Laplace's equation

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This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

Laplace's equation is an equation such that the sum of all of the non-mixed second order partial derivatives of a function, f, equaling zero. All functions that have a Laplacian (usually denoted $\nabla ^{2}f$ by physicists and $\Delta f$ by mathematicians) of zero (and thus are a solution to Laplace's equation) are called harmonic functions.

In $\mathbb {R} ^{3}$, Laplace's equation in Cartesian co-ordinates is:

$\nabla ^{2}f={\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}=0$

or in the mathematicians' notation, $\Delta f={\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}=0.$