Mathematics-Online lexicon: Laplace Operator and Harmonic Functions

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	Mathematics-Online lexicon:
	Laplace Operator and Harmonic Functions

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overview

Let

be a twice differentiable scalar function. Then

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

is called the Laplace-Operator.

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

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

A typical example of a harmonic function is

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

()

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automatically generated 7/ 5/2005