I've searched everywhere on Google for a somewhat formal proof for the integration by substitution -- a.k.a $u$-substitution or change of variables -- in double integral with the Jacobian determinant method but I can't find any. Anyone knows any sources for the proof?
In case if the theorem is not familiar it goes something like:
$$ \int\int_R f(x)d^2x=\int\int_Tf(G(u)) |\det DG|\, d^2u $$