Definition:Divergence Operator/Riemannian Manifold/Definition 1

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

Let $\mathbf X : \map {\CC^\infty} M \to \map {\CC^\infty} M$ be a smooth vector field.


The divergence of $\mathbf X$ is defined as:

\(\ds \operatorname {div} \mathbf X\) \(:=\) \(\ds \nabla \cdot \mathbf X\)
\(\ds \) \(=\) \(\ds \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X}\)

where:

$\star_g$ is the Hodge star operator of $\struct {M, g}$
$\d_{\d R}$ is de Rham differential.


Also known as

The divergence of a vector field $\mathbf V$ is usually vocalised div $\mathbf V$.


Also see

  • Results about divergence can be found here.