Does this mean that I no longer use the notation T = Σij Tij ei ^ ej ? Is this a tentative alternative notation to the italic notation? Things are out of control! Does the 11.18 edit log entry tell me anything? Yes, it says that my T^ notation replaces the italics notation! OK, just wrote up a meta edit log so I can track what I have done over the entire history of this doc project. After doing that, I did cleanup on the folder, removing as many doc files to subfolders that I think don't have a bearing on the doc future. Sat Jan 16, 2016 I am reading Appendix D, and I like it up to the start of D.3, which is where trouble will begin. // Spent the day doing bra-ket refamiliarization. Just doing it for k = 1 in the non-dual ket world was complicated for me but I think I have a good start. However, my cold is going downhill fast, signing off at 7 or so. Weds Jan 20. Have continued in sickness to ponder Appendix D, writing some auxiliary docs on the topic. I am wondering if it is useful or meaningful to claim that the tensors and tensor functions are the same animal in a different basis. I don't think that is quite right. If you use a ket that is all ei basis vectors, you do get this Tii....i = <T | ei,ei, .... ei > . TI = <T | eI > (D.1.5) This is like vi = <v|ei> and is nothing much to write home about. Take the case of k = 2: Tij = <T | ei, ej> Now suppose you write |v1> = Σa(v1)a|ea> and |v2> = Σb(v2)b|eb> as two general vectors in V. Notice that these are those "vector transformation" combinations. Like saying v = vx + etc. That is v = Σiviei Then you have | v1, v2> = Σa(v1)aΣb(v2)b | ea, eb> Whereas | ei, ej> is a basis vector in V2, | v1, v2> is a "most general element" in V2. Then <T | v1, v2> = Σa(v1)aΣb(v2)b <T | ea, eb> = Σa(v1)aΣb(v2)b Tab and this is our "tensor function" T(v1,v2) = ΣaΣb (v1)a(v2)b Tab What is it? It is the tensor Tab weighted by the components of the two vectors. The two vectors are arbitrary vectors selected from V, they could both be the same if you wanted. This object is the closure of an abstract rank-2 dual tensor T ϵ V*2 with an arbitrary but pure element of V2, namely v1v2 . OK, this is good stuff. I just reviewed Chapter 2 a bit, but there is nothing dual until Section 2.11 where I would like now to read more carefully. I may want to make some changes. fe I have started a rewrite of my dual vector space stuff. It may end up as a new Chapter 3 depending on how long it comes out being. This is being done in a separate file. Jan 21, 2016. I finished my full rewrite of Section 2.11 on Dual Spaces and Vector Functions. Although I had to use the ugly 3 dot equation numbers (2.11.a.3), it is still a very major improvement, and I think removes the need for Appendix D completely. The Dirac notation is fully explained, transformations are in there, things are finally I feel solidly nailed down. I now have to ponder the entire rest of this document in light of what this new Section 2.11 has to say. I am not sure how this is going to turn out. Jan 22, 2016. Reread all of Sec 2.11, it flows pretty well. Now I will move ahead and try to do some continuity work getting things integrated. I read Section 3.1 which is a review of outer products. I then follow my own instructions and skip 3.2 on the Kronecker, and I move on. to 4.1/ 4.1 is OK, does not seem totally stupid after the fancy 2.11 section. Reviewing and going slow. 4.2 is OK, does dual version of section 4.1 4.3 I have to switch to an external document to get current version! (a) OK, I am feeling better again (b) OK (c) OK, geometric connection, all good stuff. (d) OK, symmetric and ordered sums concept (e) OK, some strange dot products 4.4 Still in separate doc, now redo 4.3 but we are in the dual space V*2. (a) OK (b) OK (c) OK (d) OK changed title to Tensor Functions from Components. I just installed the above two sections into v6 main doc. Am tired. Tomorrow will do a full read of Ch 4 and use that as momentum to do Ch 5 right afterwards. Sign off 7 PM. Jan 23, 2016. 4.1.4,5 using ki for scalars, maybe change that ***** I used s in 1.1.2// Change made. 4.1 OK and good 4.2 OK and good 4.3 (a) OK and good (b) OK and good (c) OK and good, the geometry connection (d) OK and good (e) OK and good 4.4 (a) OK and good (b) OK and good and short (c) OK and good and short (d) OK and good and short now 10:50 AM 5.1 OK and good 5.2 OK, and added Dirac notation which explains the dot product! 5.3 OK 5.4 OK and good 5.5 OK and good 5.6 OK and good Pause for comments: This stuff is not time-wasting repetition going from k= 2 to k = k. It is much more, involving the general theory of T(V) . I like it, I hope other people will like it. Now, Chapter 5 is about tensor products in Vk . Let's rethink again the idea of a "translation table" approach, perhaps I know more from the Dirac notation. I will read on anyway for now I have just finished at 5 PM a new rewrite of Chapter 6 which is much less copy-paste and more interesting and I think a lot shorter. The tensor function result is fully incorporated. Did one or two edit passes, and then installed this as Ch 6 in v6 main doc. All done. It is 6 pages, Ch 5 was 11 pages, so it is a compacted version compared to what I used to have! On now to Ch 7 which is non-dual k wedge. These next two chapters are the heavy duty chapters. I am a little unclear where my most current Ch 7 is located. I guess it is the in-line main doc version. Chapter 7 inline review 7.1 OK, refers a lot to Appendix A which I reviewed on Jan 14, 9 days ago. ( App A = 24 pages!) 7.2 OK. Lots of relatively new stuff here. Done for today. // Resuming 1.24.16 Sunday. 7.3 OK, meaty details STOP. There is something fishy about (7.4.2). The first two lines are OK. The third line is illogical! Let write it out in more detail Qjj....j = Σii....i Tii....i (ei^ ei^ .....^ ei)jj....j = Σii....i Tii....i Alti(ei ei ..... ei)jj....j = ... I think the (7,2,9) proof is also bad. // OK this was a good find and I have fixed things up by adding more to Appendix A concerning AltI and AltJ . Next: I need that fancy "ordered sum theorem". It is sitting in original App A and I need to put it somewhere. // Added it solo at the end of Appendix A, which is on all things P. It is (A.9.1). I now continue with the Chapter 7 edit pass: Chapter 7 inline review 7.1 OK, refers a lot to Appendix A which I reviewed on Jan 14, 9 days ago. ( App A = 24 pages!) 7.2 OK. Lots of relatively new stuff here. Done for today. // Resuming 1.24.16 Sunday. 7.3 OK, meaty details 7.4 OK and lots more meaty details. 7.5 OK, and a huge amount of notational detail. 7.6 OK and very short 7.7 OK on multiindex, I like it and it is short 7.8 OK and this is a section of Major results. 7.9 (a) OK after I cleaned up an AltJ and AltI ambiguity similar to that above. (b) OK, special case where T or S is a scalar (c) OK, on graded commutativity (d) OK. Notice that the wedge product of tensors is non-trivial!! (e) OK, whew!!! That is one LONG chapter, and I live in fear of the Chapter 8 translation! I did pick up few good repairs on this edit pass of Ch 7. I don't like the eq num system, but there it is. Wandering through Nov 20 Chap 8. Stalling here: The symmetric expansion is very straightforward. Let Tii...i = (α1)i (α2)i ... (αk)i = (α1 α2 ..... αk)ii...i <T| ei, ei .... ei> = <α1,α2...αk| ei, ei .... ei> = (α1)i (α2)i ... (αk)i or T = (α1 α2 ..... αk) . // α1 ϵ V*, T ϵ V*k (8.5.1) OK, that is Dirac-consistent. But then..... I have slowed down now on this Chapter 7 to Chapter 8 conversion. Write up a little piece on the Alt operator in Dirac notation. Unsure how this all is going to work. Mon Jan 25, 2015 Let's try once again to "get happy" with Ch 8 Nov 20. What the reader and I want is a cookbook of facts, not some general Dirac translation rules, but I need to know those underlying rules to make sure things are right! Chapter 8 Nov 20 review 8.1 OK and I think "just right", minimal but states the facts 8.2 OK and also just right. Not sure what red ** marking equations means. 8.3 OK, for "component" I always "do both" e and v arguments with eqnum.a 8.4 OK, first time I have made it through this section. All seems OK. 8.5 OK, unbolded all α's, it seems OK, did all details except refs. 8.6 OK, short, that ratio and graph thing. 8.7 OK, unbolding all α's, seems all OK and not to tough 8.8 OK, so far so good, has "ok to here" mark at the end! 8.9 (a) OK, took various edits to get things to be parallel with 7.9 (a) (b) OK, just quoted simple cases summary, just fine (c) OK, short summary, just fine (d) OK, took a while, did full detailed translation (e) OK, just quoted the little rule Done, at least to this point. How does Spivak arrive at his normalization? Suppose he did this: v1^ v2^ .....^ vk = 1 ΣP (-1)S(P) ( vP(1) vP(2) ..... vP(k)) = 1 [ (v1 v2 ..... vk) + all signed permutations ] . = k! Alt(v1 v2 ..... vk) = 1 Σii...i εii...i (vi vi ..... vi) ir = 1 to k . (7.1.3) My tensor expansion would be like so: Σii....i Tii....i (ei^ ei^ .....^ ei)jj...j = Σii....i Tii....i k! AltI(ei ei ..... ei)jj...j // (7.3.8) = k! [Alt(T)]jj...j ≡ [T^]jj...j T^ = k! Alt(T) (7.4.2) That is, suppose I define T^ in this manner. Then what happens when I do T^ ^ S^ ? I am staring at (7.9.a.3) and see no difference there. So I still get T^^S^ = ΣI (TS)I e^I But then I would have (ei ^ ei ^ .... ^ ei) = k! Alt(ei ei .... ei) e^I = k! Alt(eI) (7.3.8) Then Alt(TS)J = AltJ(TS)J = ΣI (TS)IAltJ(eI)J // (5.6.5) and (A.5.10) that Alt is linear = ΣI (TS)IAltI(eI)J // use (A.8.26) since (eI)J is totally antisymmetric in I and J = ΣI (TS)I (1/(k+k')!) (e^I)J // (7.3.8) = (1/(k+k')!) (T^^ S^)J // (7.9.a.5) so T^^ S^ = (k+k')! Alt(TS) . (7.9.a.7) T^ = k! Alt(T) S^ = k'! Alt(T) I know that Alt(TS) = Alt(Alt(T)Alt(S)) so then Alt(TS) = Alt(T^S^) / [k! k'! ] Then we end up with T^^ S^ = (k+k')! Alt(TS) = (k+k')! / [k! k'! ] OK, I have written it all up, this was a major breakthrough for me, for the first time I understand where all those strange Spivak factors are coming from! Chapter 7 now has two new subsections which I will now have to reflect into Chapter 8! Hurray! Jan 26 Tues 2016. Added (f) and (g) to Chapter 8 Nov 20 to match those added yesterday to Chapter 7. Added a Dirac section to Appendix A. Am getting close to a final assembly. What about Appendix D? // I decided to keep it, and I added a section debunking the transformation similarity, and another section with my earlier comments about wavefunctions. What about Chapter 9 on wedge theory approach? Chapter 9 9.1 OK and in fact quite good. 9.2 OK and it all comes back. I think everything is written, I am ready to assemble! // Assembly complete, 194 pages. Idea: Review old matrix notes on and and see if anything should be added. DONE Idea: add section showing all major notations. Done, but needs editing. Jan 27 Weds 2016. So what stuff is in my hand-written binder notes on and ? There are 13 pages of notes. p 1: summary, I will have to look at the various sections. p 2: rules for operators. I red-checked the ones that agree with wedge doc claims. p 3. Direct product of two matrices. I think the connection here is this: hand written notes: (AxΓ)iα,jβ = AijΓαβ wedge doc: (ST)ii',jj' ≡ SijTi'j' (ST)ii',jj' ≡ SijTi'j' . (3.2.15) In wedge doc I call this a definition. So I can say that the "direct product of two matrices" discussed in my hand written notes corresponds to the Kronecker product. Lower on the page I talk about direct product of two vectors as (AxΓ)iα = AiΓα and this is my general wedge doc outer product, so it is covered. a p 4. I talk in notes about a source D&S, perhaps a book long gone. No page notes refer to such a source. I talk about a direct product matrix times a direct product vector. I do have this in wedge doc [(ST)(vw)]ii' = Σjj'(ST)ii',jj' (vw)jj' . // = (xy)ii' (3.2.16) where the right sides is exactly such a thing. In the hand notes I have (AxΓ)iα,jβ (aγ)jβ and that is a match, so this concept is in fact covered in wedge doc. I do allow 4 arbitrary dimensions. What about the idea of (S1+S2) T = S1T + S2 T ? This is in the hand notes. I added a few lines at the end of the Kronecker doc section showing and proving these facts. Good! What about this claim I make in hand notes on page 4 : (S1S2)(T1T2) = (S1T1)(S2T2) In think the S's have to be square so that S1S2 exists. Then is the above true? [(S1S2)(T1T2)]ii',jj' = (S1S2)ij(T1T2)i'j' = (S1)ik (S2)kj (T1)i'k' (T2)k'j' = (S1)ik (T1)i'k' (S2)kj(T2)k'j' = (S1T1)ii'kk' (S2T2)kk'jj' OK, I have just tacked this property on as well to the end of Section 3.2, good! p 5: Nothing new here. p 6 is on direct sums which stuff is in my Appendix B. In my Appendix B, I have space V and W, and I have vectors v and w. I don't have any operators like S. I would like to see some matrices here maybe. Task: Add a new section at the end of Appendix B on the direct sum of matrices!!! DONE So far I only have the direct sum of vectors and of spaces. p 7 OK, added a new rule, but I skip the combined rule, too messy. p 8. Decided not to include my strange * operator, more suitable for Lie Groups. OK, I have finished my scan of the matrix hand notes and have added a Section 9 to Appendix B, it is all done! Status at 7 PM on Jan 27, 2016. The document is ready for final proofing and then pagination and them putting out on the web. // Wrote an Overview and Summary, reordered the notation sections somewhat, added section page breaks. It is now 205 pages, quite respectable I would say! Needs references. Then needs a LOT of checking, especially equation numbers. Jan 28 Thurs 2016. Letting the doc cool off. I want to look back now at Sjamaar. I will read some meta notes now. I am in Chapter 2. He talks about α = differential form and then α2 and αp . In wedge doc I have these two expansions of a k-form (Sja says degree k) T^ = Σii....i Tii....i (λi^ λi .....^ λi) T^ = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) I never try to square one of these, but I do tensor product, so I could try that with the same element. Then I would get T^^ S^ = ΣI (TS)I λ^I I ≡ I, I' = i1,i2...ik+k', λ^I ≡ (λi^ λi .....^ λi) . (8.9.a.5) so T^^ T^ = ΣI (TT)I λ^I I ≡ I, I' = i1,i2...ik+k', λ^I ≡ (λi^ λi .....^ λi) . (8.9.a.5) = ΣITITI' λ^I = ΣITITI' (λ^I^ λ^I') Example: (T^^ T^)(v1,v2,v3,v4) = Σiiii Tii Tii (λi^ λi^ λi^ λi) There seem to be some symmetry issues here. Another notation. (T^^ T^)(vJ) = Alt(TT)(vJ) OK, what would Alt(TT) produce in my generic world? Alt(TT)iiii = ΣP(-1)P (TT)iiSii I already wrote this out, [T^^ T^]I = [T^^ T^]iiii = (1/4!) ΣP(-1)S(P)TiiTii = (1/24) [ TiiTii - TiiTii + TiiTii - TiiTii + 20 more terms ] (7.9.a.10) I don't think this vanishes. Simpler case of vectors DOES vanish however V^V = 0. My commutation rule was this Alt(ST) = (-1)kk'Alt(TS) so Alt(TT) = (-1)kkAlt(TT) So if k = 0, 2, 4... then this just says Alt(TT) = Alt(TT). But for k = 1,3,5...it says Alt(TT) = 0 !! So that is the general answer and agrees with vectors. 3*3 = 9, 5*5 = 25, always odd. So this is a useful fact I might add somewhere. But what does Sja really mean when he says α2 ? I will just quote a few Sja things here: His sum seems to match my symmetric sum. But then he uses the same notation to represent an increasing sum, and of course that is a problem for me since I have T^ = Σii....i Tii....i (λi^ λi .....^ λi) T^ = Σ1≤i<i<....<i≤n Aii...i (λi ^ λi .....^ λi) The coefficients are different in the two cases and I find that (1/k!) A = Alt(T) A = k! Alt(T) I never followed up on my products using the ordered notation! So beware this problem. // I see this meaningful only in symmetric notation // a result that I already have with T^^ S^ What about this one: For me this would mean (T1)^^(T2)^^...^(TN)^ = ΣI (T1IT2I .... TNI) e^I = ΣI (T1T2....TN)I e^I OK, this is all for a separate doc, not here in the edit log! Feb 9, 2016 I am resuming now after a long Cod break of 11 days, have to get the flywheel in the mud turning again which will take some time. I note the section above. I am now reading through the Sjamaar meta notes while looking at his PDF on the left side of the screen. I am adding comments of today's date to those notes. 1. Be sure to add equations doing the wedge powers of a tensor, such as T ^ T. I omitted this from wedge doc, but Sja talks a lot about such things. ******* Thinks like Tp ≡ T ^ T....^ T. A theorem then says that Tn+1 =0. Done reviewing Ch 2 of Sja meta notes. Feb 10, 2016 Continued with Sja, but I am blocked now by this basic equation: Why can you say dxi = ei = λi and have things like dxi(v)? I just do not GROK the connection between the dual space and differential forms. I looked at Sja, at Spivak, at Benn Tucker. Most diff forms discussions don't even mention the dual space business with λi and those that do are too complicated for me to translate, it would take 6 months for me to get into Benn Tucker's book. So I am simply missing this very major connection and will have to spend maybe a week searching the web to find a person who explains it. It seems to me that for a given basis ei, the dual basis ei is completely determined, so why can you willy-nilly say ei = dxi ? Maybe you are working only in Cartesian space and you are saying dxi = |dxi| ei and so you have this extra scalar factor |dxi| just sort of hanging around? I need to find someone who speaks English and who understands why this connection is not obvious to the newbie reader. Feb 11, 2016 I have started a new doc called "relation between diff form and dual space". This log will continue in that doc for a while. I made some progress there, but decided I really want to do a complete and detailed review of Sja at this time. Also, just found the Flanders book on line. Sjamaar Detailed raw notes review. Chapter 1: Introduction 1.1 manifolds, OK 1.2 manifolds described by equations, φ(x) = c, tangent space TxM, OK 1.3 using parameters like θ,φ to define a surface, very short section, OK 1.4 configuration spaces as examples of parameter spaces defining manifolds, OK Chapter 2: Differential forms in Cartesian Space 2.1 elementary properties of differential forms, all OK 2.2 the exterior derivative "d" on a form, various rules and conclusions, all OK 2.3 forms being closed or exact; the angle form; all OK 2.4 the meaning of *dxI (Hodge star operator), OK 2.5 writing grad, div, curl and lap using *, d and α symbols. OK Chapter 3: Determinants and Pulling Back Forms 3.1 determinants treated in terms of certain axioms, all OK 3.2 general theory of pullback forms, all OK Feb 12, 2016 I spend this day firming up Bucks' presentation of integration over curves. I proved a few small theorems and added them to the Buck notes. Buck's talk about two basic type of integration over curves. The first type they discuss is "integration of a scalar function over a curve with respect to arc length". In that type, if you set that scalar function to f = 1, the integral gives the arc length. The second type they discuss is the integral of a 1-form over a curve, as defined by the pull back idea. This is the line-integral type integration, and is very different from the arc length type integration. It helped me to get that cleaned up. Feb 13, 2016 I now resume the Sjamaar review with Ch 4 which is exactly about integration of 1-forms. Chapter 4: Integration of 1-forms 4.1 the pullback in new notation, the angle form, respeeding (reparametrizing), work ex, all OK 4.2 integration of exact 1-forms and Ex 2.6, very good OK 4.3 global angle function from angle form and winding numbers, all OK Chapter 5 : Integration of k-forms and Stokes's Theorem 5.1 Notation review, respeed idea in k dimensions, k-cubes and k-chains, all OK 5.2 study of boundary ∂c of a k-cube c, magic signs, ∂(∂c) = 0 all cases; all OK 5.3 degenerate k-cube, integral of k-form is zero on same boundary of a degenerate k-cube is a degenerate (k-1)-cube. if c = ∂b + c' where c' is degenerate, then c is boundary of b seems all OK 5.4 Stoke's Theorem and how it generates all my integral theorems, tricky stuff but OK Chapter 6: Manifolds 6.1 embedding idea, the graph idea, tangent space, atlas, examples: OK Manifold is a surface which allows embedding at any point, atlas of charts to cover OK 6.2 Another view for surfaces defined by c = φ(x) : c regular says surface = manifold Feb 14, 2016 I plan to continue the Sja review, but I do note this concept: The cotangent space is a the space of functionals defined on the tangent space". Each of these spaces is a function of position x on the manifold of interest. Maybe that is a fiber bundle. The tangent space basis vectors, if called ei , would certainly be ei(x) and thus we would also have ei(x). I will resume this after I finish my Sja review. I think you could say ei(x) = the columns of matrix (Dψ) where ψ is the function which defines the manifold, so there is some connection between ei and a derivative. I spent most of today going back over Chapter 6, and in particular 6.2 which gives an alternative definition of a Manifold. I spent much time trying to relate this approach to Lagrange doc and Bucks, but I concluded that, although there are similarities (in that you look for where a matrix drops below full rank), the situations really are not directly connected. So this ended up being a nice waste of time. But it did force me to relearn the c = φ(x) : c regular says surface = manifold development. The first approach of 6.1 is just covering the manifold with small embeddings. Chapter 7: Differential Forms on Manifolds 7.1 the issue over overlap of i and j maps and pullbacks and stuff, seems OK 7.2 THIS is where Sja does all that λi stuff and where he puts the rabbit into the hat by casually saying (ei)T = dxi = λi with no explanation whatsoever! In this way he associates a k-form with an element of the dual space, and once we have that, we have alternating functions and wedges and all that familiar stuff. I did not finish reviewing 7.2 but will do that tomorrow. Here is the big mystery I need to resolve, why we are magically allowed to associated λi with dxi. I am also wondering: OK, you want to associate forms with elements of the wedge dual space, fine. Yes, that brings in the alternating functions. But what does this association of forms with dual space vectors do for me? How does it help anything? Of what use is it? All these questions continue to burn away like inextinguishable candles, perpetual sparklers. Why not have the forms be elements of the non-dual wedge space? Feb 15, 2016 In the AM session I went over the Chapter 7 review once again, doing more details. Let's now return to my comment above of yesterday: "The cotangent space is a the space of functionals defined on the tangent space". This really says that we are defining a differential form at x to be an functional (vector) of this cotangent space at x. That then leads to an alternating function associated with that form. Say it again: on a manifold, a differential form at x is associated with an element of the dual space which is the cotangent space to TxM. Sjamaar never uses the phrase "cotangent space" but it certainly seems relevant. Chapter 7 was very "rocky" for me, but at least it is now reviewed. It talks about two approaches to defining differential forms on manifolds. The first method just says you have a separate pullback on each patch of the manifold so αi = ψi*α on Ui, and there are some consistency conditions for the overlap regions. The second method makes this association of forms with dual space elements with the ansatz that you set λI = dxI. I think in fact Sjamaar just treats this as an ansatz and with this ansatz he shows that this approach replicates results of chapters 2 and 3, but he never addresses words of explanation to this ansatz. It is just quietly assumed. You could I suppose do all of this second approach using λI notation, and when you are done and comparing to Chap 2,3 you could make the connection λI = dxI . I am certainly hoping to find a more direct presentation somewhere. Instead of breaking off and doing that now, I will continue with my Sjamaar review since it does no harm and since I invested very heavily in his monograph (perhaps 60 days invested?) Chapter 8: Volume Forms 8.1 formula for volume of an n-piped in N dimensional space, V = OK 8.2 orientation for manifold of codimension 1 ("hypersurface") , adding normal vector etc OK 8.3 surface measure on hypersurfaces: F (*dx) = (Fn) μM means F dA = (F ) dA OK OK on Chapter 8. I drew a blank on the title, but then found this is sort of "measure" on surface when applied to a surface. His n-piped volume formula is quite novel to me and I see how it relates to tensor doc. This chapter then finally ties back to Buck's early integrals which give area, or which integrate a function over an area. I think Sjamaar has here fitted those integrals into the general k-form integration theory. These chapters all need "meta meta notes". Chapter 9: Integration on Manifolds 9.1 "manifold with boundary", examples, pair of pants, OK 9.2 manifold as union of cubes with no overlap problems, OK 9.3 therefore we can just rewrite Stokes and all it implies for any manifold M, OK! Feb 16, 2016 Chapter 10: Applications to Topology 10.1 retraction theorem, Brouwer's fixed point theorem, OK 10.2 maps which are homotopies (morphing a manifold to another), OK 10.3 manifolds with holes, homotopic loops, Poincare conjecture: in dimension n, is any simply connected manifold 1-1 mapable to the unit sphere? OK n=3 was the tough case, proved 2003, so OK for all n now, no longer a conjecture This concludes my review of Sjamaar, where I am ignoring the two appendices. I should write up some meta meta notes on the whole thing soon. But now I am ready to go back to the ei = λi = dxi problem which I have started in "relation between...", so this log now goes idle for a while. Feb 19, 2016. Got Sja and Spi on the same page for 88A. Now I will try to correlate Sja's two independent developments of the pullback idea. First I want to firm up "the determinant connection" which I currently outline in Sja Ch1-5 raw notes. // Did this, but did it wrong, will repair tomorrow. Feb 20, 2016 Cleared up some technical details: (1) Why the Spivak tensor function equation (also appearing in Sja p 88) gives the same pullback result as the Sja Ch 3 approach to pullbacks. (2) Why the Spivak normalization is used, directly related to this pullback tensor function equation. (3) How the (Dφ)IJ stuff works and how you get "the determinant" in dyI = Σ'J det( (Dφ)IJ) dxJ . All good things to get written down. Feb 21, 2016 Status. I am still unhappy with the notion of integrating forms. It is still illogical that dxi is λi in one place, but is a calculus differential in another place. This discrepancy is most blatant when you look at a simple line integral, integrating a 1-form as in Sja Ch 4. For example, in the pullback you have dx = (dc/dt) dt but somehow dxi = λi but dt is a normal calculus differential. It is very hazy to me what the pulled back coordinates are. Are they wedged? Are they differentials so you can integrate normally, or are they also dual space basis vectors? This is just not made clear in Sja Ch 4 on integrating 1-forms. By not showing the wedge symbols, Sja confuses everything. [ correct! ] A related problem I have is that I am still really unable to read the Spivak or Flanders sources and these sources remain fuzzy to me. These are the two main books that Sja quotes in his Preface. Maybe I should look again at these books. Spivak opens with all that tensor function stuff, but I never learn much about why he does that. I need better notes on these two books. Progress. Amazingly, today I made great progress on both the items mentioned above. I now know exactly the answer to the first question I have long been seeking. AND I made huge (yoooge) progress in Spivak notes, getting all the way through Stoke's theorem on chains and the end of Chapter 4 (up to page 109 of 159). He is all making sense to me now, thanks to Sja's paper which does more details and examples. I would call this a breakthrough day and tomorrow I want to consolidate my winnings of today. Feb 22, 2016 Forms question: Spivak claims that if you integrate a form over a block or a cube in Rn then the integral of a form equals the corresponding calculus integral which is of course just a real number. This still seems a bit odd to me. For a line integral, eg, you start with integral of fi λi over a curve, and you end up with a pulled back integral over dt with an integrand that is a function of t. But this dt is really a form λi as well, I think? In a surface integral, things more obvious, you end up with a cube integral of dx1 ^ dx2 where presumably these things are still functionals, so there is a function dx1(v) and so on. OK, I have now cleared up this mystery with results explained in "clarify pullback...doc". This took all day, I did a 1-form, then a 2-form and then a k-form. This material will end up somehow in an added section of wedge doc. Time now 4:30 PM, the above took all day. Feb 23, 2016 I am ready to start into the beginning of my differential forms Chapter of wedge doc. I want to be very current on wedge doc up to that point, so I have to review yet again at least various sections. 2.10 on tensor expansions (a) OK (b) OK 2.11 on dual spaces and Dirac (a) OK (b) OK (c) OK (d) OK (e) OK (f) OK so this is all OK despite annoying equation numbers I repaired things as needed I will now skip to Chapter 5 and come back later to earlier ones 5.1 OK 5.2 OK even though mostly a duplicate of 2.10(b) as stated. 5.3 OK 5.4 OK 5.5 OK very short 5.6 OK done with Chapter 5, no problems were encountered On now to Chapter 6 opening text OK 6.1 OK 6.2 seems OK 6.3 OK 6.4 OK 6.5 OK here I "equate" V*k with what might be called V*kf . *** 6.6 good. OK On now to Chapter 7 wedge in the non-dual space opening text OK 7.1 OK, fixed up the normalization comment 7.2 resume here manana -- OK Feb 24, 2016 resume 7.3 OK 7.4 OK 7.5 OK 7.6 OK 7.7 OK 7.8 OK 7.9 (a) OK (b) OK (c) OK (d) OK (e) OK (f) OK (g) OK I have made lots of small changes in Section 7 in this pass, so I need to do the all-important Chapter 8 review very carefully, referring back to Chapter 7. Doing side by side with Ch 7 on left Ch 8 on right. Chapter 8 opening material -- needs a rewrite, just one paragraph. 8.1 OK 8.2 OK 8.3 OK resume tomorrow starting 8.4 Feb 25 resume STOP Bugz noted: Bug #1: I say this: T = Σii....i Tii....i (ei ei ..... ei) . T ϵ Vk (5.2.1) (7.4.1) T^ = Σii....i Tii....i (ei^ ei^ .....^ ei) . T^ ϵ Lk (7.4.4) T^ = Σ1≤i<i<....<i≤n Aii...i (ei ^ ei ^ .... ^ ei) . T^ ϵ Lk (7.4.7) These are most general elements in spaces Vk and Lk . How do you describe these spaces in English. Vk seems to be the space of "rank-k tensors". Tii....i are the components of such a tensor T. I then really have to go to the last line above where I know A is totally antisymmetric and A = Alt(T) in terms of its components. So I could say that Lk is the space of totally antisymmetric rank-k tensors, where the tensor is T^ and the tensor components are AI. Maybe I should call the tensor A and not T^. But then later on when you see something called A or B or C, how do you know these are TA tensors? Bug #2. I thought I once claimed that there was only ONE totally antisymmetric tensor of rank k, Aii...i(x) = a(x) εii...i where a(x) is a scalar. Ah, but be careful! The above statement is only valid if k = n. So for k < n, there are in fact more than one totally antisymmetric tensor. For example if k = 2 and n = 3 we have Aii ir = 1 to 3 has 9 coefficients You cannot write this as a(x) εii because εii has no meaning. You cannot have an ε tensor with 2 indices where each takes 3 values. So there is no bug #2 and there are lots of rank-2 TA tensors. Bug #3. In (7.1.3) I state that v1^ v2^ .....^ vk = (1/k!) Σii...i εii...i (vi vi ..... vi) Well the sums here really do run from 1 to k, not 1 to n, so THIS usage is OK, there Bug #4. I say these things T^ ≡ Alt(T) . (7.4.3) T^ = Σii....i Tii....i (ei^ ei^ .....^ ei) . T^ ϵ Lk (7.4.4) We refer to this type of expansion as a symmetric expansion, and we know it is redundant since the symmetric sum includes each true basis vector k! times. According to (A.5.9), we know from (7.4.3) that Fact: T^ii....i is a totally antisymmetric tensor. (7.4.5) Well, I think all these bugs are now OK. Question: Can you use Dirac notation in Wedge world? |e^I> ≡ |e11> ^ |e12>^ .....^ |e1k> ϵ Lk <e^I| ≡ <e1| ^ <e2|^ .....^ <ek| ϵ Lk T^ = |T^> = ΣI TI |e^I> T^ = <T^| = ΣI TI <e^I| This looks OK to me. Backing up now. I am reviewing chapter 4, fixing up space names, and adding Dirac sections at the end of each of the sections 4.1 4.2 4.3 4.4. Getting the reader prepared. Added Dirac finales to the first three, fourth still to go. Feb 26, 2016 Resuming on adding Dirac to Section 4.4. // Done. I have now used V*2f and Λ2f to denote the spaces of tensor functions, I think this improves clarity instead of saying they are just the same. Added comment on || in Dirac notation. I am now ready to resume my review of Chapter 8 where left off above Chapter 8 opening material -- needs a rewrite, just one paragraph. 8.1 OK 8.2 OK 8.3 OK 8.4 OK 8.5 OK took a long time to get right 8.6 OK 8.7 OK 8.8 OK 8.9 (a) OK (b) OK (c) OK (d) OK (e) OK (f) OK (g) OK This is a bit of milestone, the Chapter 8 review is done! Before diving into Ch 10 on diff forms, I want to review the new Sjamaar just so I know what has changed. // Well maybe I have already done this, I see a doc. I was using a PDF compare thing and gave up, but I will try it now just manually. // Done, I just added notes to my existing document. He did change the λ to β and μ, sadly, but OK. I claim doc is 96% the same ignoring small reorderings. There is some new material but it does not affect me much. So I am now ready to go on a new Chapter 10 starting tomorrow! Feb 27, 2016 Got rolling on Chapter 10 after initial writer's block, in a separate doc of course. Feb 28, 2016 Resume Chapter 190. I started off today by reviewing and editing yesterday's work. I think the writeup is "pretty good" IMHO. Feb 29, 2016 Worked on the pullback stuff section 10.6 in file chapter 10. Mar 1, 2016 Updated to Chapter 10 v 1 where I switched n↔m to be compatible with both Sjamaar and Spivak. Had to update the 3 pictures as well onto a new vsd page. Question: In the pullback, do we pull back only xλi which are in the tangent space at x? If so, then the (Dφ) matrix would be square! I feel the ground rolling again. Here is what I would have to say in this case: φ*(xλi) ≡ Σj=1n (Dφ)ij tλj = Σj=1n Rij tλj i = 1,2...n . (10.6.2) and then (Dφ) is square and n x n. I think this is all wrong! You can't really tell from Spivak who writes This shows that the sum goes to n, but how many i are there? It must be m. Bug: I think I should be using the usual ei for Rm and not my tricky xei which vary from point to point. I think I should be using the "constant" ei vectors. Ouch. Maybe that is why Sja got rid of his tangent space discussion in 2nd edition. Let's now go to Chapter 10 v2. But things are a mess in the head right now. Look at p 68 Sjamaar and you see that the tangent vectors in x-space are (Dφ)e1 and (Dφ)e2 where ei are the t-space basis vectors. Do I know this fact? Look first in Sja notes. Yes, it's all there. My Ch 10 is just falling apart in my hands right now. I started writing before I knew what I was doing! I thought I had it all down pat. Mar 2, 2016 Well, the bad news is that my Section 10.6 is a disaster. The good news is that I THINK I can describe the pullback idea completely in the language of transformations and tensor doc, and that is where it really belongs. The pullback is not some new thing out of the blue. We are just transforming a form from one space to another space. That really is all there is to it, now I just have to find a way to write that down. Will do this now in a new separate doc. // I tried but could not make it go today, and am out of time at 5:15 PM with run and Pub coming. Mar 3, 2016 I write |αx> in terms of |xei> in an expansion I don't have to write here. We know that these xei are in x-space and that there are m of them which form a basis there for Rm. There are also some |tej> in t-space, and there are only n of these. Are these two sets of basis vectors related in some way? If I think of Rm an add-on to the smaller Rn then I could claim that |xei> = |tej> for i = 1,2...n On the other hand, the rule v' = Rv for vectors translated to vx = R vt should say |xei> = | R tei> or xei = R tei which is a completely different claim. This led to a large dose on the subject of inverting non-square matrices, I have a new math section now on that topic, learned a lot, hope to bring it to bear on the above problem tomorrow. Today I did 6 handwritten pages trying to lay out a plane for my |αx> derivation, it is still fighting me. Mar 5, 2016 Yesterday I think I detangled the web and today I will try to write it up. I am reviewing Chapter 10 in the original version first. // Got sidetracked on my frames doc since a Poland email came in. // Finished that. Mar 6, 2016 Will attach Chapter 10 once more with gains from 2 days ago. I just carefully reread sections 10.1 through 10.5 and am ready to roll on a new Section 10.6 on pullbacks using my "latest and greatest". // I did in fact get a good start and decided to diffuse also into a Section 10.7. Keep adding more theorems to wedge doc! Had to give up on the RT notation which is just as well. Now 9 PM, signing off. te1 xe1 r1 = φ*T(xe1) Mar 7, 2016 Continuing on Section 10.6, it got heavily edited today. I think that the xei of this chapter which span the tangent space are the SAME as the tangent base vectors of tensor doc. I will try to prove that right here. In tensor doc I write in 7.18.1, (en)i = Sin = Rni This says that the en are the rows of R**. Ouch! Translating this statement to t and x space, you get (ten)i = Sin = Rni but this conflicts with my saying that (ten)i = δni In wedge doc this is true in two cases: (un)i = δni (en(e))i = δni OK, I have now to shut down all operations on Chapter 10.6 and "clarify" what I really mean by the basis vectors there in terms of tensor doc and its brief summary in wedge doc and the detailed wedge doc presentation on ambiguity of components. Right now this is a complete blur. Mar 8, 2016 Today I got Chapter 10.6 v1 doc into reasonable shape, showing what I want to show. I feel, however, that it lacks clarity and simplicity. There is some simpler way to understand the meaning of φ* which in some sense is just a matrix like R. Mar 9, 2016 I started into "is phi star symmetric" and ran aground with a linear algebra problem that I don't understand. I have some basic misunderstanding of linear algebra here. This problem has a long history with me going deep back into quantum mechanics and operators, I have encountered this confusion many times. I don't think it matters whether you use Dirac notation or vector/matrix notation. The confusion is present in any notation, but of course I don't know what that confusion really is. This problem really has nothing to do with wedge doc or differential forms. It is a modularized confusion within linear algebra itself. I have brought Chapter 10.6 to a stable form, showing all the things I wanted to show. However, I feel that some major piece of knowledge is missing which would make the section about 10x simpler if I knew what it was. For example I end up with what looks like φ*| vn,vn...vn> = | Rvn,Rvn...Rvn> . This is completely mysterious right now, but I know it will be very simple when I have it figured out. Today is March 9 and I have run out of time because I have to do the MRL taxes very soon and get it mailed to her and all that stuff. I have to buy the tax package, etc etc. So I am now shutting down wedge operations probably for 2-3 weeks. Then I can resume trying to detangle this mystery. A related problem is to get the transpose T operator more under control, with a covariant and matrix version looking the same and various ambiguities floating around in various docs. Another related problem is all of tensor doc when n ≠ m. Also I have to update tensor doc for the many errata I have found. So shutting down for a while. Mar 13, 2016 Finished the MRL and PHL taxes and they are all shipped off to the main PO. Took I guess 4 days. Today I resumed on wedge doc work. I had a go-round with the names of xun versus xen : changed then restored so my current working file is Section 10_6 v3. I have this continuing confusion with φ* = R, and that is what I am trying to resolve, even going to a separate "paradox document" as I do when the going gets tough. Worked on this all day to 3:30 PM and basically got nowhere, but did clarity the 10_6 writeup. I will continue on this post run or walk. Mar 17, 2016 Got distracted by this paradox thing, ended up writing a new Section 7.19 in tensor doc (all installed and done). I reviewed some of my paradox docs and think things are now resolved. I then finally am back to trying to edit on Section 10_6 v3 . I am at this point, (xui) = R(tui) or (xui)j = Σa=1n Rja (tui)a = Σa=1n Rja δia = Rji i = 1,2...n j = 1,2..m . (10.6.8) Question: is this correct or not? My new detailed data in 10.6 shows that xui plays the role of u'a and the above is basically u'a= Rua u'(e')a= R(e',u)u(u)a = Rij u(u)a // from tensor doc (7.19.25) so I think the above equations are good as I state them in (10.6.8) . I now pause in Section 10_6 v3.doc and transfer to "paradox v1 3_14_16.doc" just to review the paradox and how I resolved it. I want to make sure I don't get caught again! Mar 20, 2016 I think I am starting to win now on Section 10.6. As of 11AM I have a first cut rewrite of this section and am about to start Section 10.7. // At 3 PM I have gotten through 10.7 and it is all working. That was a very long digression from when I first reached that point maybe 12 days ago. Next I want to go back to Chapter 2 and do cleanups and additions, because it is now out of date relative to tensor doc. Chapter 2 review OK opening material 2.1 OK but I skip confusion from (2.1.15) to end of this section, will do that later. 2.2 OK except for red 2.3 OK, about the en basis vectors. 