Let $U \subseteq \mathbb{R}^n$ be a bounded [open set](/page/Open%20Set) with $C^1$ [boundary](/page/Boundary) $\partial U$, and let $u \in C^1(\overline{U})$. Then for each $i = 1, \dots, n$,
\begin{align*}
\int_U \frac{\partial u}{\partial x_i}(x) \, d\mathcal{L}^n(x) = \int_{\partial U} u(x) \, \nu^i(x) \, d\mathcal{H}^{n-1}(x),
\end{align*}
where:
- $\mathcal{L}^n$ denotes $n$-dimensional Lebesgue measure on $\mathbb{R}^n$,
- $\mathcal{H}^{n-1}$ denotes $(n-1)$-dimensional [Hausdorff measure](/page/Hausdorff%20Measure) on $\mathbb{R}^n$, restricted to $\partial U$,
- and $\nu(x) = (\nu^1(x), \dots, \nu^n(x))$ is the outward unit normal to $\partial U$ at $x$.
