Are you talking about the substitution theorems for multi-variable calculus?
ellipse is the 1st region.... there are theoretical proofs of the Jacobian transformation using vector analysis? the higher powers of the Jacobian are used for general relativity
this is an easy integral you can do by inspection where u^2+v^2=1 and you can change to radial coordinates or use square root limits
I've worked through the proof of approximating the image region R by secant vectors in Stewart in change of variable integrals but was a little unclear...I've heard that Hubbard or others are good texts....but maybe it's better to spend time learning other things like complex analysis?....