Laplace operator
Assume, we are on the sphere with metric
φ is an eigenfunction to the Laplace operator on the sphere for any {,,}∈\0.
The eigenvalue is
The same computation with respect to a different coordinate system is
The code for the Laplacian is fairly simple:

In the next lines, we set up a parametrization f:→ of the 3sphere in , compute the induced metric tensor, and check that φ:→R defined below, is an eigenfunction of the Laplace operator.
Now, back to the . What is the explicit partial differential equation an eigenfunction function of the Laplace operator satisfies on ? We give the answer with respect to the two different coordinate systems introduced above.
Created by Mathematica (December 22, 2006) 