Laplacian

Let $(x_1,\mathrm{\dots },x_n)$ be Cartesian coordinates for some open set $\mathrm{\Omega }$ in ${ℝ}^n$. Then the Laplacian differential operator $\mathrm{\Delta }$ is defined as

$$\mathrm{\Delta }=\frac{{\partial }^2}{\partial x_1^2}+\mathrm{\cdots }+\frac{{\partial }^2}{\partial x_n^2}.$$

In other words, if $f$ is a twice differentiable function $f:\mathrm{\Omega }\to ℂ$, then

$$\mathrm{\Delta }f=\frac{{\partial }^{2}f}{\partial x_1^2}+\mathrm{\cdots }+\frac{{\partial }^{2}f}{\partial x_n^2}.$$

A coordinate independent definition of the Laplacian is $\mathrm{\Delta }=\nabla \cdot \nabla$, i.e., $\mathrm{\Delta }$ is the composition of gradient and codifferential.

A harmonic function is one for which the Laplacian vanishes.

Notes

An older symbol for the Laplacian is ${\nabla }^2$ – conceptually the scalar product of $\nabla$ with itself. This form is more favoured by physicists.

Derivation

Click here¡http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj”¿ to see an article that derives the Laplacian in spherical coordinates.

Title	Laplacian
Canonical name	Laplacian
Date of creation	2013-03-22 12:43:48
Last modified on	2013-03-22 12:43:48
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	18
Author	matte (1858)
Entry type	Definition
Classification	msc 31B05
Classification	msc 31B15
Related topic	DAlembertian
Related topic	Codifferential
Defines	Laplace operator