Definition:Divergence Operator/Riemannian Manifold

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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:

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


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

Definition 2

Let $\struct {M, g}$ be an oriented Riemannian manifold.

Let $\rd V_g$ be the volume form of $\struct {M, g}$.

Let $X$ be a smooth vector field on $M$.

Then the divergence of $X$, denoted by $\operatorname {div} X$, is defined by:

$\paren {\operatorname {div} X} \rd V_g := \map d {X \mathop \lrcorner \rd V_g}$

where $\lrcorner$ denotes the interior multiplication, and $\map \rd {X \mathop \lrcorner \rd V_g}$ is the exterior derivative of $X \mathop \lrcorner \rd V_g$.

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