volume element v2 REVIEWED

Volume Element v2 PhL 5.5.16 5.5.16 : This is where I wrote the new tensor doc "nested cofactor" section, see bottom below. It is all installed and happy. I want to resume what I was doing in "Measure in Sja" doc in reading Ch 8 of tensor doc. I think I am on track with this idea, dAn = J (Πi≠ndx'i) en (8.4.a.1)  |dAn| = |J| |en| (Πi≠ndx'i) = |J| (Πi≠ndx'i) dAn = dA'n dA'n ≡ Πi≠ndx'i dAn = dA'n dA'n ≡ Πi≠ndx'i where you cross out row n and column n. In this world, I have a skewed n-piped in x-space and an aligned n-cube in x'-space, and dAn is the vector area of a face on the skewed n-cube. If x-space is Cartesian, then g = 1 and I know that ' = STS // developmental notation, covariant g' In standard notation, how do I write this? See (7.5.6), g'ab = Sa'aSb'b ga'b' and if g = 1, this just says g'ab = Sa'aSa'b g'aa = Sa'aSa'a = (ST)aa'Sa'a = (STS)aa no sum on a. But this is not really the object I want!! I don't want a diagonal element of g', I want the cofactor of a diagonal element. Lagrange does say minor(M32) = Σaaaa...a εa2aaa...a M1aM2aM4aM5a ...... Mna . (B.3.3) minor(M22) = Σaaaa...a εa2aaa...a M1aM3a3M4aM5a ...... Mna . (B.3.3) a2 sum missing M2a missing This is very painful, this is of course a reflection of all that ∂M stuff, faces of faces. How about cof(g'nn) = [cof(g')]nn ? g'nn g' = cof(g'nn) ? Maybe this last is useful. It says that what I want = cof(g'nn) = g'nn g' g' = det(g'**) Then go to, g'ab = Raa'Rbb'ga'b' = Raa'Rba' so that g'nn = Rna'Rna' = Rna'(RT)a'n = (RRT)nn no sum on n Then what I want = cof(g'nn) = (RRT)nn det(g'**) This path is not panning out. Resume on 5/6/16 Consider this form of the area on an n-piped (dAn)i = (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x (Πi≠ndx'i) // en factor and index missing (8.4.c.1) Then consider (dAn)2 = (dAn)i(dAn)i For get the differentials and look just at the vectors and assume Cartesian space, so (dAn)2 = εiabc..x (e1)a(e2)b.... (eN)x εia'b'c'..x' (e1)a'(e2)b'.... (eN)x' = εiabc..x εia'b'c'..x' (e1)a(e2)b.... (eN)x (e1)a'(e2)b'.... (eN)x' where n is missing everywhere. I claim that Σi εibc...x εib'c'...x' = ΣP (-1)S(P) δb,P(b') δc,P(c') ..... δx,P(x') // n missing and therefore εiabc..x εia'b'c'..x' = ΣP (-1)S(P) δa,P(a') δb,P(b') ..... δx,P(x') And then (dAn)2 = ΣP (-1)S(P) δa,P(a') δb,P(b') ..... δx,P(x') [ (e1)a(e2)b.... (eN)x (e1)a'(e2)b'.... (eN)x'] = Σabc..x ΣP (-1)S(P) [ (e1)a(e2)b.... (eN)x (e1)P(a)(e2)P(b).... (eN)P(x)] // liberties... where n is missing. = Σabc..x ΣP (-1)S(P) [ Ra1Rb2.... RxN RP(a)1RP(b)2....RP(x)N] // no n = ΣI ΣP (-1)S(P)[ RIZ RP(I)Z] But in "volume element.doc" I show (not exactly right maybe but generally) that this is = det(RTR) but with n missing! = cof (RTR)nn = cof(g'nn) and things are sort of tied together. Try again on 5/7/16. In R3 I can at least make this connection. Consider the face if a 3-piped spanned by e1 and e3. Then on the one hand I would say that the area of that face was related to the square root of det(STS) where S = (e1,e3) So this "little S" is missing e2. Now I can write out this det thing to get det(STS) = det [ (e1,e3) ] = det = e1e1 e3e3 - (e1e3)2 Alternatively, the area is related to the square root of cof(g'22) = the exact same determinant, since g'ij = ei ej So at least I have one case where the two are equal and I see why they are equal. But I would like to have a more general connection between these two objects!!! Try another example. Consider a 4-piped which has a face spanned by e1, e3, e4. The two methods for the area of this face would be det(STS) where S = (e1,e3,e4) S = "little S" with piece missing cof(g'22) = [cof(g')]22 = cof [(STS)22] S = full 4x4 S = cof [ (e1,e2,e3,e4) ]22 = det [ (e1,e3,e4) ] OK, I think I see the light. Let's try to add some stuff to tensor doc. What I have now is in blue, and I will attempt an edit! dV' = J2 dV'  |dV'| = J2 dV' . (8.4.e.5) Do not edit stuff below, it was installed into tensor doc on 5.15.16 and subsequently edited a bit. __________________________________________________________ (h) Nested Cofactor Formulas and STS notation Descending Leaf Hierarchy The object dAn is the "area" of a face on a differential N-piped spanned by the full set of vectors ei but not including en. 