Green's Identities

Theorem
Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary $\partial M$.

Let $\hat g$ be the induced metric on $\partial M$.

Let $\map {C^\infty} M$ be the smooth function space.

Let $u, v \in \map {C^\infty} M$ be smooth real functions on $M$.

Let $\rd V_g$ be the Riemannian volume form.

Let $\nabla^2$ be the Laplace-Beltrami operator.

Let $\grad$ be the gradient operator.

Let $N$ be the outward-pointing unit normal vector field on $\partial M$.

Let $\innerprod \cdot \cdot_g$ be the Riemannian metric the metric $g$.

Then:


 * $\ds \int_M u \nabla^2 v \rd V_g = \int_{\partial M} u N v \rd V_{\hat g} - \int_M \innerprod {\grad u} {\grad v}_g \rd V_g$


 * $\ds \int_M \paren {u \nabla^2 v - v \nabla^2 u} \rd V_g = \int_{\partial M} \paren {u N v - v N u} \rd V_{\hat g}$