Laplacian

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols



∇
⋅
∇


{\displaystyle \nabla \cdot \nabla }
,




∇

2




{\displaystyle \nabla ^{2}}
(where



∇


{\displaystyle \nabla }
is the nabla operator), or



Δ


{\displaystyle \Delta }
. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(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, where the operator gives a constant multiple of the mass density when it is applied to the gravitational potential due to the mass distribution with that given density. Solutions of the equation Δf = 0, now called Laplace's equation, are the so-called harmonic functions and represent the possible gravitational fields in regions of vacuum.
The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling a variety of physical phenomena. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection.

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