Change of variables formula for multiple integrals

Change of variables formula (one standard Riemann form): Let $U,V\subseteq\mathbb{R}^n$ be open and let $\Phi:U\to V$ be a $C^1$ diffeomorphism . Let $E\subseteq U$ be a set such that $E$ and $\Phi(E)$ are “nice” for Riemann integration (e.g., bounded with boundary of measure zero ). If $f$ is Riemann integrable on $\Phi(E)$, then $f\circ \Phi\cdot |\det D\Phi|$ is Riemann integrable on $E$ and $\int_{\Phi(E)} f(x)\,dx = \int_E f(\Phi(u))\,\bigl|\det D\Phi(u)\bigr|\,du.$

This theorem is the rigorous basis for coordinate changes such as polar, cylindrical, and spherical coordinates, and it explains why the Jacobian determinant appears in such transformations.

Proof sketch: First prove the linear case: if $\Phi(u)=Au$ with $A\in GL(n)$, then volumes scale by $|\det A|$. For smooth $\Phi$, on small boxes $\Phi$ is well-approximated by its derivative $D\Phi$; the Jacobian determinant controls local volume distortion. One then partitions $E$ into small pieces, applies near-linearity on each piece, and passes to the limit using uniform continuity and additivity.