Sec 8 flailing

Comments on Section 8 PhL 2.20.12 Notes added 2.29.12. This long doc represents a lot of flailing about many aspects of Section 8, and it eventually led to a full rewrite of that section. It is still tough to explain "the meaning" of dA' and dV' in x'-space since the geometry cannot be drawn, and this is a major flailing subject discussed below. I don't think there is any reason to read the stuff below ever again, I think it is all "resolved". I just now added more to my Section 8 subsection on the transformation of arbitrary vectors, areas and volume. The vectors are never an issue, the confusion is always about the area and volume. I show that dA = g (dx[1]) x (dx[2]) ... x (dx[N-1]) dV = g det (dx[1], dx[2], ... dx[N]) dA' = g' (dx'[1]) x (dx'[2]) ... x (dx'[N-1]) dV' = g' det [dx'[1], dx'[2], ... dx'[N]] = dA' dx'[N] where only the permutation tensor ε is used. This gives an operational way to think about dA' and dV' in x'-space. You compute dA' from the "usual" cross product, then throw in a factor of g'. To get the volume, you then dot dA' with the last vector, this dot being done with g'ij. So if someone asks "what is the meaning of dV' and dA' in x'-space", the two lines above at least give something you can look at. You can do the cross product as usual and muliply by the scalar. But if you want a geometric picture showing these quantities, it is little hard to explain why you need to mulitply by g' to get the area and volume. __________________________________________________________________________________ I am unable to really understand my own presentation here. I think something somehow is wrong, but I don't know what it is, so now is the time to find out. The section has "rested" for several months which is probably a good thing. Question 1. What is the meaning of quantity dV' which appears for the first time in section (d) ? At least I think I know that dV is the volume of the n-piped in x-space. This is something concrete that I can at least imagine. I just argue that dV = | dAn dx(n) | is a vector density of weight -1 and I think conclude that dV' = J dV = g'1/2 dV but I don't know that dV' means. Is it the volume of the orthogonal n-piped in x'-space? Perhaps it is this dV' = | dA'n dx'(n) | ?? // see new Section 8, pretty much this is right I think I do know that dx'(n) = R(x) dx(n) I claim in this same section (d) that dA'n = g'1/2R dAn so that begs the question: What does dA'n mean? Suppose N = 3. Is it wrong to say that dA'1 = dx'(2)dx'(3)e'1 = dθ dφ e'r Maybe I need to think of something like spherical coordinates specifically to de-mush this stuff: dA'r = dx'(2)dx'(3)e'1 = dθ dφ e'r dA'r dA'r = (dθ dφ)2 e'r e'r = (dθ dφ)2 g'ab (e'r)a(e'r)b = (dθ dφ)2 g'rr dA'θ dA'θ = (dφ dr)2 e'θ e'θ = (dφ dr)2 g'θθ= (dφ dr)2 r2 => dA'θ = rdφ dr I think I should ponder all of this in simple polar 2D coordinates. All of my confusion exists there. If I can nail it there, then I will be happy in sphericals and in generals. Polar coordinates: dA'n = g'1/2R dAn g'1/2 = ρ dA = area of grey patch shown on right, has units L2 R = S-1 = *************************************************************************8 I had better do sphericals since dA at least is different there from dV. 1. Up to section d, I think all makes sense for sphericals. In example 2 I write out all the dAn objects for this system. These are genuine areas with L2 units, unlike the dA'n objects. I could add dV to that section. What is the definition of dAn please? We start with An in the previous section where An is supposed to be the true area of an n-piped face. I have dozens of hazy questions all at once here. (1) Does |J| = g'1/2 = |det(S)| have dimensions? Well, r2sinθ is the volume factor, so it is J and so yes, it has dimensions. Then look at dV = g'1/2 Πidx'i for sphericals. Πidx'i is L, so L3 yes it works. (2) Does en have dimensions? Does en have dimensions? From appendix A E1 = det(R) (-1)1-1 e2 x e3 = det(R) e2 x e3 E2 = det(R) (-1)2-1 e1 x e3 = det(R) e3 x e1 E3 = det(R) (-1)3-1 