I'm having some problems with the derivation of the Jacobian determinant when used to describe co-ordinate transformations. As I understand it, the Jacobian determinant should relate the areas defined by two vectors in both co-ordinate systems. As the vectors are not necessarily perpendicular, the area is calculated using the cross-product, giving:
(Examples of the derivation can be found here and here.)
My problem is that is not a cross product and so doesn't describe the area of a parallelogram in the u-v co-ordinate system. So, as far as I can see it, one of two things is happening:
1) du and dv are assumed to be perpendicular, and so the area is just the product of the sides of the rectange, dudv.
2) du and dv are assumed to be very small, so that the area approximates a rectangle
So yeah, any ideas? What am I missing?