Laplace Operator

In mathematics, the Laplace operator, or Laplacian, is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space, commonly written using the notations delta, nabla dot nabla, or nabla squared. It is named for the French mathematician Pierre-Simon de Laplace, who first applied it in the study of celestial mechanics, where he showed that the Laplacian of a gravitational potential arising from a mass density is a constant multiple of that density. Functions that solve Laplace's equation are called harmonic functions and describe possible gravitational potentials in regions with no mass; the operator also appears in Poisson's equation for electric and gravitational potentials, in the diffusion equation for heat and fluid flow, in the wave equation, and in the Schrodinger equation of quantum mechanics. Beyond physics, the Laplacian is used in image processing and computer vision for tasks such as blob and edge detection, is regarded as the simplest elliptic operator underlying Hodge theory and de Rham cohomology, and serves as the infinitesimal generator of standard Brownian motion.