Definition:Laplacian/Riemannian Manifold
 | This article, or a section of it, needs explaining. In particular: Establish (a) separate definitions of Laplacian based on $\nabla^2$ and $\map {\operatorname {div} } {\grad f}$ approach, and (b) 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. Note we have Laplacian on Scalar Field is Divergence of Gradient which is a start. a) I think there is a problem here, because you chose to define $\nabla^2 f$ as $\map {\operatorname {div} } {\grad f}$. In your case, "$\nabla^2$" is a just symbol denoting some operator, not an actual operator like $\nabla \cdot \nabla$ or $\nabla^\mu \nabla_\mu$. b) No such equivalence exists because what I wrote applies to arbitrarily curved spaces, while you are talking about flat space. A better way to put it is to say that Laplace-Beltrami operator in flat 1-, 2-, 3-dimensional space in some coordinates reduces to your textbook result. Or maybe we should stop trying to write an equality between Laplacian and Laplace-Beltrami operator. Just make separate pages and add that sometimes they are the same, sometimes different, so some authors abuse the nomenclature and refer to the Laplace-Beltrami operator as the Laplacian. Well basicall then i don't know, so im not in a position to resolve this. So as to implement the bare modicum of consistency, would it then be appropriate to amend all instances of Definition:Laplacian by means of the $\operatorname {div} \grad$ definition, and be done with it? We can always use the derivation that under certain rare circumstances $\operatorname {div} \grad$ is the same as $\nabla^2$, which incidentall is indeed just a symbol denoting some operato, but then again everything is just a smbol denoting and operator and not the operator itself. $+$ is just a symbol denoting addition, it's not addition itself. Or did I miss something important? 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. |
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
In the context of Riemannian manifolds, the Laplacian can be seen as $\Delta f$, particularly in older texts.
The $\nabla^2$ form is to be used 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.
Also known as
The Laplacian is also known as the Laplace operator, Laplace's operator or Laplace-Beltrami operator.
The last name is usually used in the context of submanifolds in Euclidean space and on (pseudo-)Riemannian manifolds.
Source of Name
This entry was named for Pierre-Simon de Laplace.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds