Laplace operator

In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable.

This corresponds to div(grad φ), hence the use of the symbol del to represent it:

It is also written as Δ.

In two-dimensional Cartesian coordinates, the Laplacian is:

In three-dimensional Cartesian coordinates:

In cylindrical coordinates:

In spherical coordinates:

The Laplacian is linear:

The following holds also:

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