Laplace's equation

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In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:

where \nabla^2 is the Laplace operator and \varphi 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 steady-state heat equation.

Contents

Definition

In three dimensions, the problem is to find twice-differentiable real-valued functions \scriptstyle f, of real variables x, y, and z, such that

In Cartesian coordinates

In cylindrical coordinates,

In spherical coordinates,

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