tensor doc fiddle

Tensor doc fiddling I know things like this Theorem 2: |(Πxi≠nei)| = (8.4.f.2) So here one of the ei has been omitted. This thing is the cofactor of a matrix element g'nn. Since this is a diagonal matrix element, I could write this same result as . But a minor is a determinant, so I could write this as cof(g'nn) = min(g'nn) = det(g' with row n and col n removed) In developmental notation the covariant metric tensor g' is called ' . If x-space is Cartesian so = 1, then we know from (5.7.6) that ' = STS. Thus we can write min('nn) = det(' with row n and col n removed) = det(STS with row n and col n removed) From (3.2.7) we know that S = [e1, e2, e3 .... en ] ≡ S(n). Start over: We know from (5.11.3) that e1e1 e1 e2 e1 e3 ...... e1 eN e2e1 e2 e2 e2 e3 ...... e2 eN ' = e3e1 e3 e2 e3 e3 ...... e3 eN (5.11.3) ........ eNe1 eN e2 eN e3 ...... eN eN I am confused about n versus N in Section 8 of tensor doc, so trace this a bit. N = dim of the N-piped in the drawing n = variable that runs 1,2,3....N for example we have dx(n) = en dx'n (8.2.1) // edges So in sections on edge and area transformations, this is what n means: n = 1,2....N. Now in Section 8 (f) I have some theorems for general n, and here n could be any integer. So Theorem 2 says |(Πxi≠nei)| = for any n in range n = 1,2.....N Now finally we get down to (h) which is the nested cofactor thing. Prior to this I have shown that dAn = dA'n (8.4.g.4) which is the "volume" of faces n and np of the N-piped. So n = 1,2....N allowed here. Now the face N is the face you get with eN is removed, so special case says |(Πxi≠Nei)| = for any n in range n = 1,2.....N Plan B. Let's just try to edit Section 8.4 (h) (h) Nested Cofactor Formulas The object dAn is the "area" of a face on an N-piped. This face, which is itself an (N-1)-piped, in turn has its own "areas" which are (N-2)-pipeds, and so on, so there is a hierarchy of "areas" of dimensions N-1 all the way down. The area ratios of corresponding areas under transformation F are determined by equations similar to that above. For example, the mth face of face n of an N-piped has area ratio . Here cof(g') is an NxN matrix, and [cof(g')]nn ≡ qnn is a number. [cof(q')]mm ≡ qnn is a number, and then cof[qnn] This at least makes some sense since the matrix cof(g') has dimension N-1, so its cofactor matrix has dimension N-2, and so on. For N=3, these faces would be line segments and one would have cof(g'33) =  cof[cof(g'33)]22 = g'11 = h'12  = h'1 (8.4.h.1) and h'1 is in fact the edge length ratio given above in (8.4.g.2). Plan C. I want a better way to describe an area of an area. So far, all I have is this:  |dAn| = |(Πxi≠nei)| dA'n . dA'n ≡ Πi≠ndx'i (8.4.f.4) and this is the magnitude in Cartesian view of an area of faces n and np of the differential N-piped. If I think of dAn,m as the magnitude of the area of face m of face n of the N-piped, then  |dAn,m| = |(Πxi≠n,mei)| dA'n . dA'n ≡ Πi≠n,mdx'i (8.4.f.4) Example: Let N = 4. then |dA1| = |e2 x e3 x e4| dx'2dx'3dx'4| |dA1,3| = |e2 x e4| dx'2dx'4| Why can't I write the vector area? On that I have right now dAn = J dA'n en area vector of face n of N-piped Plan D. Go back to properties of an N-piped. Maybe this does not belong in Chapter 8. It seems to me my massive Appendix B ought to include information about areas of areas and it does not! Consider rvolumeN = Σnαnen 0 ≤ αn ≤ 1 rface(i) = Σn≠iαnen 0 ≤ αn ≤ 1 i = 1,2...N rface(ip) = Σn≠iαnen + ei 0 ≤ αn ≤ 1 i = 1,2...N (B.3.d.3) An = |det(S)| En = σ (-1)n-1e1 x ... x eN // en missing = σ (-1)n-1 Πxi≠n ei σ ≡ sign[det(S)] = sign[det(R)] (B.5.d.10) How about this idea: rface(j) of face(i) = Σn≠i,jαnen = rface(i) of face(j) This at least expresses the locus of the face I want. The two faces indicated are the same, and I have checked this a bit with figure (B.3.c.1) of tensor doc.