The Divergence Theorem on Riemannian Manifolds

The divergence theorem generalizes the classical Gauss theorem to Riemannian manifolds, relating the integral of the divergence of a vector field over a domain to the flux through its boundary.

Setup

Definition4.7Divergence of a vector field

Let $(M, g)$ be an oriented Riemannian manifold with volume form $\mathrm{dvol}_g$ . The divergence of a vector field $X$ is the smooth function $\operatorname{div}(X)$ defined by

\mathcal{L}_X \mathrm{dvol}_g = \operatorname{div}(X) \, \mathrm{dvol}_g,

where $\mathcal{L}_X$ is the Lie derivative. In local coordinates, $\operatorname{div}(X) = \frac{1}{\sqrt{|g|}} \partial_i(\sqrt{|g|} X^i)$ .

DefinitionFlux form

Given a vector field $X$ on an oriented Riemannian $n$ -manifold, the flux $(n-1)$ -form is $\iota_X \mathrm{dvol}_g$ , where $\iota_X$ denotes interior multiplication. By Cartan's formula, $d(\iota_X \mathrm{dvol}_g) = \mathcal{L}_X \mathrm{dvol}_g = \operatorname{div}(X) \, \mathrm{dvol}_g$ .

The Theorem

Theorem4.6Divergence theorem on Riemannian manifolds

Let $(M, g)$ be a compact oriented Riemannian manifold with boundary $\partial M$ , and let $\nu$ be the outward unit normal along $\partial M$ . For any smooth vector field $X$ on $M$ ,

\int_M \operatorname{div}(X) \, \mathrm{dvol}_g = \int_{\partial M} g(X, \nu) \, \mathrm{dvol}_{\partial M},

where $\mathrm{dvol}_{\partial M}$ is the induced volume form on $\partial M$ .

Proof

By Cartan's formula, $d(\iota_X \mathrm{dvol}_g) = \operatorname{div}(X) \, \mathrm{dvol}_g$ . By Stokes' theorem,

\int_M \operatorname{div}(X) \, \mathrm{dvol}_g = \int_M d(\iota_X \mathrm{dvol}_g) = \int_{\partial M} \iota_X \mathrm{dvol}_g.

At each point $p \in \partial M$ , decompose $X = g(X, \nu)\nu + X^\top$ where $X^\top$ is tangent to $\partial M$ . Then $\iota_X \mathrm{dvol}_g |_{\partial M} = g(X, \nu) \, \mathrm{dvol}_{\partial M}$ (since $\iota_\nu \mathrm{dvol}_g |_{\partial M} = \mathrm{dvol}_{\partial M}$ and $\iota_{X^\top} \mathrm{dvol}_g$ restricted to $\partial M$ vanishes on $(n-1)$ -tuples of tangent vectors). $\blacksquare$

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Applications

ExampleGreen's identities

Let $(M, g)$ be a compact Riemannian manifold with boundary, and $u, v$ smooth functions. Taking $X = u \nabla v$ :

\operatorname{div}(u \nabla v) = u \Delta v + g(\nabla u, \nabla v),

so the divergence theorem gives Green's first identity:

\int_M (u \Delta v + g(\nabla u, \nabla v)) \, \mathrm{dvol}_g = \int_{\partial M} u \frac{\partial v}{\partial \nu} \, \mathrm{dvol}_{\partial M}.

Subtracting with $u$ and $v$ interchanged yields Green's second identity:

\int_M (u \Delta v - v \Delta u) \, \mathrm{dvol}_g = \int_{\partial M} \left(u \frac{\partial v}{\partial \nu} - v \frac{\partial u}{\partial \nu}\right) \mathrm{dvol}_{\partial M}.

RemarkIntegration by parts on manifolds

The divergence theorem provides the foundation for integration by parts on manifolds, which is indispensable in the calculus of variations, PDE theory, and mathematical physics. Weak formulations of elliptic PDEs on manifolds all rest on this identity.