Definition:Divergence Operator/Riemannian Manifold/Definition 2
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Definition
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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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds