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Introduction

Laplace's equation
Laplace's equation is a partial differential equation. Partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.
Laplace's equation is named after Pierre-Simon Laplace who first studied its properties. The solutions of Laplace's equation 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. The general theory of solutions to Laplace's equation is known as potential theory. In the study of heat conduction, the Laplace equation is the steady-state heat equation.