Laplacian

The Laplacian for a scalar function is a scalar differential operator defined by

(1)

where the are the scale factors of the coordinate system (Weinberg 1972, p. 109; Arfken 1985, p. 92).
Note that the operator is commonly written as by mathematicians (Krantz 1999, p. 16).
The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation

(2)

the Helmholtz differential equation

(3)

the wave equation

(4)

and the Schrödinger equation

(5)

The analogous operator obtained by generalizing from three dimensions to four-dimensional spacetime is denoted and is known as the d'Alembertian. A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined. The square of the Laplacian is known as the biharmonic operator.

A vector Laplacian can also be defined, as can its generalization to a tensor Laplacian.
The following table gives the form of the Laplacian in several common coordinate systems.

coordinate system
Cartesian coordinates
cylindrical coordinates
parabolic coordinates
parabolic cylindrical coordinates
spherical coordinates

The finite difference form is

(6)

For a pure radial function ,

			(7)
			(8)
			(9)

Using the vector derivative identity

(10)

so

			(11)
			(12)
			(13)

Therefore, for a radial power law,

			(14)
			(15)
			(16)

An identity satisfied by the Laplacian is

(17)

where is the Hilbert-Schmidt norm, is a row vector, and is the transpose of .
To compute the Laplacian of the inverse distance function , where , and integrate the Laplacian over a volume,

(18)

This is equal to

			(19)
			(20)
			(21)
			(22)
			(23)

where the integration is over a small sphere of radius . Now, for and , the integral becomes 0. Similarly, for and , the integral becomes . Therefore,

(24)

where is the delta function.