Let $\Omega\subset\Bbb{R}^n$ be open and $\chi:\Omega\to\chi[\Omega]\subset\Bbb{R}^n$ a $C^1$ diffeomorphism between open subsets of $\Bbb{R}^n$. Then, for any Lebesgue-measurable [function](/page/Function) $f:\chi[\Omega]\to[0,\infty]$, we have
\begin{align*}
\int_{\chi[\Omega]}f(x)\,dx &=\int_{\Omega}(f\circ\chi)(y)\cdot |\det D\chi(y)|\,dy.
\end{align*}
