\begin{definition}[Definition:Divergence Operator/Riemannian Manifold/Definition 1]
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:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf X 
      | o = := 
      | r = \nabla \cdot \mathbf X
}}
{{eqn | r = \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X}
}}
{{end-eqn}}
where:
:$\star_g$ is the Hodge star operator of $\struct {M, g}$
:$\d_{\d R}$ is de Rham differential.
\end{definition}