Converting Laplacian to spherical coordinates.

[scorpion990]

Hey! I'm self-studying a bit of quantum chemistry this summer. My introductory P.chem book (David Ball) doesn't specifically show the conversion of the laplacian operator from Cartesian to spherical coordinates. I don't really feel satisfied until I've actually derived it myself... So... Question:

Is there an easier, non "brute-force" method of converting the laplacian from Cartesian to spherical coordinates? I know that I can rewrite it using the multivariable chain rule and evaluating dozens of partial derivatives, but I'd rather not. From my experience, there has to be an intermediate theorem which shortens the process.

I'd prefer something that is not too overly theoretical (I'm still a freshman!) I've taken undergraduate math up until differential equations, and I'm familiar with some elements of applied math.

Thanks!

 

[D H]

See this wiki page: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates"

 

[scorpion990]

That doesn't seem to show how to derive the formulas...
I started the conversion, but I'm not patient enough to go through will all of it... I guess I could take div(grad f), but then I'll need to know how to convert the standard basis vectors from cartesian into spherical coordinates. Err..

EDIT: I found a source which does the brute-force way:
http://planetmath.org/encyclopedia/DerivationOfTheLaplacianFromRectangularToSphericalCoordinates.html#foot1096

...and I'm not going to bother following the derivation step by step. There must be a higher-level simplification.