I am having hard time conceptually grasping how the bivariate change of variable technique works in statistics. The technique can be summarized as follows: Given random variables $X$ and $Y$, and transform of the variables $U=f_1(X)$, $V=f_2(Y)$ the joint distribution of the transformed variables can be found as:
$$f_{U,V}(u,v) = f_X(f_1^{-1}(u,v))f_Y(f_2^{-1}(u,v))|J|$$
Where $|J|$ is the determinant of the Jacobian. The inverse has to be single valued.
Can anyone provide a detailed proof for why the method works and where the equation comes from?
