Change of variables formula

Change of variables formula: Let $U,V\subseteq\mathbb{R}^n$ be open sets and let $\Phi:U\to V$ be a $C^1$ diffeomorphism . If $f:V\to\mathbb{R}$ is such that the multiple Riemann integrals below exist (for example, if $f$ is continuous with compact support in $V$), then

where $\det D\Phi(u)$ is the Jacobian determinant of $\Phi$ at $u$.

This is the multivariable generalization of the one-dimensional substitution rule , and it is fundamental for computing multiple Riemann integrals under smooth coordinate changes.