## Laplacian

The Laplacian operator for a Scalar function is defined by

 (1)

in Vector notation, where the are the Scale Factors of the coordinate system. In Tensor notation, the Laplacian is written
 (2)

where is a Covariant Derivative and
 (3)

The finite difference form is
 (4)
For a pure radial function ,
 (5)

Using the Vector Derivative identity
 (6)

so
 (7)

Therefore, for a radial Power law,
 (8)

A Vector Laplacian can also be defined for a Vector A by

 (9)

in vector notation. In tensor notation, A is written , and the identity becomes
 (10)

Similarly, a Tensor Laplacian can be given by
 (11)

An identity satisfied by the Laplacian is

 (12)

where is the Hilbert-Schmidt Norm, is a row Vector, and is the Matrix Transpose of A.

To compute the Laplacian of the inverse distance function , where , and integrate the Laplacian over a volume,

 (13)

This is equal to
 (14)

where the integration is over a small Sphere of Radius . Now, for and , the integral becomes 0. Similarly, for and , the integral becomes . Therefore,
 (15)

where is the Delta Function.