2.4 OK 2.5 OK 2.6 OK 2.7 OK 2.8 OK 2.9 OK 2.10 (a) OK (b) OK 2.11 (a) OK (b) OK (c) OK , but holding below (2.11.d.6) because we have the T and T issue ****** Here is something that I seem to have forgotten (it appears in Section 2.9 and appeared earlier when I wrote all the 16 equations:) em = Rmiui = a "vector transformation" Why is this not in my kinematics package? And what are its implications for Section 10.6 ? Maybe this simplifies something somewhere. And I never completed my comment about "component transformation". Mar 21, 2016 6 PM and have been unable to start work! Maybe I can clear up the item mentioned just above. In Picture A I have the data block all listed. The above says |em> = Rmi|ui> = | ui> <ui|em> = | ui> Rmi and there it is. I added a whole Section 2.11 (g) on the covariant transform stuff, then I can quote it later. I added pieces to Section 10.6 v4 relating to S and R and to vector versus component equations and also added all the completeness statements. Those loose ends are now done. Tomorrow I will proof 10.6 v4 and see what I think. Trying to get all the pieces of wedge doc to wedge together properly. Mar 22, 2016 Proofing 10.6 v4 and now I am ready to face up to this question which I put inline there: Question: What now happens to the equations quoted above RRT = RTR = 1 SST = STS = 1 RS = SR = 1 RT = R-1 = S ST = S-1 = R . (2.11.g.3) (10.6.1) when R is no longer square? (1) The chain rule says: Σb=1n = δac or Σb=1n Rab Sbc = δac or RS = 1m Σb=1m = δac or Σb=1n Sab Rbc = δac or SR = 1n so it seems that RS = 1m and SR = 1n so don't write SR = RS = 1 since that mixes the two 1's. (2) Writing RS = 1m implies that S = (R-1)right and R = (S-1)left. Writing SR = 1n implies that S = (R-1)left and R = (S-1)right Therefore S = (R-1)right = (R-1)left = R-1 R = (S-1)right = (S-1)left = S-1 and the inverses are well-defined as two-sided inverses. (3) What about S = RT which is the little reverse tilt rule. This is (7.5.13) in tensor doc. I read through the proof. It requires that gaa' = Saa" Sa'b" g'a"b" which says g is a rank-2 tensor even though spaces have different dimension. How much of tensor doc is still valid when m > n ??? I have no idea really! I would have to read through the whole thing! So I guess I will go digress on this subject for a while. (4) Consider RRT = 1 ? This is the orthogonality relation. Are the orthog relations still true in the non-square context? Write this one out and see: Rab(RT)bc = Rab Rcb = Rab Sbc = (RS)ac = δac RRT = 1 so I think this is all true. ********** all systems stop *********** Maple tells me that det(AAT) = 0 for any non-square matrix A, but I am unable to prove this with all the tools I have! It is not 0 for a square matrix. Where do I already have notes on such matrices? Some notes in the non-square matrix theorem folder, but nothing on this. Mar 24, 2016 Yesterday I did something long deferred. I went through the first 6 chapters of tensor doc (the development chapters) and studied what ideas survive and what ideas do not survive if the R matrix is tall (non square with more rows than columns). In a nutshell, here is what I found: These Objects do Not Exist (DN) These Objects do Not Exist (SN) 'mn en En g'mn en en Sij Sij = Rji Vi In summary, almost all of tensor doc is "disabled" in this case. The following objects still exist Rij g'mn Vi gmn , gmn un e'n u'n un e'n u'n Then today I made v5 and v6 of Section 10.6.doc and I think enough of tensor doc survives to make the development go through regarding differential forms and the pullback notion. Yesterday also I tried to find an n x n R-matrix relating x-space to Tx'M, in order to reactivate all of tensor doc for this small but square R matrix "Applying tensor doc to differential forms.doc" . I was unable to make this go. Before I follow through on that in v6 edit, I want to try to make an enlarged m x m R matrix and rescue tensor doc in THAT manner instead. I am motivated by the fact that I think the Bucks do this from time to time in their treatment of transformations, sort of adding artificial dimensions. So I will not attempt that. Well I did that in "Applying tensor doc to differential forms Plan B" doc and it went OK, but I don't see any motivation to use this method in Section 10.6 on pullbacks. But it did seem to reactivate tensor doc if that was the goal. I then started editing Section 10.6 v6 with the following point of view: avoid anything that is disabled in tensor doc for a non-square tall R matrix! I spent a lot of hours today editing the start of this document to bend it into the shape I want. It went pretty well, but then I ran into a paradox: something is wrong with doing matrix elements and expansions of rank-2 tensors in Dirac notation! I want to say <e'i | R | uj> = Rij = [R(e',u)]ij where R = Σab [R(e',u)]ab | e'a> | ub> but you can see that this does not work. What you really need is this R = Σab [R(e',u)]ab | e'a> < ub| so this operator R has basis vectors where one is a normal basis vector and the other is a dual basis vector. I have never even considered such an animal! But let's try it our right here <e'i | R | uj> = Σab [R(e',u)]ab <e'i | e'a> < ub| uj> = [R(e',u)]ij How would this work for a regular rank-2 tensor M? M = Σab [M(u,u)]ab | ua> < ub| But this is not what I have written in wedge doc! This is some special Dirac space M operator??? So this is a major issue and I have to stop editing in 10.6 v6. Something is confused in the earlier machinery! // Took a break and am back, maybe not so bad as I thought. (1) Recall from tensor doc that <a | M | b> ≡ <a | M b> = a (Mb) = ai(Mb)i = ai[ Mijbj] = ai Mij bj = bj Mij ai = bj (MT)ji ai = bj [MTa]j = b (MTa) = <b| MTa> = <b| MT |a> . (7.9.17) and we can always put into simple matrix/vector notation. There can be no Deep Mysteries here. (2) Near (E.5.2) I have the dyadic notation and I write A = Σij αij bibjT = a matrix where bi are some basis vectors. In Dirac you would write this as A = Σij αij | bi><bj| and that is what I am looking for. But how do you compare this with earlier in Appendix E, A = Σijαij (bibj) (E.2.11) A = Σijαij |bi> |bj> (E.2.11) This A is a vector in a tensor product space. It is not a matrix. Another example A = ΣijAij (uiuj.) Aijk... = covariant components of A in x-space A = ΣijAij |ui> |uj> I refer often to this last A as a "tensor" without specifying a basis. A is a rank-2 tensor, and the numbers Aij are the covariant components of this tensor in the ui basis. Look now more carefully at what is said in Appendix E. I write in (E.3.3) that A = dyadic = (ab) ≡ ab = a tensor product space thing (E.3.3) Aij = dyadic matrix element = (ab)ij ≡ aibj = ai (bT)j = (abT)ij (E.5.1) Are these compatible? On the first line I would write Aij = (ab)ij = [ab]ij = aibj so yes, things are in fact compatible. Looking at the above two lines, it seems that you could say A = (abT) but that is only valid in the sense of a matrix, that is, each side of this last equation is a matrix. But in A = ab the object A is a tensor product of two vectors, it is not a matrix. So my notation is weak. In the Dirac world, look again at <a | M | b> ≡ <a | M b> = a (Mb) = ai(Mb)i = ai[ Mijbj] = ai Mij bj = bj Mij ai = bj (MT)ji ai = bj [MTa]j = b (MTa) = <b| MTa> = <b| MT |a> . (7.9.17) Here M is really a matrix, it is not a direct product of vectors in some tensor product space. OK, I have now cleared this up and added the explanation to the end of tensor doc Section E.5, so no equation numbers were changed. This was a good fix up , and the dyadic notation ab really is ambigious. Will resume editing Section 10.6 v6 manana. Mar 24, 2016 Today I have "simulated" in Section 10.6 v7 how Section 10.6 would flow if I fully implement the elbow room idea. I certainly adds a lot of complexity. I somehow suspect it is not necessary, and the way I did it is probably too artificial to be useful. Will ponder tomorrow. Mar 25, 2016 Today's plan. I think the simple non-elbow-room world will fly after all. I hope to show this by studying a simple example and maybe I can show that S exists after all so the basic plan for a pullback will go through as I had it in earlier versions. Not sure how this will come out. Bucks often do isolated examples with transformations, but I want to make my huge systematic machine fly. Results: (1) I did a simple example (see non-square matrix folder) and I was able to compute S and I found explicitly that SR = 1 and RS ≠ 1. (2) I had to re-understand my "chain rule identity" claim and it is in fact not true in the non-square case where the smaller sum is involved, see "summary" at the end of the doc "confusion about the chain rule". It is always valid for the full rank square situation. Therefore, I can now go back to an earlier version of Chapter 10.6 ! v7 where I did the huge elbow room embedding thing, fascinating but not useful v6 where I saw that S and a lot of other things don't exist, when they do exist But I did more examples, and learned more things. In v6 where I say "does not exist" I should say instead "is not unique". I then looked into which "pullbacks" are unique and which are not. It varies depending on whether the basis vectors are up-label or down-label, and whether they are covariant or contravariant. I was plugging away again in the "rewrite" of tensor doc for non-square R matrix, but did not finish that task. that would be a good task to finish I think, manana! Today's examples were VERY helpful. Mar 28, 2016 Several work days have passed and I finally have Section 10.6 v8 where the pieces seem to be in place and where my various doubts are assuaged. So today I will proof the earlier sections of Ch 10 (with edits) to see if 10.6 fits properly into the puzzle. I want x-space to be Rm in Section 10.1. Section 10.1 OK Decided to maintain hats on dx^I so some comments removed. Section 10.2 OK Uses same xei notation for tangent space vectors, but later I will be using xui . NB. Section 10.3 OK derivatives of forms Here we have a few calculations, replaced λI by λ^I. Section 10.4 OK commutation of forms Section 10.5 OK commutation of forms Closed, exact, Poincare, angle form So I am now read up on what goes before my new Sections 10.6 an onward. Section 10.6 opening OK (a) OK with blue provisionals (b) OK and an excellent section with those two examples IMHO (c) OK, an eye opener on non-square matrix theorems, very good and worth including (d) OK, so the only complication of non-square is non-uniqueness of certain package items (e) OK, did a little repair work here. Section 10.7 OK, it has held together! Section 10.8 OK, just my translation from x'x to xt, I give ample details! Section 10.9 opening section OK (a) first path: OK (b) second path OK This was a very good edit pass on all of the above. In what doc did I make my Spivak references to the pullback? // OK, I have reconciled at the end with both Sjamaar and Spivak. BUT I have forgotten to prove the basic properties of φ* I need to add that somewhere. Started into rough work on this in a scratch doc. φ ψ ψ o φ Λk(Vt) Λk(Vx) Λk(Vy) Mar 29, 2016 Made the Second Path to the tensor function pullback equation VERY much simpler. Added useful new equations to Ch 5,6,7,8 for operators acting on product spaces Proved all the basic properties of φ* . I notice that I am missing something of the form dy = det dx, i am sure it is hidden in my stuff somewhere and I should make it be explicit. Want reader to recognize all his old friend equations. Mar 30, 2016 Did more theorems in Appendix A to nail down floating ideas that keep distorting away from reality. You have to always nail down any little piece you can because there are thousands of pieces. I see now that I am going to need to really revamp Section 10.9 more, Things are in a bad ordering, and I now see important pieces that I omitted (but which are right there in Sjamaar). Getting those App A theorems nailed down I think is a great help. I also see that I have to face the "measure ATA" business that Sjamaar brings up. When I try to write calculus differentials like dxdy in sphericals I get confused. This is something "not nailed down" so it just flaps around in the wind and causes trouble. Mar 31, 2016 Started version 10 and rearranging things. Lost and Found Department Somewhere I commented on the distinction between a Dirac operator M and a matrix M, but today I cannot find that comment. I looked for a few hours, while also evaluating search tools. My conclusion is that if you want to locate something like M in a file (special font), you can only do that within Word on a specific file. If you want to search multiple files, maybe a third-party package can do it, but none are free. You could write a VBA program I made a little test file and looked at it in hex, but the formatting is separate from the text no doubt with pointers, so text search can never find such a thing. So OK, what exactly would I like to say on the subject of Dirac operators? <a | M | b> ≡ <a | M b> = a (Mb) = ai(Mb)i = ai[ Mijbj] = ai Mij bj = bj Mij ai = bj (MT)ji ai = bj [MTa]j = b (MTa) = <b| MTa> = <b| MT |a> . So that would mean v' = (Mv) = (M)v (v')T = (Mv)T = vT MT |v'> = |Mv> = M|v> <v'| = <Mv| = <v|MT . (2.11.g.4) What am I trying to say here? M is a matrix which acts on vector v , but not clear what the basis is if you do that. Axis-aligned basis is always a good default, which is the Chapter 2 u basis. Then [Mv]i = Mijvj When you write v' = (Mv) = (M)v |v'> = |Mv> = M|v> what are you trying to say? <ui| M| uj> = Mij M is an operator which can act to the right on a vector, and to the left on a dual vector. It is not a matrix. It is a bidirectional operator in a Hilbert Space. Where in wedge doc do I show Dirac operators? // OK, a whole day of hours has passed, I have written it all up in a much improved fashion in a new section which is 2.11 (h). So what about scripting all Dirac operators in wedge doc? The first occur in Section 2.11 (h) which I just wrote. Let's scan after that point. Nothing through end of Section 6. All references are only to vectors! But now first occur in my recent section 7.9 (d) . I could repair that as follows P [ | (T1)^> | (T2)^> ... | (T2)^> ] = P | (T1)^> P | (T2)^> ... P | (T2)^> . (7.9.d.14) In other words, the action of P on the larger space is defined in terms of its action on the spaces which make up the tensor product. This result holds as well for the wedge product of N tensors, P [ | (T1)^> ^ | (T2)^> ^ ... ^ | (T2)^> ] = P | (T1)^> ^ P | (T2)^> ^ ... ^ P | (T2)^> (7.9.d.15) Proof: P [ | (T1)^> ^ | (T2)^> ^ ... ^ | (T2)^> ] = P [Alt ( | (T1)^> | (T2)^> ... | (T2)^> ) ] = Alt [P ( | (T1)^> | (T2)^> ... | (T2)^> ) ] = Alt [ ( P | (T1)^> P | (T2)^> ... P | (T2)^> ) ] = P | (T1)^> ^ P | (T2)^> ^ ... ^ P | (T2)^> . Nothing to it really. Same fix in Section 8.90 (d) ending. Nothing else in the existing doc before Chapter 9 added. so really no changes to make but the two just noted. How about Chapter 10? Sections 10.1 thru 10.5 have no use! So I will now start editing v10 with script for Dirac ops! Editing 10.6 (a) done 10.6 (b) done, no changes 10.6 (c) done, no changes 10.6 (d) done, no changes 10.6 (e3) done, one tiny change 10.7 done 10.8 done 10.9 got started April 1, 2016 I have worked all day, it is now 3 PM, and conclude that my entire Chapter 10 needs a complete reordering, it is just out of logical order very badly. In section 10.2 I am talking about forms on manifolds, but I have not said where this surface is coming from, I have not said that R is non-square, it is ridiculous. We need a full rewrite here from the git go. Confusion: In Section 2.11, why do I say λi = <ei| and λi(v) = <ei|v> = vi? I have just explained earlier in Chapter 2 how these are the tangent base vectors and not the axis-aligned vectors. And later in that section I am using |V> = Σa Va |ea> where the contravariant components really do go with the tangent base vectors e. Then around (2.11.e.13) I am depending on the fact that λi = <ei| again. Something is amiss here, better put Chapter 10 on hold and clear up this mess in Chapter 2. I am now back to the famous Section 2.11 which has now just malfunctioned badly. April 4, 2016 I think I have figured out the confusions above, and have been rewriting Chapter 2's sections 2 -7 rather heavily. I complete got rid of Picture F and all those (u) and (e) labels. I realize now that I don't need "two conventions" for the meaning of an index, I can just add primes in the expansions of later chapters on the components. My plan is to completely restabilize all of Chapter 2, and then see if it then satisfies all the needs of the rest of the doc. Right now it is half stabilized. I have gotten rid of the "paradox" I originally stated to motivate the reader to read Chapter 2. In the end, it just is not very interesting and is distracting and seems less significant now that (u) and (e) stuff is gone and I just stick with x-space and x'-space. As of 9:25 AM I am good through the start of Section 2.8. That was the major change area I think, and I now continue on. Section 2.8: changed eq nums, removed references to (e) and (u) stuff, a bit shorter now, all OK. Section 2.9 is OK, small edits. Section 2.10 will require more cosmetic changes now that (e) and (u) are gone. // It is done, and I have done various repairs, mainly getting the prime on the expansions coefficients. We are now ready to launch into the dual space Section 2.11 where I expect some problems. Section 2.11 (a) OK (b) OK (c) slowing down above 2.11.c.2 . I need to change this to basis-aligned only!!! Otherwise λi(v) = <ei|v> = vi is incorrect using my now-standardized single-convention notation. I could say λi(v) = <ei|v> = v'i but I don't think that will help. Decisions here have major impact later since e is everywhere! Maybe skip ahead and look to see what this will mean, say chapter. Ouch!! Go copy out Chapter 4 and edit it changing e→u everywhere and see what that looks like. WOW, it is just impossible to do. I really am going to have to change conventions and state clearly that I am doing so. // But I battled through 4.1 anyway, maintaining the single convention. Ouch! Let's now continue into 4.2. I am flip-flopping on whether to change conventions! April 5, 2016 Wrote some notes called "the basis vector conundrum" as if a reader critic were commenting on my wedge paper and its notation. I conclude that for chapters 4.5.6.7.8 I really have two options: 1) continue to use ei , but say that they really mean ui. 2) change all the ei to ui The third option of trying to make the ei be tangent base vectors has not lost its appeal to me. I am trying out option 2 above in sandbox wedge v2.doc on Chapter 4 only. This chapter should encounter all structures in all "four worlds. I have done the edits, and I will now proof the new Chapter 4 and see what I think! Chapter 4 opening material : OK, the preview of the four sections to come 4.1 OK. I deleted one set of comments and replaced it with another, and deleted references to T(e) in two places. But generally it "reads OK" I would humbly say. 4.2 OK, one deletion, in the dual world not so much changes 4.3 OK, no changes 4.4 OK Conclusion: Changing all ei to ui is the right thing to do. I don't want to do this e and u defined carefully in Chapter 2 "e" stands for u in Chapters 4.5.6.7.8 (so I can avoid all the edits). e is restored to its Chapter 2 meaning in Chapter 10. Hopefully I am seeing some sunlight now! With this in mind, I want to go back to Chapter 2 again! Now is 2 PM. // At 3 PM I reached the start of 2.5 and am holding for a real world break. Section 2.11 (a) OK (b) OK (c) OK, did lots of changes from e to u and v'i to vi. (d) getting stuck here on my active vs passive pictures and how they relate to anything! April 6, 2016 Working now on the active/passive interpretation. Decided to add a picture back in Section 2.7 instead of having it in Section 2.11. I think I have something wrong then in 2.11. I look in tensor doc, and my explanation there seems wrong too! So let's back up and maybe edit the tensor doc text!! // Well, that was not needed. I have removed the transformation section from 2.11 and installed it earlier in section 2.7 on transformations. So should section (d) in 2.11 be deleted? I will start again reading 2.11 Section 2.11 (a) OK (b) OK (c) OK, did lots of changes from e to u and v'i to vi. (d) I have removed most of this section on xforms but some questions remain Next Problem: I write two equations, both seemingly on the same footing: λi(v) = <ui|v> = vi = ui v λi = <ui (2.11.c.5) α(v) = α v compare to Vn = un V = u'n V' // (2.7.1) or (2.2.6) and the evil question returns once again: are these things scalars or vectors? Despite all my efforts today and new stuff added in Section 2.7, the question still confuses me. The question seems inextinguishable, it never goes away. and this really is the component of a vector, not a scalar, and I have dealt with this issue earlier with my new piece on the three experiments. There I talked about Vn = un V = u'n V' // (2.7.1) or (2.2.6) looking like a scalar, and I think that exact same issue arises here. New approach, have to test every single word in every sentence, it is super subtle. Confusion about vectors and scalars Consider this fact: un V = u'n V' Being the dot product of two vectors, this object transforms as a scalar under x' = F(x) as shown in (2.2.6). We might express this fact by writing s(n) = un V s'(n) = u'n V' s(n) = s'(n) n = 1,2....N . What we have here is a set of N scalars, s(n) for n = 1,2..N, where n is just a label. One would never claim that these scalars s(n) transform as a vector, which would require that s'(n) = Rnms(m). We don't have such a relation; what we have is s'(n) = s(n). Now it happens that s(n) = s'(n) = Vn where Vn is the component of a vector. Notice that : V'n = RnmVm true s'(n) = Rnms(m) false; what is true is s'(n) = s(n) s(n) = Vn s'(n) = V'n false The point is this: just because the scalars s(n) take the values Vn does not mean that the scalars s(n) transform as vectors, nor does it mean that the vector vn transforms as a scalar. ***** It happens that the value of this scalar is s = s' = Vn . This is just some real number like π. The fact that the scalar s = s' has the value Vn does not imply that the components Vn of the vector V transform as a scalar under x' = F(x) . In fact V'n = RnmVm and these components transform as a contravariant vector. April 7, 2016 Not happy with vector vs scalar, and cannot make the Section 2.11 stuff on λi work right. Starting on new doc now called "the lambda i puzzle" and will work there for a while today. I did some work on the doc and came up with some very good reasons to define λ'i = <e'i| and I now have a new version of Section 2.11 (c) where I was having problems. Try resuming the edit sequence Section 2.11 (a) OK (b) OK (c) OK, did lots of changes from e to u and v'i to vi. (d) gone (e) OK, replaced all e with u here, as already done in Chapter 4 // "replaces" done, read done (f) OK after changing e to u Stop at (2.11.g.5) . I think I have updated this with <a | M | b>, so I have stale text here. Go find that updated text now. I have been editing in v3 which I thought was the latest. Update is not in v2, and it is not in blank version. Not in wedge v2. Found it! It is in Dirac operators. OK, installed (g) and (h). Resume the above Section 2.11 (a) OK (b) OK (c) OK, did lots of changes from e to u and v'i to vi. (d) gone (e) OK, replaced all e with u here, as already done in Chapter 4 // "replaces" done, read done (f) OK after changing e to u (g) OK (h) OK So at 12:45 I have finally gotten through Chapter 2. But I now have to adjust for the missing section. To do this complex editing process, I will start in this way replace all .d. by D there were none since old (d) deleted replace all .e. by .E. in entire Chapter 2 18 replacements made replace all .f. by .F. 18 replacements made replace all 11.g. by 11.G. 4 replacements made replace all 11.h. by 11.H. 20 replacements made Next comes the back-up edit sequence replace all 11.E. by 11.d. in entire Chapter 2 18 replacements made replace all 11.F. by 11.e. in entire Chapter 2 18 replacements made replace all 11.G. by 11.f. in entire Chapter 2 4 replacements made replace all 11.H. by 11.g. in entire Chapter 2 20 replacements made Should be all done now. Big Question: Here is my current Section 2.11 contents where each subsection has really ugly equation numbers like 2.11.d 17. Question: Should I redo this as Sections 2.11 through 2.17 ?? I will try that in a scratch area right now and see what I think. Wait. This loses parallelism with Section 2.10 on the expansions. And it loses the idea that all this new stuff is related to Dual spaces. Question: is there some simpler way to do the equation numbers? (2.11.e.12) (2.11e.12) (2.11(e).12) (e.12) (2.11.e12) Perhaps this last one (2.11.e12) (2.11.e13) Let's try that in scratch and see what it looks like. A single edit does it I think A set of edits to do this replace 11.a. with 11.a replace 11.b. with 11.b replace 11.c. with 11.c replace 11.d. with 11.d replace 11.e. with 11.e replace 11.f. with 11.f replace 11.g. with 11.g Now read through and see what you think. // I like it much better. (2.11.g19) (2.11.h7) There are then only 3 "fields" for the reader to absorb, which is the same as everywhere else in the doc. Let's now review my edited copy of Chapter 4 to make SURE I am happy with what I am doing with e and u and all that stuff. // I see this is directly edited into wedge v2, so read it there. OK, I think this is the way it is going to be. Let's upgrade now to wedge v3 before doing the Ch 2 install and further edits. Done Now first install the new EQ nums at the end of the scratch Ch 2. Done. Now comes a long set of Chapter edits, replacing e by u. This will take a long time but it has to be done. Maybe in so doing I will find problems. Just do it in place in wedge v3. 5.1 done 5.2 done 5.3 done 5.4 done 5.5 done 5.6 done  That was not too bad. Now scan the chapter. Done, found a few missed. On to Ch 6: 6.1 and open are OK 6.2 already fixed for some reason! 6.3,4,5 already fixed 6.6 OK These was not much to do in Ch 6 because I use λ everywhere and this did not change. On to Ch 7: 7.1 OK 7.2 OK 7.3 OK 7.4 OK 7.5 OK 7.6 OK 7.7 OK 7.8 OK 7.9 OK, this was a very long section to edit! 8.1 OK 8.2 OK 8.3 OK 8.4 OK 8.5 OK 8.5 OK 8.6-9 OK not much in the dual chapter due to λ's Chapter 9: nothing Appendix A: OK, there were just a few way out in the middle of this appendix Appendix B: OK, left some generic e's in the middle, changed others Appendix C: OK Appendix D: OK Not done yet. Got through Chapter 10 v2 right now. Well this needs slow reading, I will hold off. So OK, what should I do next? I guess I have to nail down Chapter 10, rather do that that proof the entire monster again with all its u's in place of e's. No doubt I missed some. What about u's in Chapter 3? DONE, there was some stuff in there. April 8, 2016 Starting into Ch 10 review. This is Chapter 10 v2 and does not include 10.6, Chapter 10 10.1. Differential Forms Defined OK 10.2. Differential Forms on Manifolds OK, but I did a lot of editing here, using x' = F(x). 10.3. The exterior derivative of a differential form OK! 10.4. Commutation properties of differential forms OK 10.5. Closed and Exact, Poincaré and the Angle Form OK, I have derived some of the basis "standard facts" of differential forms and made linkage back to the earlier chapters of the book, both Chapter 2 for those tangent base vectors, and Chapter 8 on the wedge space activity. The next topic is pullbacks, and I have to now see which of my many Section 10.6 writeups I want to start with and then edit. Section 10.6 v 10 10.6 opening material, OK after lots of editing (a) OK, now there are no references to Tensor, it all comes from Chapter 2 ! (b) OK, doing lots of edits. (c) OK, non-square matrix stuff, few edits (d) OK, revised kinematics package (e) OK This doc 10.6 v 10 continues on with a section 10.7 which I will now review: Section 10.7 is not complete, this must be where I paused to go back and redo things in Chapter 2 with the e and u stuff. So we now slow down and get this section into some reasonable order. 6:30 PM. I am having trouble proving that R and F* are operator. How about start with k = 1: <s1α + s2β | R = <R[s1α + s2β] | = <s1Rα + s2Rβ | = s1<Rα | + s2<Rβ | = s1 <α | R + s2 <β | R I don't like my notation very much, and all my linearity proofs "beg the question". Manana. At least today I got up all the way into Section 10.7 without pitfalls. April 9, 2016 (1) Added more in Section 2.11 about linearity of the R operator, hope it is finally nailed down. (2) Continued work in Section 10.7, getting down all the "facts" about the pullback operator F*. I have stayed in "my own" tensor doc notation so far. Not sure if I will keep it this way. Need to do those little basis vector things dφi which I wrote up at the end of 10.6 doc. Then not sure where I am headed in 10.7. Need to get the torus pictures in there and all that. Well, PM session, I will reread Section 10.1-10.6 and then install all of it. Then comes 10.7/ 10.1 very excellent, OK 10.2 very excellent as well, OK 10.3 very excellent again, three in a row, OK 10.4 just fine, OK, on commutation properties. 10.5 very good, ready to install so OK 10.6 opening material OK (a) states the cloud of equations surrounding our xform,. OK (b) very good examples! It is OK. (c) linear algebra for non-square matrices, a true vignette! OK (d) the kin package mods for non-square R, OK (e) OK, about selection of basis vectors April 10, 2016 OK, I am going to install all this stuff right now into tensor doc v3, I need to move forward ! DONE! I continue in a separate doc with Section 10.7. Worked on this Section 10.7 and beyond, all day long. Not done! April 11, 2016 Revamping the later parts of yesterday's work. Once again I come across this loose end that needs attention soon: Task Assignment for Soon ******** In tensor doc, show how you compute partial measures like drdθ in spherical coordinates. This involves the det(ATA) thing that Sjamaar mentions. Since I will be updating tensor doc soon, this needs to be installed somewhere in tensor doc. It will be a jacobian-looking thing. It involves those nested cofactor things. I just don't want to stop right now and do this because it might take a week. But come back and do it!!! The application then will be this: I have this fact dFi ^ dFi = Σ1≤j<j≤n det dxj ^ dxj (10.8.8) where x'i = Fi(x) and I want to compare this to a "normal calculus result", dx'i dx'i = [ ?????? ]iijj dxjdxj2 where i = 1,2...n in Vn At 10 AM I am finished with first drafts of Sections 10.7 and 10.8. Let's proof these both right now before moving on. Everything is in the F notation, no φ yet. Proofing: 10.7 OK, I like it, had to made several edits for missing arguments 10.8 OK and just fine 10.9 also OK, this is the change from x,x' to t,x as a minimalist thing. I decided at this point to go through Sjamaar and seem which of his equations I can correlate. In doing this, I found a bug in my attempt to verify his (2.4), so this has shown that something I have somewhere is wrong, need to debug this problem right away, don't want some basic error floating around!! April 12, 2016 I think I finally resolved the dα cases that Sjamaar shows, and I have a general method, it took several hours. I will add this at the end of Section 10.3 on external derivatives. I did some rewriting of section 10.3 but I see now that I have more rewriting to go in 10.1 where the same issues come up. Things are in the wrong location here and there. // I have now rewritten both 10.1 and 10.3, both sections being quite short, renumbered it all. Now I want to expand 10.3 showing how to write dα in other ways. // Did this, did the three examples for k = 1, k = 2, k = 3, verified the first two results against Sjamaar. It is done. This was in Section 10.3. Breaking at 4:30 PM for a while this Tuesday. April 13, 2016 I "finished" sections 10.7 and 10.8 and 10.9 in separate doc and I have installed them as well as Appendix F. I think I now have several integration/repetition problems that need to be ironed out. My section 10.2 is about manifolds, but I have ignored material written in "clarify pullback" As of noon I have added more stuff here and there, such as motivation for the exterior derivative at the start of that section, were I thrown in Stokes and Hodge stuff all as motivation. At 4:30 PM I am moving on to Section 10.10 and I have written up the 1-form case and have started on the 2-form case. I have a few problems. Why don't I have an abs value around the derivative in the 1-form case the way Bucks so. Why does my 2-form result agree with Arapuro in Buck Ch 7 notes, but not with Buck page 368 (7-3) . And there is still the absolute value confusion here as well. Maybe you add abs value as part of the "second definition". There are several loose bolts floating around here, I am not surprised. April 14, 2016 I read some Buck notes, started into "A review of Buck curve and surface integration" and think I have clarified things. This led to a rewrite of section 10.10 (a) on the curve, and I now start again on the surface. I have made good progress by 8:15 PM on the two sections in 10.10, it is starting to make a lot more sense now! Still have to add the general k-form section. Then maybe wedge doc is complete. But I still have that ATA area business to deal with in tensor doc and maybe in wedge doc as well. April 17, 2016. I have been tuning up Section 10.10 and it seems pretty good now. But a major question remains not even addressed: Why do these k-form integrals reproduced the "classical" results for things like line integrals and surface integrals. As a start into this subject, I am first going to deal with the ATA measure stuff that Sjamaar mentioned and which cued me in to an addition to tensor doc. This is all happening now in "measure in Sja... doc". April 19, 2016. Took day off yesterday to pack for the Maze. Good time to review things. I think I have successfully described "technical aspects" of tensor products, wedge products, and differential forms. I have shown how anything can be "computed". But interpretation is weak or non-existent. Here are some questions that need to be answered. 1. How would someone use the "integral of a differential form" to do something practical? 2. Why would you want to "integrate a functional" over a surface? 3. How do you do "classical integration" over surfaces with no mention of differential forms. Need some kind of "measure" on the surface. 4. Is there a connection between classical integration and differential form integration? Or are they completely unrelated. I suspect they are related. In my "measure in Sja" I have made some small progress today. April 27, 2016 Back from the Maze, and nothing is working right. I am back to an old question: 5. How do you visualize a 1-form when you have R2 on the left and R3 on the right? This would be a case where k < n where k = 1 and n = 2. Back now to the meaning of the integral of a differential form. The highly mathematical Loring Tu book gets around to integration of forms and makes this definition, Here he is just integrating an n-form in Cartesian space. One should interpret this object ω = f(x) dx1 ^ dx2 .... ^ dxn as a functional yes. But then you "evaluate that functional" like this: ω(A) = ∫A ω = a real number It is a different kind of functional evaluation than what I call a tensor function. It is my "second definition". You see how he removes the wedge products with the ≡ (he writes as :=) definition above. What is the mapping here? ω : A → R The domain of this mapping is the space of regions of Rn. This is a new idea for me. Question: Who tells me that the differential form α is an element of the dual space Λk ? Did I just make this up? Sjamaar on page 83 does associate dxi with eiT and he calls this a constant 1-form. April 29, 2016. I think I am ready to move forward on integration, but first I am rereading Chapter 10 to this point, and I am thinking of removing the φ business. 10.1 OK and very good IMHO 10.2 Diff forms on manifold 10.3 Motivation = OK and good, External Derivative = STOP. Suppose λ^I is on a manifold in Rn. Then it would be x-dependent and then my argument in (10.3.8) is not correct because I treat the λ^I as constants. If they are functions of x, the theorem is not true, but then I suspect the definition of d is also not correct. I will ignore this little detail. For Rn it is a non-issue because we are not dealing there with a manifold. Definition of the Exterior Derivative OK but not great Expressing dα in standard form OK and good comparisons with Sjamaar 10.4 Commutation Properties -- OK, short and sweet 10.5. Closed and Exact, Poincaré and the Angle Form -- OK 10.6 Transformation Kinematics (a) Axis-Aligned Vectors and Tangent Base Vectors : The Kinematics Package -- OK (b) What happens for a non-square tall R matrix? -- OK (c) Some Linear Algebra for non-square matrices -- OK (d) Implications for the Kinematics Package -- OK (e) Basis vectors for the Tangent Space at point x' on M -- OK 10.7 The Pullback Operator R and properties of the Pullback Function F* -- OK action packed 10.8 Alternate ways to write the pullback of a k-form The ordered sum form of a k-form pullback -- OK The dFi form of a k-form pullback -- OK Summary all in cosmetic notation -- OK 10.9 A Change of Notation and Comparison with Sjamaar and Spivak [ I have to do this painful thing to get comparisons with Sjamaar!! ] opening text -- OK Drawings for the new spaces -- OK Summary all in cosmetic notation -- OK Comparison with Sjamaar -- OK, good in fact The tensor function pullback -- OK, and good and this concludes section 10.9 As usual, this review took about half a day, getting the big wheel in syrup rolling again. I have now started a brand new Section 10.10 which is talking about surface and line integrals without mentioning differential forms, sort of providing the Buck background. I think it is going well. April 30, 2016. I have now finished the new section 10.10 which discusses surface and curve integrations without reference to "differential forms" though I do mention "pull backs". Needs proofing before continuing. I will probably end up adding more later and incrementing equation numbers. Well I skipped the proofing for now and dove straight into section 10.11. May 2, 2016. Have now finished sections 10.10, 10.11, 10.12, 10.13 all on diff forms, may add that to the paper title. Maybe I am finally DONE! with the content of this monograph after many months. I want to review Buck and others and make sure I am OK and have not misstated things or omitted important things. Need to add the ATA measure stuff to tensor doc and maybe to wedge doc. May 4, 2016. The plan is this: proof the four sections noted above, install them, and THEN start into the RTR issue. Section 10.10 Well, it took me from 6 AM to 5:30 PM just to get this one section re-edited and to get the surface and line integral sections as parallel as possible. I just don't know why editing takes so long. Equation numbers constantly be shifted. Nothing is installed yet, heading off to the Pub soon. May 5, 2016. Starting today with proofing of Section 10.11. This did not take long, and I see no reason for major edits. I did repair references to Section 10.10. Section 10.12 review. But STOP. Equation numbers in 10.9 are screwed up! Go back now and proof that whole section and then make all forward corrections. // I managed to made an ad hoc fix to equation numbers, so no need to review at this time, so return to Section 10.12 review, Section 10.12 General Review of k-form integration OK Integration of a 1-forms OK Section 10.13 done It is now 3 PM and I am ready to install 10.10 through 10.13, finally!!! Done ! So now it is off to the RTR measure problem once again. May 6, 2016. Got a proof of the idea that the volume factor squared is RTR doing this in geometry. But I don't quite have the even simpler result that Sjamaar gets where his R is made of the basis vectors. I will continue on this tomorrow, but I do feel I got a piece of this puzzle at least written down. Sat May 7, 2016 I did a major edit on my new Appendix F, took several AM hours. Then added Section F.6 stating and proving a theorem I have been pondering. Then added Section F.7 showing how the measure relates to det(RIJ) stuff. Still no contact with measures in tensor doc. But then I made that contact! I started an edit of one section of tensor doc which I now see had an error in the use of "cof", so I will fix that error and add the STS stuff. This was the last missing piece on this general topic!! Sun May 8, 2016 Did edit on tensor doc section on "nested cofactors" and it is ready to install. Added notes there making connection to boundaries and Stokes and Sjamaar, added wedge doc and Sjamaar to tensor doc refs. Final gasp: I am trying to make a few more "connections" to items in Sjamaar volume form chapter including Hodge *dx and am stalled out doing this. A little extra detail I would like to add into my Appendix F just to have it there. Will resume on this after feeding. OK, I now have these few more connections in Section F .7 written up in a temp doc right now. However, I just ran into a snag: (1) My Hodge * sign rule seems to disagree with Sjamaar page 24 old notes. (2) This sign rule is messing up the end of my new Section F.7 since I cannot say dx^' = *dx' . So I will resolve this manana. I have never tested that sign rule much! Mon May 9, 2016 I repaired the snag and broke appendix F into App F and App G. I think I am really done! Time to proof this latest work Appendix F: The Volume of an n-piped embedded in Rm opening material OK F.1 Volume of a 2-piped in R3 OK F.2 Volume of a 2-piped in R4, R5 and Rm OK F.3 Volume of a 3-piped in R4 OK F.4 Volume of a n-piped in Rm OK F.5 Application: The differential volume element of the tangent space Tx'M OK This was a fast and effortless review. Continuing on: Appendix G : The det(RTR) theorem and its relation to differential forms G.1 Theorem: det(RTR) is the sum of the squares of the full-width minors of R OK // this proof is in fact very clean I think. Uses powerful earlier App A results. G.2 The Connection between Theorem F.6 and Differential Forms opening section before hypersurface case OK hypersurf section with Example OK So I'm Bernie Sanders and I approve of these two Appendices. Time to install these two appendices! // Installation complete. We are 325 pages. Starting Differential Forms Review of Chapter 10. Once again! 10. Differential Forms opening statement good 10.1. Differential Forms Defined OK, even very good, small edits OK 10.2. Differential Forms on Manifolds OK, I like it, informal and nonrigourous. OK 10.3. The exterior derivative of a differential form Motivation section is good, no eq nums. OK rest of section is good, general formula then examples. I do worry about my symmetric theorem when λi varies with x. 10.4. Commutation properties of differential forms short and sweet, OK 10.5. Closed and Exact, Poincaré and the Angle Form pending a lookup stopping at 8:30 PM for the day. Tues May 10, 2016 10.5. Closed and Exact, Poincaré and the Angle Form OK 10.6 Transformation Kinematics opening comments OK (a) Axis-Aligned Vectors and Tangent Base Vectors : The Kinematics Package OK ! (b) What happens for a non-square tall R matrix? OK and very good. (c) Some Linear Algebra for non-square matrices OK and excellent. (d) Implications for the Kinematics Package, just a catalog of what is unique, OK (e) Basis vectors for the Tangent Space at point x' on M OK, not bad 10.7 The Pullback Operator R and properties of the Pullback Function F* OK, a long section!! egg break 10.8 Alternate ways to write the pullback of a k-form The ordered sum form of a k-form pullback OK and good The dFi form of a k-form pullback OK, made improvements Summary all in cosmetic notation OK, grinding fine detail, gather it up 10.9 A Change of Notation and Comparison with Sjamaar and Spivak Drawings for the new spaces OK Summary all in cosmetic notation OK Comparison with Sjamaar OK The tensor function pullback OK, Spivak comparisons, should help somebody! 