\[ \DeclareMathOperator{\div}{div} \]

Let \((M,g)\) be an oriented Riemannian manifold with boundary. For any compactly supported vector field \(X\) on \(M\),

\begin{equation*} \int_{M} (\div X)dV_g = \int_{\partial M} \langle X, N\rangle_g dV_{\tilde{g}}, \end{equation*}

where \(N\) is the outward-pointing unit normal vector along \(\partial M\) and \(\tilde{g}\) the induced Riemannian metric on \(\partial M\).

See also Link to heading

• integration by parts on compact Riemannian manifolds
• divergence theorem