Definition:Laplacian

Definition

Scalar Field

Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $\map U {x_1, x_2, \ldots, x_n}$ be a scalar field over $\R^n$.

Let the partial derivatives of $U$ exist throughout $\R^n$.

The Laplacian of $U$ is defined as:

$\ds \nabla^2 U := \sum_{k \mathop = 1}^n \dfrac {\partial^2 U} {\partial {x_k}^2}$

Vector Field

Let $\map {\R^n} {x_1, x_2, \ldots, x_n}$ denote the real Cartesian space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

Let $\mathbf V: \R^n \to \R^n$ be a vector field on $\R^n$:

$\forall \mathbf x \in \R^n: \map {\mathbf V} {\mathbf x} := \ds \sum_{k \mathop = 0}^n \map {V_k} {\mathbf x} \mathbf e_k$

where each of $V_k: \R^n \to \R$ are real-valued functions on $\R^n$.

That is:

$\mathbf V := \tuple {\map {V_1} {\mathbf x}, \map {V_2} {\mathbf x}, \ldots, \map {V_n} {\mathbf x} }$

Let the partial derivative of $\mathbf V$ with respect to $x_k$ exist for all $f_k$.

The Laplacian of $\mathbf V$ is defined as:

\(\ds \nabla^2 \mathbf V\)

\(:=\)

\(\ds \sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf V} {\partial {x_k}^2}\)

Riemannian Manifold

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 Laplace-Beltrami operator of $f$ is defined as:

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

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.

Also see

• Results about the Laplacian can be found here.

Source of Name

This entry was named for Pierre-Simon de Laplace.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Laplacian or Laplace operator
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Laplacian (Laplace operator)