10.10 Integration of functions over surfaces and curves INTEGRATION OVER SURFACES Example 1 OK, I like it since no need to do a pullback. Example 2: OK Example 1a,2a OK Example 1b,2b OK INTEGRATION OVER CURVES Example 3 OK, I like it since no need to do a pullback. Example 4: OK Example 3a,3a OK Example 4b,4b OK Final comments are very important. This section 10.10 is brutal but I think good. It shows complexities that have nothing to do with "differential forms". 10.11 Integration of differential k-forms over Surfaces OK 10.12 Integration of 1-forms OK 10.13 Integration of 2-forms First definition: OK Second definition: OK Comment on orientation OK Further processing: OK Generalization from φ: R2→ R3 to φ: R2→ Rm OK Generalization to φ: Rn→ Rm with k = n OK Time 3:15, that was a long haul! Did a quick read of G.2 to make sure OK continuity from the ending of the above 10.13 where I give a reference to App F and G. Weds May 11, 2016 Look back through Ch 10 and deal with "red markings". Only marking concerns one ref and the Spivak mapping picture. // Done, it was easy to add, and I added it in my manifold description section. Next: do a proofing of Appendix D which I have not looked at for a long time! Appendix D: A Unified View of Tensors and Tensor Functions D.1 Tensor functions in Dirac notation OK D.2 Basis change matrix OK, did various edits here D.3 Transformations of tensors and tensor functions OK, I like the negative discussion D.4 Tensor Functions and Quantum Mechanics OK, will put no refs for QM, but checked <x|ψ> Scan all of D for eq num seq. Section D.1 is messed up. Global fix. Replace (D.1.n) by (D.1.n-2) for n = 5 and up. DONE. Next: do a proofing of Appendix B which I have ignored for a long time Appendix B: Direct Sum of Vector Spaces 1 OK 2 OK 3 OK 4 OK 5 OK 6 OK 7 OK 8 OK 9 OK This is in fact a nice little gem of an appendix on an isolated subject, I had forgotten about it. The relevance is in the direct sum of vector spaces to get the tensor algebras. No need to proof this item ever again, it is ready to ship! Scan all of B for eq num seq. DONE and OK. Now a quick pass on Appendix E since related to Chapter 10. Appendix E: Kinematics Package with x' = F(x) changed to x = φ(t) OK I checked all eq nums, got things aligned, etc. No further review necessary. Status of Review Process 5-11-16 (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs I guess it is time for Appendix A which will be a long haul. But I first looked at the Overview and did some updating there. It now seems OK, but I may change it later, so Overview Status of Review Process 5-11-16 11:15AM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Notation Status of Review Process 5-11-16 12:20PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs References Status of Review Process 5-11-16 3:05PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Appendix A review summary OK A.1 Rearrangement Theorems and a Determinant Theorem OK, lots of meat here. A.2 The Alt Operator OK, a long list of boring stuff! Just the facts, Ma'am. A.3 The Sym Operator -- copy paste edit of previous, read it all, proofs too, OK A.4 More on Alt and Sym and the decomposition of functions OK, painful bug good. opening overview OK (a) Alt Equations (translated from Section A.2 above) OK (b) Sym Equations (translated from Section A.3 above) OK (c) Alt/Sym and Other Equations (translated from Section A.4 above) OK A.5 Application to Tensors OK (a) Alt Equations (translated from Section A.2 above) OK (b) Sym Equations (translated from Section A.3 above) OK (c) Alt/Sym and Other Equations (translated from Section A.4 above) OK A.6 The permutation tensor ε OK A.7 The wedge-product-of-vectors Alt equation OK A.8 Application to Tensor Functions opening material OK (a) Alt Equations (translated from Section A.2 above) OK (b) Sym Equations (translated from Section A.3 above) OK (c) Alt/Sym and Other Equations (translated from Section A.4,A.6 and A.7 above) In the process above, I have decided to remove the following items from Appendix A: _______ Suppose now that function f has k arguments and function g has k' arguments. We then define a meaning for the tensor product in our generic function space as follows Definition: Tensor product: (fg)(1,2,....k+k') ≡ f(1,2...k) g(k+1,k+2....k+k') (A.2.19) _______ Definition: Tensor product: (TS)ii...i ≡ Tii...i Sii...i, where the ranks of tensors T,S are k,k'. (A.2.19) (A.5.22) _______ Definition: Tensor product: (TS)(vi,vi....vi) ≡ T(vi,vi....vi) S(vi,vi....vi), where the ranks of tensors T,S are k,k'. (A.2.19) dupe (A.8.22) But I need A.2.19 near C.1.19 ouch! This needs work. Can I do the tensor product generically and then painfully show that it is right in the main body chapters? This needs pondering, time now 6 PM, so things on hold for a while. OK as of 9PM I have implemented a plan to deal with the above and I can tomorrow continue with proofing of Appendix A where I left off above. Thurs May 12, 2016 Getting rid of (A.2.19), new name is (A.10.1) in a new section. Did single cross ref update. Start over on Appendix A, faster reading this time A.1 Rearrangement Theorems and Determinants STOP. Snag encountered in discussion of det(MT) at the end of App A.1. I decided to add my proof in Appendix A and this results in the following global adjustment (A.1.20) → (A.1.24) OK, all cleaned up. So start again on Appendix A. A.1 OK A.2 OK A.3 OK A.4 OK A.5 OK A.6 OK, ends with seemingly irrelevant statement A.7 OK A.8 OK, very long, lots of Facts, a bit dizzying A.9 OK, this is my old ordered sum theorem which has not changed in a long time A.10 OK Now scan all of Appendix A and confirm good eq num sequences: DONE, OK after two bugs below fixed: Bug: There are two (A.5.12)'s. If I renumber after, then I have to do this global change Change replace (A.5.n) → (A.5.n+1) for n = 12 and higher, but check for n = 12 carefully! DONE. Bug: There are two (A.8.12)'s. If I renumber after, then I have to do this global change Change replace (A.8.n) → (A.8.n+1) for n = 12 and higher, but check for n = 12 carefully! DONE, and there were many to do! Finally I have finished proofing this long 31 page appendix that is jam-packed with theorems! So here is my status: Status of Review Process 5-12-16 2:10PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Appendix C Review opening text OK pending C.1 Theorem One. OK, did lots of edits however! C.2 Theorem Two, OK but again lots of edits. Sets to lists, etc. C.3 Theorem Three, OK. C.4 Summary OK Check eq num seq for all of C: Goof up after (C.4.12) . So Bug Change replace (C.4.n) → (C.4.n-3) for n = 16 and higher. DONE, only a few hits. New status: Status of Review Process 5-12-16 4:45PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Mow lawn and back at 7 PM. Note: I am NOT checking Tensor eq nums in this pass. Chapter 2 Review Opening text: OK 2.1 R, S and how tensors transform : Picture A OK. This is VERY good, and I doubt I will ever write a better review of this subject. 2.2 The metric tensors g and g' and the dot product. OK and very good. 2.3 The basis vectors en and en OK, still good 2.4 The basis vectors un and un OK and very good 2.5 The basis vectors e'n and u'n and a summary OK 2.6 How to compute a viable x' = F(x) from a set of constant basis vectors en OK, but have to fix the code!! ****** so g is replaced by gp = g' DONE 5.13 Bugs encountered below (2.7.3) !!! Global change: replace (2.7.n) → (2.7.n-4) for n = 8 and higher. DONE. Continuing after fix: 2.7 Expansions of vectors onto basis vectors OK and pretty good. 2.8 continue manana Please review or make the Maple/Visio index for wedge doc! Then I can fix the above code! Fri May 13, 2016 detail: page tensor doc extra 1.vsd has a certain STS composite equation which I think goes into my little tensor doc update on nested cofactors, but I cannot find that update right now *********** I will handle that when I do the tensor doc update. Just build the Maple and Visio and other-books-referenced index, took several hours. Fixed g to gp in the code for section 2.7. Resuming now review of Chapter 2, 2.8 The Outer Product of Tensors and Use of OK 2.9 The Inner Product (Contraction) of Tensors 2.10 Tensor Expansions (a) Rank-2 Tensor Expansion and Projection OK (b) Rank-k Tensor Expansions and Projections OK Check all Ch 2 eq num seq to this point: Done, found and fixed two problems. So continue now. 2.11 Dual Spaces and Tensor Functions opening material OK (a) The Dual Space V* in Matrix and Dirac Notation OK (b) Functional notation OK (c) Basis vectors for the dual space V* OK, long but good I think (d) Rank-2 functionals and tensor functions OK (e) Rank-k functionals and tensor functions OK (f) The Covariant Transpose OK (g) Linear Dirac Space Operators Pause to consider claim I make: (aTM)b M acts to the left, and note that (aTM) = (MTa)T [ (* * *) ] How can this possibly be, since in the regular matrix sense you put MT in there?? [(MTa)T]i = (MTa)i = (MT)ijaj = Mjiaj (aTM)i = [ajMji] = Mjiaj OK, I added a "footnote" showing how this works. It has to do with tilted matrix notation. I think it is OK. So now I continue along , (g) Linear Dirac Space Operators OK, long and difficult and strange, however! OK That concludes Chapter 2. All the time I am fixing small errors as I proof, not documented here. Check 2.11 eq nums: OK. Comment: Chapter 2 is 65-18 = 47 pages long. Section 2.11 is 65-44 = 21 pages of this. I still think it should properly be a single unified chapter. But 2.11 is so long it needs subsections, and that leads to the four point eq nums, I will just live with that!! Status is now as follows. I am holding off on the pure math sections till the end. Status of Review Process 5-13-16 3:37PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Chapter 3 Review 3.1 Outer Products Reviewed: Compatibility of Chapter 1 and Chapter 2 OK, but repetitive. 3.2 Kronecker Products OK, boring, off the main path, but it is done. Check eq num for Ch 3: done and all OK. Status: Status of Review Process 5-13-16 4:48PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Chapter 4 Review 4.1 The tensor product of 2 vectors in V2 OK, no problems, really just review. 4.2 The tensor product of 2 dual vectors in V*2 OK, but painful changing of many ch 2 eq nums 4.3 The wedge product of 2 vectors in L2 (a) Definition of the wedge product of 2 vectors and the space L2 OK (b) How big is the space L2 compared to the space V2? OK (c) Wedge products and determinants: the geometry connection OK (d) Components OK (e) Dot Products 4.4 The wedge product of 2 dual vectors in Λ2 (a) Definition of the wedge product and the space Λ2 OK (b) How big is the space Λ2 compared to the space V*2? OK (c) Wedge products and determinants OK (d) Tensor Functions OK, a little rocky but it flies. Check all Chapter 4 eq num: (4.1.10) is missing! Fixed, no refs. More problems in Section 4.3, fix tomorrow. I am not sure whether or not I am using "parallel numbering". Also not sure all that Dirac notation really buys anything in wedge world. Sat May 14, 2016 Verified that section 4.1 eq num fix is OK. There is no parallelism between 4.1 and 4.2 on eq nums, so there should be no holes in 4.2 eq nums. And there are none, so OK. Section 4.3 is all new, being on wedge for first time, so not parallel to anything, should have no holes. But things are messed up starting at (4.3.20a). So work needed here. Major fix: (4.3.20a) → (4.3.20) (4.3.20b) → (4.3.21) (4.3.n)→(4.3.n-2) for n = 24 through 35 I will make one pass searching on (4.3. to fix all this stuff: DONE! There is an close parallel between Section 4.3 and Section 4.4, so I have to make the same "major fix here" as above. Start just with this: (4.4.20a) → (4.4.20) (4.4.20b) → (4.4.21) Then search on (4.4.20 and fix anything. Done. Now map the parallelism to fix remaining eq nums after (4.4.20) in Section 4. Just do these one at a time since parallelism is not exact, (4.4.23) → (4.4.22) parallel check, and global fix DONE (4.4.25) → (4.4.23) parallel check, and global fix DONE (4.4.25) → (4.4.23) parallel check, and global fix DONE (4.4.28) → (4.4.25) parallel check, and global fix DONE (4.4.29) and (4.4.30) → (4.4.26) parallel check, and global fix DONE (4.4.31) → (4.4.27) parallel check, and global fix DONE (4.4.32) → (4.4.28) parallel check, and global fix DONE (4.4.33) → (4.4.29) parallel check, and global fix DONE All done with this mess. make a new wedge copy. Let's browse all of Chapter 4 one more time. DONE, fixed a few more things. Make new copy. Status of Review Process 5-14-16 8:10AM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Onward! Chapter 5 Review opening text OK 5.1 Pure elements, basis elements, and dimension of Vk OK 5.2 Tensor Expansion for a tensor in Vk ; the ordinary multiindex OK pause to check eq nums for these two sections: DONE. 5.3 Rules for product of k vectors OK 5.4 The Tensor Algebra T(V) OK 5.5 Comments about tensors OK 5.6 The Tensor Product of two or more tensors in T(V) OK pause to check eq nums for 5.3 thru 5.6: DONE. I made certain items red to make sure I correct them the same way in the parallel Chapter's to follow! Reviewing Ch 5 was not as bad as I thought it would be. So Status of Review Process 5-14-16 9:00AM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs make new copy. Chapter 6 Review opening text OK 6.1 Pure elements, basis elements, and dimension of V*k OK 6.2 Tensor Expansion for a tensor in V*k ; the ordinary multiindex OK 6.3 Rules for product of k vectors OK 6.4 The Tensor Algebra T(V*) OK 6.5 Comments about Tensor Functions OK 6.6 The Tensor Product of two or more tensors in T(V*) OK Go back and clear red markings from Ch 5 and Ch 6: done Check all eq num in Ch 6: OK Again, was not bad. Definitely was worth while "writing out" this Chapter. Status of Review Process 5-14-16 10:00AM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs make new copy. Chapter 7 Review opening text OK 7.1 Definition of the wedge product of k vectors OK, did some clarifications. 7.2 Properties of the wedge product of k vectors OK check eq nums to this point: OK 7.3 The vector space Lk and its basis OK 7.4 Tensor Expansions for a tensor in Lk OK 7.5 Expansions for the wedge product of k vectors OK 7.6 Number of elements in Lk compared with Vk. OK 7.7 Multiindex notation OK 7.8 The Exterior Algebra L(V) OK 7.9 The Wedge Product of two or more tensors in L(V) (a) Wedge Product of two tensors T^ and S^ OK check all eq nums from 7.3 to this point: all OK, so continue on (b) Special cases of the wedge product T^^ S^ OK (c) Commutivity Rule for the Wedge Product of two tensors T^ and S^ OK (d) Wedge Product of three or more tensors OK, found some errors and fixed. (e) Commutativity Rule for product of N tensors OK (f) Theorems from Appendix C : pre-antisymmetrization makes no difference OK (g) Spivak Normalization OK check eq nums starting with 7.3 to the end: THAT was very painful! There are SO many equations to check, you go nutso. But it is done. So Status of Review Process 5-14-16 6:50PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Question: What happened to my intro where I talk about direct versus tensor products and Shankar and such things!! Did it vanish? No, it is in Chapter 1 at the start, OK. Sun May 15, 2016 Chapter 8 Review opening text OK 8.1 Definition of the wedge product of k vectors OK, did some clarifications. 8.2 Properties of the wedge product of k vectors OK 8.3 The vector space Lk and its basis OK 8.4 Tensor Expansions for a tensor in Λk OK 8.5 Various expansions for the wedge product of k dual vectors OK 8.6 Number of elements in Λk compared with V*k. OK 8.7 Multiindex notation OK 8.8 The Exterior Algebra Λ(V) OK check eq nums to this point: OK, lots of holes 8.9 The Wedge Product of two or more dual tensors in Λ(V) (a) Wedge Product of two dual tensors T^ and S^ OK (b) Special cases of the wedge product T^^ S^ OK (c) Commutivity Rule for the Wedge Product of two dual tensors T^ and S^ OK (d) Wedge Product of three or more dual tensors OK and painful (e) Commutativity Rule for product of N dual tensors OK, major mods (f) Theorems from Appendix C : pre-antisymmetrization makes no difference OK and painful! (g) Spivak Normalization OK check eq nums for 8.9 : Question: What does ** mean on some equation numbers? Well, while developing Ch 8 I marked equations which now refer to tensor functions, while in Ch 7 the corresponding equation is different because it refers to tensor components. I just deleted all these ** markings since they just add confusion. The reader can figure it out. Status of Review Process 5-15-16 10:45AM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs It happens that the matrix for 1 (1/2) can be brought into block diagonal form by a similarity transformation where the diagonal blocks are (1/2) and (3/2), and so one writes 1 (1/2) = (1/2) (3/2). Chapter 1 Review 1.1 The Tensor Product as a Quotient Space OK 1.2 The Tensor Product in Category Theory OK eq num check: OK Status of Review Process 5-15-16 1:30PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs Chapter 9 Review 9.1. Development of Lk as Vk/S. OK 9.2. Development of L as T/I OK eq num check: OK Status of Review Process 5-15-16 2:00PM (red means done) Overview, Notation, 1,2,3,4,5,6,7,8,9,10 A,B,C,D,E,F,G Refs So this long ( 7 days!!) edit review of wedge doc is finally finished! There are of course many bad references in there still, grammar problems, and so on. But you have to stop at some point and put it out there. Before doing wedge pagination, I want to finish my tensor doc update prior to ITS pagination. I may find something that affects wedge doc. And I want to do a long review of all my many docs and pdf's to see if I have missed something or made some gross mistake. OK, I did all known edits on tensor doc, I think it is OK. Should do spell checks on new areas! Just checked the new nested cofactors area, spelling is good. And checked the E.1 area as well, both OK. Question: Why do I have STS in tensor doc, but RTR in wedge doc?? Please come up with a very good answer to this (I may have answered it elsewhere, but out of time tonight). Mon May 16, 2016 I spent a lot of time this AM answering the Question posed above. This resulted in a longer front end for Appendix F of wedge doc, and then an extra comment in tensor doc. I think it is now completely clarified! Trying now to nail down my wedge overview. OK. Final review of a few things. Ready at 11:15 to start review of support documents before doing pagination. I ran into Lang's comments on cosmetic notation and I am again uncomfortable about things, so the train comes to another halt. Here is what I am proposing to say in Section 10.2 _____________________ Regarding our "cosmetic notation" dxi ≡ λi of (10.1.9), the reader can take some support from Lang [1999] page 131, (10.2.5) Recall that λi(v) = vi from (2.11.c.5) so one could write dλi(v) = dvi. Taking v = x, one then gets dλi(x) = dxi which is Lang's first notation above. It is his second comment about writing differential form ω that concerns dxi ≡ λi . We would write his last line as ωx' = Σ'IfI(x') λI where λi = <e'i|, ____________________ But this raises several questions and red flags! I was thinking that λi = <e'i| was a function of space, so you might write λx'i = <e'i(x')| but in Rm we don't have the e'i being functions of space since they are just the constant axis-aligned basis vectors. But I suppose with a metric tensor g' you would say e'i = g'ij(x') e'j and so yes, then e'i is in general e'i(x'). But if x'-space is Cartesian, then e'i has no x'-dependence. All my work is quite vague on the metric tensor but I fear it may be very important in other applications. So I am taken back now to consider what I really mean by this statement: αx' = Σ'I fI(x') λ'^I = Σ'I fI(x') dx'^I = Σ'I fI(x') <e'^I| ϵ Λ'k ≡ Λk(Rm) (10.2.3) Earlier in wedge doc I have λ'i as a basis vector for space V'* and there is no question of spatial dependence. And in x-space I have λi basis element of V* where λi = <ui| which underpins everything I say basically! This is a functional. λi = <ui| = (ui)T . Question: If you write λi = <ui| = (ui)T in x-space, could this be a function of x? I do know that (ui)j = δij ui = gij uj (ui)j = δij So if x-space is some general metric space with gij = gij(x), then ui = gij(x) uj. Go back to this λi(x) = <ui|x> d[λi(x)] = d <ui|x> = <ui|dx> + <dui|x> = dxi + <dui|x> So you can only say d[λi(x)] = dxi when dui = 0 which means affine connection = 0 which means Cartesian space. OK, as if 4:45PM I got happy again with Lang's stuff and added a comment on it at the end of Section 10.2, not as quoted above. I can now continue on this review of all my docs. Tues May 17, 2016 Reviewed all docs in the Section 10 subfolder. In doing so, decided to add my circle manifold picture into wedge doc. But that is about all. I relive all the pain I had during development! Found nothing that is a showstopper. So both wedge and tensor docs stand ready for production. Task: check all tensor doc eq references in wedge doc! These end in a prime usually. DONE, searched also on Tensor, made a few repairs! I am now going to simply read certain selected sections of wedge doc: Overview and Summary OK symbols browzed and OK, proofed this recently. Tensor Product vs Direct Product very good, added Shankar page numbers etc 1.2 The Tensor Product in Category Theory . I like it. Chapter 2 2.1 OK, the reader should have a sense of ease since everything is just laid out one item at a time. 2.2 OK on metric tensor, I like it, just the facts maam 2.3 The basis vectors en and en Still maintaining quality and readability I think. 2.4 The basis vectors un and un continues good. 2.5 The basis vectors e'n and u'n and a summary OK still good 2.6 How to compute a viable x' = F(x) from a set of constant basis vectors en This is quite trivial, but I want an example of how en determines x' = F(x), as discussed earlier. 2.7 Expansions of vectors onto basis vectors I like the scalar paradox heading off at the pass. 2.8 The Outer Product of Tensors and Use of  OK, rabbit into hat for Skip to 2.10 Pause; What about d(A^B)? This appears page 21 old Sjamaar but I have omitted it! // OK I added it at the end of the external derivative section. Taking a break now. Did some more fast reading, but I just glaze over quickly. I need to get on with production. I can always do repairs later on in Updates. Weds May 18, 2016 Today I just "reviewed" Chapters 4,5,6,7,8 and 10, made a few changes. I then reviewed Appendix F and G, since they are relatively new, and the Refs list. I put my own ref on an equal footing with other web refs both in wedge and tensor doc, advising reader of web search in the overall refs heading. So no more "Phil Lucht Documents" which did seem tacky. Did a cleanup pass on Appendix D, several good changes made. App A.10 relatively new, so read through it,OK. Did a final review of the Overview and Summary, added a reference to the engine room. I think I am done reviewing and ready to start doing production work. Wedge Production Spelling check: Overview thru end of Chap 3: found and fixed about 5 spelling errors Ch 3 start to 5.1.5: none 5.1.5 to 7.2.3: about three errors fixed 7.2.3 through 8.4.4 : found and fixed one 8.4.4 through 10.2.4: none found 10.2.4 to 10.9.4: none 10.9.4 to 10.13.25: none 10.13.25 to A.9.4: found one A.9.4 to the end: none DONE. This was painful but did not last too long, maybe an hour. Add section next page dividers: done Add headers for each section. done Pagination: TOCL: done Overview: done Notation: done Ch 1: done Ch 2: done, was slow going, a big chapter. Ch 3: done, short and fast Ch 4: done, went pretty fast, each of the 4 main sections gets a page start Ch 5: done, went fast Ch 6: done, very fast Ch 7: done Ch 8: done Ch 9: done Ch 10: done App A: done App B: done App C: done App D: done App E: done App F: done App G: done Refs: done, exactly one page! Pagination complete at 8:15 PM!! Took several long hours. Thurs May 19, 2016 At 5:30 AM did some more perusal and a few small edits. Redated May 19 and I am now ready for my first PDF export using jump drives. Power up Alta, do the export directly off the USB dribve. Pile driver starts up. There are 148 bookmarks. // Export failed in its usual 1 of 3 mode, so I will run it again. We are 335 pages. Second pass is OK. I bring back the flash drive to peruse things on Black. Things to fix install title and author in properties, done check bookmarks for missing or misaligned: done, none were missing, none were misaligned! peruse and look for trouble: Bug: My coptic TNR symbol ϵ which I used in place of just cost me 6 hours of pain! In a nutshell, here is the story: 1. This coptic symbol exists in TNR 5.20 font on win7, but does not exist in TNR 3.00 font on Alta where I need to do my PDF export. 2. You cannot update the TNR font on Alta because it is used by the system, and I suspect the 5.20 might not even be compatible so would be dangeous. 3. When you display your doc on Alta as a Word doc, you see this problem -- the character shows up as a box. That is because Alta cannot find the character. Then Adobe punts and puts a blank. 4. I tried embedding fonts including system fonts in my file, but that did nothing to fix this problem. 5. I found a coptic font, but don't like their version of the ϵ symbol. 6. Therefore, I am going to update wedge doc right now by replacing ϵ with everywhere. Then I have to run through and fix thrown eq nums because the symbol is wider than the ϵ symbol. 7. To do the replace, you paste ϵ into the find window, but then you have to copy and then use ^c in the replace with window! This is an old bug. Do this in a test version of the doc please! There were 254 replacements! Now I have to go through and check on alignment by searching on . It took a while to go through all 254 locations! There were about 20 thrown equation numbers and a few other alignment issues. It is done in a scratch file. Going for 2nd PDF export! Started around 5:30 AM, is now 2 PM! 148 bookmarks, same as last. Done. Back on Black: scan for blank pages: done, no blank pages check section headers general perusal: it looks good, but I want to update tensor doc and get them both out at the same time. Sun May 22, 2016 I went ahead and released the above May 19 version to the web (but not researchgate). I then decided to add Appendix H which has a gold mine of stuff about Hodge and differential operators. So here are the edits I will do Install Appendix H after first spell checking it and after first doing a detailed review. DONE. Add heading for Appendix H, paginate it, make sure last headers are all OK. DONE Add overview of Appendix H, will have no pagination effect DONE Add a few more symbols to the Notation list that related to this Hodge stuff DONE Change dx^Z to the more logical dV in Appendix G, review that appendix, I have suspicions DONE Make sure new TOC ends well. Put in the new date of release. DONE My suspicion has to do with the way I define vectors in Appendix G versus the *dx type vector. Mon May 23, 2016 Accidentally make last night's wedge doc edits in the "before eps" version, so today transferred all those changes (hopefully got all) over to the correct doc. Now proofing the new appendix H before install. Appendix H H.1 OK, added Corollary, referred to Sjamaar exercise 2.15 H.2 OK on gradient H.3 OK on Laplacian H.4 OK on divergence H.5 OK on curl H.6 OK on Maxwell I am ready to install App H: first make a section break! DONE. Now go through list above. I want now to review Appendix G.2 where I have suspicions. Appendix G.2 -- I think it's OK, made clarifying edits. Paginate H now: DONE. Did some review and I am now ready for another release. Alta: now 155 bookmarks. Finished, did cycle, but found a bug reviewing my errata list! Did another cycle, all is well, stored a new release area, updated to xmission, checked it, all done for now! Time is 3:30 PM 5/23/16. Fri June 3, 2016 Resuming post-Torrey maintenance trip and subsequent recovery. I edited in the simple errata, and now I want to do some stuff in App H that is a little fancier. First step then is to proof Appendix H just to remember what is going on there. How can I work in the idea that div curl F = 0 and curl grad f = 0 ? Done. Repaginate Appendix H. Done. Upated links date. Updated paper date. All errata are done. I am ready to release, but may do some more Buck review first. Sat June 4, 2016 Fixed errata in App F, and found more in Ch 8. This could go on forever, there will always be errata in a 348 page document. I will next do some Buck stuff before doing a new PDF. // I have finally finished Buck Ch 7, added a few more things (noted on errata log) and at 5 PM I am ready to do a June 4 release. There was no June 3 release. Making PDF on Alta right now. // Did another cycle, and sent out to the xmission site, all done I hope. Now 6:15 PM. Ready for researchgate too I think. Sun June 5, 2016 On researchgate I was easily able to update my tensor doc, and then I released wedge doc there with a long abstract. It is done, it is done, it is done !!!!!!