Definition:Divergence Operator

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:

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

Real Cartesian Space
The usual context in which the divergence operator is encountered is real Cartesian space:

Also known as
The divergence of $\mathbf f$ is usually vocalised div $\mathbf f$.