e1 x e2 = det(R) e1 x e2 For sphericals we know that R = det(S) = r2 sinθ = J = g'1/2 er = (sinθcosφ, sinθsinφ,cosθ) |er| = 1 = h'r eθ = r(cosθcosφ,cosθsinφ,-sinθ) |eθ| = r = h'θ eφ = rsinθ(-sinφ,cosφ,0) |eφ| = rsinθ = h'φ En ≡ g'ni ei = g'nnen = h'n-2 en for any orthogonals. g'ni = contravariant er= er = L0 eθ= r-2eθ = L-1 eφ= r-2sin2θeφ = L-1 Then consider dAn = |det(Sab)| en Πi≠ndx'i = g'1/2 en Πi≠ndx'i // label on dAn is placed up to match that on en So then dAr = L2 L0 L0 = L2 OK dAθ = L2 L-1 L = L2 OK and same for dAφ (3) I am concerned about notation for area magnitudes. In section c we have only vector areas mentioned. The same true of section d. The same in section e. First magnitude of area appears in section f defined in terms of covariant or Cartesian dot product |dAn|2 = dAn dAn and then dAn ≡ |dAn| Does this perhaps conflict with this concept: dAn = Σi (dAn)iei ?? No. You have to remember that the n on dAn is a label, as is the n on en. These n's are not component indices! So I think I am OK with magnitude object dAn defined in the above manner. (4) What is the meaning of dA'n ? It first appears in section d where I write A'n = |J|-(-1) R An = |J| R An = g'1/2RAn Well, for the finite n-piped, I know that An is the area of a face. What on earth is A'n ??? Where is it defined? It first appears in my tensor density equations. A'n = |J|-(-1) R An = |J| R An = g'1/2RAn An = |det(Sab)| en = g'1/2 en Then A'n = g'1/2RAn = g'1/2 g'1/2 R en Now (R en)i = Rij(en)j (R en)i = Rij(en)j = Rij Rnj = δni = (e'n)i (5) What can I say about (e'n)i in terms of things up and down? My little tables have this (e'n)i= δni (e'n)i= δni So therefore I get in the above calculation R en = e'n and then A'n = g' e'n = J2 e'n det(S) = r2 sinθ = J = g'1/2 So in x'-space we obtain axis-aligned area vector directions. What can we say then about the magnitude of this animal? | A'n|2 ≡ A'n A'n Since this thing is a tensor density, I guess we have to say A'n A'n = J-(-2) An An = J2 {J en } { J en } = J4 g'nn = J4 h'n-2 => | A'n|2 = J4 h'n-2 => | A'n| = J2 h'n-1 | A'r| = J2 h'r-1 = (r2 sinθ)2 / 1 | A'θ| = J2 h'θ-1 = (r2 sinθ)2 / r | A'φ| = J2 h'φ-1 = (r2 sinθ)2 / rsinθ Now compute this a different way from A'n A'n = { J2 e'n } { J2 e'n } = J4 e'n e'n = J4 g'nn = same result, good. Comment: When I have written e'n = axis aligned unit vectors (e'n)i = δni ( 0,0,...1....0) I really mean that the contravariant components have this "look". The covariant components are (e'n)i = g'ni = (g'n1, g'n2, .....) which looks nothing at all like a unit vector. BUT, if g' is orthogonal, then we have (e'n)i = g'ni = (0,0,.., g'nn, .....0) = g'nn( 0,0,...1....0) = h'n2 ( 0,0,...1....0) so the covariant components in this orthogonal case are still axis-aligned, They just don't have unit magnitude. So we have found above then that A'n = J2 e'n (A'n)i = J2 (e'n)i = J2 δni det(S) = r2 sinθ = J = g'1/2 So in x'-space, this area object has covariant components which are axis aligned. (A'r)i = J2(1,0,0) r2 sinθ = J (A'θ)i = J2(0,1,0) (A'φ)i = J2(0,0,1) Apart from direction, these three have the same "size". ******************************************************* Idea for interpreting A'n. (An)i = σ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // en missing Maybe in x'-space this is constructed this way [ this is correct! ] (A'n)i = σ (-1)n-1 ε'iabc..x (e'1)a(e'2)b.... (e'N)x // en missing According to Appendix D, ε'abc... = |J|2 εabc... and we know also that (e'1)a = δ1a etc. Then we get (A'n)i = σ (-1)n-1 |J|2 εiabc..x δ1aδ2b .... δNx // en missing, δnα = σ (-1)n-1 |J|2 εi123..N // where index n missing on the ε Now, if i ≠n, εi123..N = 0 since two indices are repeated. If i = n, then do the slide. The net result is (A'n)i = σ |J|2 δni This agrees with the result obtained above, which was A'n = J2 e'n (A'n)i = J2 (e'n)i = J2 δni So at least I have SOME way to think about A'n . How about a treatment of V' now. V = | det [ e1, e2, e3 ... eN] | = εabc..x(e1)a(e2)b.... (eN)x V' = ε'abc..x(e'1)a(e'2)b.... (e'N)x = J2ε123...N = J2 So I then end up with V = |J| V' = |J|2 V' = |J| V // agrees with section d. ************************* Section 8 rewrite parts ************* _____________________________________________________________________________ (d) Objects in x'-space Objects A'n and V'. In Section (b) it was stated that, with regard to the N-piped in x-space, An = σ (-1)n-1 e1 x e2 ... x eN V = | det [ e1, e2, e3 ... eN] | = | det(Sab) | = g'1/2 = |J| As a reminder, these are the nth far face area and volume of a finite N-piped drawn with the tails of the spanning en vectors located at a point x in x-space corresponding to point x' in x'-space. One can and usually does think of en and therefore An and V as functions of the curvilinear coordinates x'. One wonders what these two objects look like in x'-space. To find out, write both objects in terms of an ε tensor as follows: (An)i = σ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // en missing V = εabc..x(e1)a(e2)b.... (eN)x In x'-space one will then have [ put primes on everything and recall that σ ≡ sign[det(Sab)] = ±1 has to do with the handedness of the transformation F and is just a constant, as is (-1)n-1 ] (A'n)i = σ (-1)n-1 ε'iabc..x (e'1)a(e'2)b.... (e'N)x // en missing V' = ε'abc..x(e'1)a(e'2)b.... (e'N)x Appendix D (e) item 3 discusses the nature of the ε tensor with regard to transformations, and shows that if one assumes that ε'abc... = εabc... = εabc...= the usual permutation tensor, then one must have ε'abc... = |J|2 εabc... |J|2 = g' = det(g'ij) Section 7 (s) states that (e'n)i = δni. Installing these two facts, the above equations become (A'n)i = σ (-1)n-1 |J|2 εiabc..x δ1a δ2b.... δNx // δn* missing V' = |J|2 εabc..x δ1a δ2b.... δNx so that (A'n)i = σ (-1)n-1 |J|2 εi123..N // index n missing on εi123..N V' = |J|2 ε123..N = |J|2 One the first line, if i ≠ n, then εi123..N = 0 because two indices must be the same (since n missing). On the other hand, if i = n, one finds εi123..N = εn123..N = (-1)n-1 ε123..n...N = (-1)n-1. Therefore (A'n)i = σ |J|2 δni σ ≡ sign[det(Sab)] A'n = σ |J|2 e'n // since (e'n)i = δni V' = |J|2 |J|2 = g' = det(g'ij) = | det(Sab) |2 These objects are then the (vector area of far face n) and volume of our x'-space N-piped. It must be said that the physical interpretation of "area" and "volume" of objects in x'-space with g' ≠ 1 is a mental challenge. Making some kind of reasonable Cartesian-space drawing ( even in 2D or 3D) of an object sitting in non-Cartesian space is frustrating, as demonstrated in Appendix C (e) for a simple 2D curvilinear system, where one has difficulty even drawing a simple vector, much less an area or volume. Despite this difficulty of geometric interpretation, the objects A'n and V' are completely well defined by the various equations stated above. Going back to the second pair of equations above, (An)i = σ (-1)n-1 εiabc..x (e1)a(e2)b.... (eN)x // en missing V = εabc..x(e1)a(e2)b.... (eN)x since the various en are true tensorial vectors (that is, weight 0), and since it is known that εabc... is a tensor with weight -1 ( as shown in App. D (d) ), then using the rule of App. D (b) item 3 concerning adding weights, one quickly concludes that An is a vector density of weight -1 and V is a scalar density of weight -1. This implies at once that A'n = |J| RAn meaning (A'n)i = |J| Rij(An)j V' = |J| V Since V = |J| (2nd equation of this section) the second line gives V' = |J|2, consistent with what was just found above. Then using the fact from section (b) that An = |det(Sab)| en = |J| en One finds that (An)i = |J| (en)i = |J| Rni and therefore (A'n)i = |J| Rij(An)j = |J| |J| Rij Rnj = |J|2 δin Bug: why is there no σ here but there was above? Fix it later!!! which again agrees with the previous result above. The covariant magnitude of A'n is determined by | A'n |2 = A'n A'n = g'ab(A'n)a(A'n)b = |J|4 g'ab δan δbn = |J|4g'nn = |J|4h'n-2 so that | A'n | = |J|2 h'n-1 Objects dA'n and dV' . By adding an appropriate set of differentials, dA'n ≡ Πi≠ndx'i dV' ≡ Πidx'i the results of the