## 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.