Change of variables Jacobian corollary

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 region for which the Riemann integral behaves well (e.g., bounded with boundary of measure zero), and let $f$ be Riemann integrable on $\Phi(E)$.

Corollary:

• The function $u\mapsto f(\Phi(u))\,|\det D\Phi(u)|$ is Riemann integrable on $E$, and
• the integrals are related by $\int_{\Phi(E)} f(x)\,dx = \int_E f(\Phi(u))\,\bigl|\det D\Phi(u)\bigr|\,du.$

Connection to parent theorem: This is the change of variables theorem for multiple integrals, often packaged as the practical rule “insert the absolute Jacobian determinant when changing coordinates.”