Difference between revisions of "Laplace's equation"
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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 | + | '''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]] | ||
Revision as of 16:02, June 6, 2010
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 
by physicists and 
by mathematicians) of zero (and thus are a solution to Laplace's equation) are called harmonic functions.
In 
, Laplace's equation in Cartesian co-ordinates is:
or in the mathematicians' notation,
