Coordinate Representation of Laplace-Beltrami Operator

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

Let $\nabla^2$ be the Laplace-Beltrami operator.

Then:

$\nabla^2 f = \dfrac 1 {\sqrt {\det g} } \dfrac \partial {\partial x^i} \paren {g^{ij} \sqrt {\det g} \dfrac {\partial f} {\partial x^j}}$

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Proof

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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