Coordinate Representation of Divergence

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Theorem

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

Let $U \subseteq M$ be an open set.

Let $\tuple {x^i}$ be local smooth coordinates.

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

Let $\operatorname {div}$ be the divergence operator.


Then:

$\ds \map {\operatorname {div}} {X^i \dfrac \partial {\partial x^i}} = \frac 1 {\sqrt g} \dfrac \partial {\partial x^i} \paren {X^i \sqrt {\det g}}$

Proof


Sources