Definition:Laplace-Beltrami Operator

This article, or a section of it, needs explaining. In particular: The presentation of this, and the symbol used to define it, are completely different from the other instances of "laplacian". Recommended a) we standardise on $\nabla^2$ unless there is a very good reason why $\delta$ trumps this, and b) establish equivalence pages for $\map {\operatorname {div} } {\grad f}$ and $\dfrac {\partial^2 f} {\partial x^2}$. (Page may already exist showing the latter in various contexts. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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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

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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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds