# Definition:Laplace-Beltrami Operator

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## Definition

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

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

Let $\grad$ be the gradient operator.

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

The Laplacian of $f$ is defined as:

$\nabla^2 f := \map {\operatorname {div} } {\grad f}$

## Also denoted as

The Laplace-Beltrami operator can be seen as $\Delta f$, particularly in older texts.

The $\nabla^2$ form is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ as it allows less opportunity for ambiguity and misunderstanding.

## Also defined as

Sometimes the Laplacian-Beltrami operator is defined with a minus sign to make its eigenvalues nonnegative.

## Source of Name

This entry was named for Pierre-Simon de Laplace and Eugenio Beltrami.