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 $f:\mathbb{R}^n\to\mathbb{R}$ is summable. Then
\begin{align*}
\int_{\mathbb{R}^n}f(x)\,\lvert Du\rvert(x)\,d\mathcal L ^n (x)
=
\int_{-\infty}^{\infty}
\Bigl(\int_{u^{-1}(r)}f(x)\,d\mathcal H ^{n-1}(x)\Bigr)\,dr.
\end{align*}
