Editing Laplace operator (section)

{{short description|Differential operator}}
{{about|the mathematical operator|the integral transform|Laplace transform|other uses|List of things named after Pierre-Simon Laplace}}
{{Calculus |Vector}}

In [[mathematics]], the '''Laplace operator''' or '''Laplacian''' is a [[differential operator]] given by the [[divergence]] of the [[gradient]] of a [[Scalar field|scalar function]] on [[Euclidean space]]. It is usually denoted by the symbols <math>\nabla\cdot\nabla</math>, <math>\nabla^2</math> (where <math>\nabla</math> is the [[Del|nabla operator]]), or <math>\Delta</math>. In a [[Cartesian coordinate system]], the Laplacian is given by the sum of second [[partial derivative]]s of the function with respect to each [[independent variable]]. In other [[coordinate systems]], such as [[cylindrical coordinates|cylindrical]] and [[spherical coordinates]], the Laplacian also has a useful form. Informally, the Laplacian {{math|Δ''f''&hairsp;(''p'')}} of a function {{math|''f''}} at a point {{math|''p''}} measures by how much the average value of  {{math|''f''}} over small spheres or balls centered at {{math|''p''}} deviates from {{math|''f''&hairsp;(''p'')}}.

The Laplace operator is named after the French mathematician [[Pierre-Simon de Laplace]] (1749–1827), who first applied the operator to the study of [[celestial mechanics]]: the Laplacian of the [[gravitational potential]] due to a given mass density distribution is a constant multiple of that density distribution. Solutions of [[Laplace's equation]] {{math|1=Δ''f'' = 0}} are called [[harmonic function]]s and represent the possible [[gravitational potential]]s in regions of [[vacuum]].

The Laplacian occurs in many [[differential equations]] describing physical phenomena. [[Poisson's equation]] describes  [[electric potential|electric]] and [[gravitational potential]]s; the [[diffusion equation]] describes [[heat equation|heat]] and [[fluid mechanics|fluid flow]]; the [[wave equation]] describes [[wave equation|wave propagation]]; and the [[Schrödinger equation]] describes the [[wave function]] in [[quantum mechanics]]. In [[image processing]] and [[computer vision]], the Laplacian operator has been used for various tasks, such as [[blob detection|blob]] and [[edge detection]]. The Laplacian is the simplest [[elliptic operator]] and is at the core of [[Hodge theory]] as well as the results of [[de Rham cohomology]].