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:

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

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems