Divergence Theorem for Riemannian Manifold

Theorem
Let $\struct {M, g}$ be a compact Riemannian manifold with boundary $\partial M$.

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

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

Let $X$ be a smooth vector field on $M$.

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

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

Let $\operatorname {div}$ be the divergence operator.

Then:


 * $\ds \int_M \operatorname {div} X \rd V_g = \int_{\partial M} \innerprod X N_g \rd V_{\hat g}$