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Mathematics-Online lexicon:

Laplace Operator and Harmonic Functions

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Let $ f = f(x,y,z)$ be a twice differentiable scalar function. Then

$\displaystyle \Delta U =
\operatorname{div}(\operatorname{grad}U) =
...}{\partial y^2}+
\frac{\partial^2 f}{\partial z^2} = f_{xx} + f_{yy} + f_{zz}

is called the Laplace-Operator.

As divergence and curl the operator $ \Delta$ is invariant under orthogonal coordinate transformations.

$ f$ is called harmonic provided $ \Delta f = 0 .$

A typical example of a harmonic function is

$\displaystyle f(x,y,z) = \frac{k}{\sqrt{x^2 + y^2 + z^2}} . $


see also:


  automatically generated 7/ 5/2005