Editing Laplace operator (section)

=== Laplace–Beltrami operator ===
{{main article|Laplace–Beltrami operator}}

The Laplacian also can be generalized to an elliptic operator called the '''[[Laplace–Beltrami operator]]''' defined on a [[Riemannian manifold]]. The Laplace–Beltrami operator, when applied to a function, is the [[trace (linear algebra)|trace]] ({{math|tr}}) of the function's [[Hessian matrix|Hessian]]:
<math display="block">\Delta f = \operatorname{tr}\big(H(f)\big)</math>
where the trace is taken with respect to the inverse of the [[metric tensor]]. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on [[tensor field]]s, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the [[exterior derivative]], in terms of which the "geometer's Laplacian" is expressed as
<math display="block"> \Delta f = \delta d f .</math>

Here {{mvar|δ}} is the [[codifferential]], which can also be expressed in terms of the [[Hodge star operator|Hodge star]] and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on [[differential form]]s {{mvar|α}} by
<math display="block">\Delta \alpha = \delta d \alpha + d \delta \alpha .</math>

This is known as the '''[[Laplace–Beltrami operator#Laplace–de_Rham_operator|Laplace–de Rham operator]]''', which is related to the Laplace–Beltrami operator by the [[Weitzenböck identity]].