![]() ![]() ![]() The Laplacian is a common operator in image processing and computer vision (see the Laplacian of Gaussian, blob detector, and scale space).The discrete Laplace operator is a finite-difference analog of the continuous Laplacian, defined on graphs and grids.The Laplacian in differential geometry.The vector Laplacian operator, a generalization of the Laplacian to vector fields.Laplace–Beltrami operator, generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold.The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation. ![]() The additional factor of c in the metric is needed in physics if space and time are measured in different units a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations, and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics. It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. It is usually denoted by the symbols ∇ ⋅ ∇ 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. ![]()
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