PlanetMath: Laplacian

(more info)

Math for the people, by the people.

Main Menu

Laplacian

(Definition)

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

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

In other words, if

is a twice differentiable function $f:\Omega\to \mathbb{C}$ , then

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

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

A harmonic function is one for which the Laplacian vanishes.

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

Click here to see an article that derives the Laplacian in spherical coordinates.

Anyone with an account can edit this entry. Please help improve it!

"Laplacian" is owned by matte. [ full author list (4) | owner history (1) ]

(view preamble)

Also defines:

Laplace operator

Log in to rate this entry.
(view current ratings)

AMS MSC:	31B05 (Potential theory :: Higher-dimensional theory :: Harmonic, subharmonic, superharmonic functions)
	31B15 (Potential theory :: Higher-dimensional theory :: Potentials and capacities, extremal length)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

No messages.

Interact