Definition:Laplacian/Scalar Field

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Definition

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:

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


Cartesian $3$-Space

In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:


Let $R$ be a region of Cartesian $3$-space $\R^3$.

Let $\map U {x, y, z}$ be a scalar field acting over $R$.


The Laplacian of $U$ is defined as:

$\nabla^2 U := \dfrac {\partial^2 U} {\partial x^2} + \dfrac {\partial^2 U} {\partial y^2} + \dfrac {\partial^2 U} {\partial z^2}$

where $\nabla$ denotes the del operator.


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.