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 this purpose we define Cof to be a matrix and cof to be a number, Cof(Mij) ≡ the submatrix of M obtained by crossing out row i and column j of M cof(Mij) = (-1)i-j minor(Mij) = (-1)i-j det[Cof(Mij)] = the usual "cofactor" . (8.4.h.1) Then Cof(g'nn) = the submatrix of g' obtained by crossing out row n and column n of g' cof(g'nn) = minor(g'nn) = det[Cof(g'nn)] . (8.4.h.2) Then for example the mth face of face n of an N-piped has area ratio . The object inside the radical is the cofactor of the m,m element of the N-1 x N-1 matrix Cof(g'nn). If we refer to this area as dAn,m we can go down the hierarchy in this manner dAn = Πi≠ndx'i face n dAn,m = Πi≠n,mdx'i face m of face n dAn,m,k = Πi≠n,m,kdx'i face k of face m of face n (8.4.h.3) and so on. As a very simple example, for N = 3 the faces dAn are line segments dx'i and one has Cof(g'33) =  cof[Cof(g'33)]22 = g'11 = h'12  = h'1 (8.4.h.4) and h'1 is in fact the edge length ratio given above in (8.4.g.2). There is another way to write these area magnitudes if we assume that x-space is Cartesian, but it is difficult to express in standard notation, so we momentarily revert to our developmental notation. In that notation, if x-space is Cartesian we can write from (7.5.6) and (3.2.7) that ' = STS, (8.4.h.5) as was written earlier in (5.11.3). Since the covariant metric tensor 'ij in developmental notation is equal to g'ij in standard notation, we shall use the standard notation below in expressing these tensor elements. The area (magnitude) dAn of face n of a differential N-piped is, from (8.4.g.4) and (8.4.e.5), dAn = Πi≠ndx'i . (8.4.h.6) The object cof (g'nn) is the minor of the above bracketed matrix with row n and column n crossed out. That reduced matrix can in fact be written Cof(g'nn) = S(n)TS(n)) = (e1, e2, ...eN) where en is missing from both vectors . (8.4.h.7) so cof(g'nn) = det[S(n)TS(n)] = det [ (e1, e2, ...eN) ] . (8.4.h.8) Therefore, dAn = Πi≠ndx'i = Πi≠ndx'i where S(n) = (e1,e2....eN) with en missing . (8.4.h.9) The same argument results in dAn,m = Πi≠n,mdx'i = Πi≠n,mdx'i where S(n,m) = (e1,e2....eN) with en and em missing . (8.4.h.10) Continuing down one more level, dAn,m,k = Πi≠n,m,kdx'i = Πi≠n,m,kdx'i where S(n,m,k) = (e1,e2....eN) with en, em, ek all missing . (8.4.h.11) In this descending hierarchy of n-piped "leaf" areas, we eliminate one tangent base vector ei at a time, and the eliminated vectors' labels are used to label the "leaf" whose area is indicated, as in S(n,m). Ascending Leaf Hierarchy One can alternatively build up the hierarchy of areas from the bottom, for example using S[n.m] to indicate the area of the 2-piped spanned by en and em in RN where the top level N-piped lives. One then has dA[n] = dx'n where S[n] = en . (8.4.h.12) In this case S[n]T S[n] = enen = h'n2 so dA[n] = h'ndx'n as expected. Next, for 2-pipeds, dA[n.m] = dx'ndx'm where S[n.m] = (en, em) . (8.4.h.13) In this case det[S[n.m]TS[n.m]] = det [ (en, em)] = det = det = g'nng'mm - (g'nm)2 = h'n2 h'm2 - (enem)2 (8.4.h.14) so dA[n.m] = dx'ndx'm . (8.4.h.15) Doing one more step, at the level of 3-pipeds in the hierarchy buildup one would then have dA[n.m.k] = dx'ndx'mdx'k where S[n.m.k] = (en, em, ek) (8.4.h.16) and now det[S[n.m.k]TS[n.m.k]] = det [ (en, em, ek)] = det . (8.4.h.17) Comment 1: Notice that the matrices S appearing above in the above STS structures are in general not square matrices because the number of components N of the ei vectors in RN generally does not match the number of ei vectors in the list which defines S. On the other hand, STS is always a square matrix and therefore always has a determinant. Comment 2: We have used the term "leaf" to refer to any n-piped appearing in the top level N-piped, with n then ranging from N down to 1 (which leaf is just point). In a more precise discussion of this subject, these leaves are called "boundaries" and in addition to having "area", these boundaries have "orientation". We have only discussed the orientation of the top level leaves by using the vector area dAn . Usually the precise discussion is couched in the language of "differential forms", and by careful consideration of boundaries and their orientations one is able to derive the generalized Stokes's theorem which reads ∫M dα = ∫∂M α (8.4.h.18) where α is a differential n-form, dα is an (n+1)-form which is the "exterior derivative" of α, M is a smooth type of surface called a manifold, and ∂M is the boundary of that manifold. This abstract theorem then encompasses all the well-known integral theorems of analysis in any number of dimensions n. See Sjamaar [2015] Sections 5.2 - 5.4 , Section 8.2, and Theorem 9.9 on page 117. The fact that is the area scaling factor appears as Theorem 8.4 on page 101 where it is called . See also Appendix F of Lucht Tensor Products where the factor appears as .