Definition:Divergence Operator/Riemannian Manifold/Definition 2

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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:

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

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