## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

## Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function onEuclidean space. It is usually denoted by the symbols ∇2 or ∆. The first notation has the advantage of recalling the simple definition of the second-order operator and and avoid the confusion with the difference operator commonly indicated with ∆. The Laplacian ∇2f(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.

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