Let $m, n$ be integers with $1 \le m \le n$, and let $A \subseteq \mathbb{R}^m$ be an $\mathcal{L}^m$-measurable [set](/page/Set). Suppose $\phi: A \to \mathbb{R}^n$ is an injective Lipschitz [function](/page/Function), and let $J\phi$ denote its Jacobian matrix, which exists $\mathcal{L}^m$-almost everywhere by [Rademacher's Theorem](/page/Rademacher's%20Theorem). If $f: \mathbb{R}^n \to [0, \infty]$ is a Borel measurable function, then the function $y \mapsto f(\phi(y)) \sqrt{\det(J\phi(y)^T J\phi(y))}$ is $\mathcal{L}^m$-measurable and satisfies the identity: \begin{align*} \int_{\phi(A)} f(x) \, d\mathcal{H}^m(x) = \int_A f(\phi(y)) \sqrt{\det(J\phi(y)^T J\phi(y))} \, d\mathcal{L}^m(y). \end{align*}