confusion re arbitrary area xform

Confusion of the Day: arbitrary area transformation PhL 2.20.12 Comment Added 2.28.12. I was now OK with how dAn transformed where dAn was an N-piped area. But I then wondered how some arbitrary dA in x-space transformed? Well, I figured that out and added to my new Section 8 in progress. But before I did that, the two "problems" below arose because I was thinking that I had to construct the arbitrary area dA somehow as a linear combination of the dAn areas. But then I realized how to construct an arbitrary dA and it has nothing to do with such a linear combination described below, so the two Problems are incompetent, irrelevant and immaterial. Problem: In 2D Cartesian space I have two edge vectors dx1 = dx dy1 = dy How do I create a diff vector from these basic lengths which points at 45 degrees? V = A dx1 + B dy1 = A dx + B dy I could then select A = k/dx B = k/dy and I get V = k ( + ) and I have solved the problem. Problem: In 3D Cartesian space I have three area patches dA1 = dx2dx3 dA2 = dx3dx1 dA3 = dx1dx2 How do I create an area patch from these basic patches which points in some given direction N? [ this is the linear combination of area patches idea commented on above.] Well, the solution is going to be dA = k N and we then get Σici dAi = k N c1dx2dx3 + c2dx3dx1 + c3dx1dx2 = k (N1 + N2 + N3 ) The solution is then c1 = kN1/ (dx2dx3) c2 = kN2/ (dx3dx1) c3 = kN3/ (dx1dx2) Warning: I have changed Appendix D so below |J| should really be J. The question posed remains a good one! Once you know that dA'n is a vector density of weight -1. the transformation rule is done! This might shorten Section 8 when I get back to it. Problem: According to tensor doc page 102, we have dA'n = |J|-(-1) R dAn = |J| R dAn = g'1/2RdAn So if someone wants to know how an arbitrary patch of area transforms, the answer is this dA'n = |J| R dAn and in the other direction dAn = |J|-1S dA'n This result is trivially simple. (1) Why then did I have to do all that work in Section 8 to determine how area transforms? (2) How is the above consistent with my Section 8 area results?