Laplace operator

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 $\nabla\cdot\nabla$ , $\nabla 2$ , or $Δ$ . The Laplacian $Δ f (p)$ of a function $f$ at a point $p$ , up to a constant depending on the dimension, is the rate at which the average value of $f$ over spheres centered at $p$ deviates from $f (p)$ as the radius of the sphere grows. 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.

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 a given gravitational potential. Solutions of the equation $Δ f = 0$ , now called Laplace's equation, are the so-called harmonic functions, and represent the possible gravitational fields in free space.