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'''Laplace's equation''' is an [[equation]] such that the sum of all of the non-mixed second order [[partial derivative]]s of a [[function]] equaling zero. All functions that have a [[Laplacian]] of zero, are called [[harmonic function]]s.
'''Laplace's equation''' is an [[equation]] such that the sum of all of the non-mixed second order [[partial derivative]]s of a [[function]], f, equaling zero. All functions that have a [[Laplacian]] (usually denoted <math>\nabla^2 f</math> by physicists and <math>\Delta f</math> by mathematicians) of zero (and thus are a solution to Laplace's equation) are called [[harmonic function]]s.
In <math>\mathbb{R}^3</math>, Laplace's equation in Cartesian co-ordinates is:
: <math> \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
</math>
or in the mathematicians' notation,
<math>\Delta f =
{\partial^2 f\over \partial x^2 } +
{\partial^2 f\over \partial y^2 } +
{\partial^2 f\over \partial z^2 } = 0.
</math>
[[category:mathematics]]
[[category:mathematics]]