Laplace's equation

Laplace's equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. The solutions of Laplace's equation are important in many fields of science, notably electromagnetism, astronomy, and fluid dynamics, because they describe the behavior of electric, gravitational, and fluid potentials.


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


Laplace's equation


Solutions of Laplace's equation are called harmonic functions.