Let $u:\mathbb{R}^n\to\mathbb{R}$ be Lipschitz [continuous](/page/Continuity) and assume that for a.e.\ $r\in\mathbb{R}$ the level [set](/page/Set) \begin{align*} \{x\in\mathbb{R}^n\mid u(x)=r\} \end{align*} is a smooth, $(n-1)$-dimensional hypersurface in $\mathbb{R}^n$. Suppose also $f:\mathbb{R}^n\to\mathbb{R}$ summable. Then \begin{align*} \int_{\mathbb{R}^n}f(x)\,\lvert\nabla u(x)\rvert\,dx &= \int_{-\infty}^{\infty} \Bigl(\int_{u^{-1}(r)}f(x)\,d\mathcal{H}^{\,n-1}(x)\Bigr)\,dr. \end{align*}