# Definition:Divergence Operator/Riemannian Manifold/Definition 1

## 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.