Editing Laplace operator (section)

== Generalizations ==
A version of the Laplacian can be defined wherever the [[Dirichlet energy|Dirichlet energy functional]] makes sense, which is the theory of [[Dirichlet form]]s.  For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.

=== 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]].

===D'Alembertian===
The Laplacian can be generalized in certain ways to [[non-Euclidean]] spaces, where it may be [[elliptic operator|elliptic]], [[hyperbolic operator|hyperbolic]], or [[ultrahyperbolic operator|ultrahyperbolic]].

In [[Minkowski space]] the [[Laplace–Beltrami operator]] becomes the [[D'Alembert operator]] <math>\Box</math> or D'Alembertian:
<math display="block">\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}.</math>

It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the [[isometry group]] of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy [[particle physics]]. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the [[wave equation]]s, and it is also part of the [[Klein–Gordon equation]], which reduces to the wave equation in the massless case.

The additional factor of {{math|''c''}} in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the {{mvar|x}} direction were measured in meters while the {{mvar|y}} direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that {{math|1=[[Natural units|''c'' = 1]]}} in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on [[pseudo-Riemannian manifold]]s.