I know the formula for a change of variables in a double integral using Jacobians. $$ \iint_{S}\,dx\,dy = \iint_{S'}\left\lvert J(u,v) \right\rvert\,du\,dv $$ where ## S' ## is the preimage of ## S ## under the mapping $$ x = f(u,v),~ y = g(u,v) $$ and ## J(u,v) ## is the Jacobian of the mapping in terms of ## u, v ##.(adsbygoogle = window.adsbygoogle || []).push({});

What bothers me is that the only proof I know of this fact involves Green's Theorem and then a chain rule followed by Green's theorem in reverse, and somehow the Jacobian magically pops up. The case for 3 variables seems even worse.

Is there a more straightforward proof which does not use Green's Theorem? I feel like it is overkill and too indirect; there should be a proof using only Riemann sums or manipulations of the differentials (chain rule).

I tried directly replacing ## dx ## by ##\frac{\partial f}{\partial u}\,du + \frac{\partial f}{\partial v}\,dv ## and similarly for ## dy ## but then the resulting integral doesn't seem to make sense: $$ \iint_{S'}\left(\frac{\partial f}{\partial u}\,du + \frac{\partial f}{\partial v}\,dv\right)\left(\frac{\partial g}{\partial u}\,du + \frac{\partial g}{\partial v}\,dv\right) $$

So I guess my question has a few parts:

(1) Is it valid to just replace dx and dy like that?

(2) Is there a proof of the formula not based on Green's Theorem (or on another FTC analog)?

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# Change of variables in double integrals

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