The single variable change of variable theorem states that, for any function $\varphi$ with integrable derivative, and a continuous function $f$:

$$\int_{\varphi(a)}^{\varphi(b)}f(x)dx=\int_a^bf(\varphi (x))\varphi '(x)dx$$

The proof is a really simple application of the chain rule and the fundamental theorem of calculus. Is there no similar calculus proof, even if tedious (assuming similar restrictions), for the double integral case:

$$\int \int_S f(x,y)dxdy =\int \int_R f(g(u,v),h(u,v)) |J|dudv$$

Considering also that a similar theorem to fundamental theorem of calculus exists for the double integrals. See for example: Proof for the "Fundamental Calculus Theorem" for two variables.

  • $\begingroup$ As I know there are no easy proofs for the multivariable case. $\endgroup$ – Mark Feb 16 at 21:38
  • $\begingroup$ If you know about differential forms, you can write $dx = x_u du + x_v dv$, and similarly for $dy$, and then compute their wedge product. From the algebraic properties of wedge product, you will see that $dx \wedge dy = |J| du \wedge dv$. $\endgroup$ – Nick Feb 17 at 18:21

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