Green's Identity

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

Suppose \((M,g)\) is a compact Riemannian manifold . The first Green’s identity holds for smooth real-valued functions \(u\) and \(v\):

\begin{equation*} \int_{M} u \Delta v dV_g=\int_{M} \langle \grad u, \grad v\rangle_g dV_g - \int_{\partial M} u \langle \grad v, 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 \(\partial M\).

This result implies the second Green’s identity:

\begin{equation*} \int_{M} (u\Delta v - v \Delta u)dV_g = \int_{\partial M} \bigl(v \langle \grad u, N\rangle_g - u \langle \grad v, N\rangle_g\bigr) dV_{\tilde{g}}, \end{equation*}