Coordinate Representation of Laplace-Beltrami Operator

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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 $f \in \map {\CC^\infty} M : M \to \R$ be a smooth mapping on $M$.

Let $\Delta$ be the Laplace-Beltrami operator.


Then:

$\ds \Delta f = \frac 1 {\sqrt {\det g}} \dfrac \partial {\partial x^i} \paren{g^{ij} \sqrt{\det g} \dfrac {\partial f}{\partial x^j}}$

Proof


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