PlanetMath: Laplace equation

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Laplace equation

(Definition)

The scalar form of Laplace's equation is the partial differential equation

$\displaystyle \nabla ^2f = 0$

and the vector form is

$\displaystyle \nabla ^2\mathbf{A}= 0,$

where $\nabla ^2$ is the Laplacian. It is a special case of the Helmholtz differential equation with

A function which satisfies Laplace's equation is said to be harmonic. Since Laplace's equation is linear, the superposition of any two solutions is also a solution.

"Laplace equation" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Other names:

Laplace differential equation

Also defines:

harmonic

Attachments:: regularity theorem for the Laplace equation (Theorem) by rspuzio
Laplace equation in cylindrical coordinates (Derivation) by bloftin

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Cross-references: solutions, function, Helmholtz differential equation, Laplacian, vector, partial differential equation, scalar
There are 33 references to this entry.

This is version 4 of Laplace equation, born on 2002-11-13, modified 2006-11-13.
Object id is 3593, canonical name is LaplacesEquation.
Accessed 12314 times total.

Classification:

AMS MSC:

26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

Pending Errata and Addenda

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