Laplace's equation

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 Δ = ∇ ⋅ ∇ = ∇ 2 {displaystyle Delta = abla cdot abla = abla ^{2}} is the Laplace operator, ∇ ⋅ {displaystyle abla cdot } is divergence operator (also symbolized 'div'), ∇ {displaystyle abla } is the gradient operator (also symbolized 'grad'), and f ( x , y , z ) {displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h ( x , y , z ) {displaystyle h(x,y,z)} , we have This is called Poisson's equation, a generalization of Laplace's equation, Laplace's and Poisson's equation are the simplest examples of elliptic partial differential equations. The Laplace equation is also a special case of the Helmholtz equation. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In Cartesian coordinates, In cylindrical coordinates,