previous section can be rewritten as follows: (dA'n)i = σ |J|2 δni dA'n σ ≡ sign[det(Sab)] dA'n = σ |J|2 e'n dA'n // since (e'n)i = δni dA'n = |J| RdAn meaning (dA'n)i = |J| Rij(dAn)j | dA'n | = |J|2 h'n-1 dA'n dV' = |J|2 dV' |J|2 = g' = det(g'ij) = | det(Sab) |2 dV' = |J| dV Differential Length. As defined in section (a) above, dx(n) ≡ en dx'n // differential N-piped edge in x-space Since this is a true vector, in x'-space one has dx'(n) = R dx(n) = Ren dx'n = e'n dx'n // Section 3 (b) e'n = Ren so that [dx'(n)]i = (e'n)i dx'n = δni dx'n = δni dL'n which is then the corresponding N-piped edge in x'-space. The covariant magnitude of dx'(n) is given by | dx'(n)|2 = dx'(n) dx'(n) = g'ab[dx'(n)]a[dx'(n)]b = g'ab δna δnb (dx'n)2 = g'nn (dx'n)2 = h'n2(dx'n)2 so that dx'(n) ≡ | dx'(n)| = h'n dx'n which shows that the nth edge of the x'-space N-piped is scaled by factor hn' relative to the corresponding edge of the N-piped in x-space (which is why h'n is called a "scale factor"). Differential Summary in x'-space: x'-space objects in terms of curvilinear coordinate variations dL'n, dA'n and dV' : dx'(n) == e'n dL'n // length dL'n ≡ dx'n [dx'(n)]i = δni dL'n dx'(n) ≡ | dx'(n)| = h'n dL'n // length magnitude dA'n = σ |J|2 e'n dA'n // area dA'n ≡ Πi≠ndx'i (dA'n)i = σ |J|2 δnidA'n dA'n ≡ | dA'n | = h'n-1 dA'n // area magnitude dV' = |J|2 dV' // volume dV' ≡ Πidx'i x'-space objects in terms of corresponding x-space objects: dx'(n) = R dx(n) |J|2 = g' = det(g'ij) = | det(Sab) |2 dA'n = |J| RdAn (dA'n)i = |J| Rij(dAn)j dV' = |J| dV **************************************************************************** Question: Is there some of this stuff that would apply to Picture A with general g and g' ? I am thinking about the Lai application. What about this idea in 3D: dx'(n) = R dx(n) dA'(12) = dx'(1) x dx'(2) = [R dx(1)] x [R dx(2)] // Lai (3.27.2) p 128 = [R e1 dx'1] x [R e2 dx'2] // my en vectors = [e'1 dx'1] x [e'2 dx'2] = e'1 x e'2 dx'1 dx'2 But my expression used above was instead this, (dA'n)i = σ (-1)n-1 ε'iabc..x (e'1)a(e'2)b.... (e'N)x Πi≠ndx'i = |J|2 σ (-1)n-1 εiabc..x (e'1)a(e'2)b.... (e'N)x Πi≠ndx'i so I would then have (dA'3)i = |J|2 σ (-1)n-1 εiab (e'1)a(e'2)b dx'1 dx'2 or dA'3 = |J|2 σ e'1 x e'2 dx'1 dx'2 I think there are two different kinds of area differentials floating around here. Type 1: dx(n) = en dx'n This has a special meaning. It is an edge of the x-space differential N-piped. Type 2: dx[n] = dxn un This is what Lai means. It is a different differential vector in x-space!!!! So I will now put a little square bracket to denote this one. Here un is a unit vector in Cartesian x-space. So start again with what Lai is saying: dA'(12) ≡ dx'[1] x dx'[2] = [R dx[1]] x [R dx[2]] = [R u1] x [R u2] dx1dx2 Let [R u1]i = Rij (u1)j = Rij δ1j = Ri1 = (u'1)i => R un = u'n where u'n are the inverse tangent base vectors. Remember that either set of tangent base vectors (forward or reverse) exist just fine in the general Picture A. So we then have dA'(12) = u'1 x u'2 dx1dx2 Now according to p 89 tensor doc, u'n u'm = gnm u'n u'n = gnn = hn2 => | u'n| = hn and '1 = u'1/hn **************************************************************************** Stop. Let's try flowing in the reverse direction. Imagine that X = x'. Then we have dX(n) = dx'n Then in Lai language, we start off with this area in x'-space dA'(12) = [dx'(1)] x [dx'(2)] = [(e'1)dx'1] x [(e'2)dx'2] = e'1 x e'2 dx'1 dx'2 Now to stick with Lai, we need x'-space to be Cartesian! Paradox: Lai talks about the before and after areas as real areas! I thought you could only have real areas in Cartesian space. And so his before and after spaces are then both Cartesian, so g = g' = 1. But since I know that g' = R g RT then I get 1 = RRT and that says R-1 = RT so R is real orthogonal. But since these are "normal matrices (not tilt down), this seems to force R = rotation. But I know his R = F is more general than just a rotation. Hence, a paradox. I am trying to wedge his discussion into my general framework, but it is not working right.