In mathematics, Laplace's equation is a secondorder partial differential equation named after PierreSimon Laplace who first studied its properties. This is often written as:
where is the Laplace operator and is a scalar function.
Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations.
The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are all harmonic functions and are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steadystate heat equation.
Contents
Definition
In three dimensions, the problem is to find twicedifferentiable realvalued functions , of real variables x, y, and z, such that
In Cartesian coordinates
In cylindrical coordinates,
In spherical coordinates,
Full article ▸
