Derivation of the Laplacian in Spherical Coordinates

[LyleJr]

Hi all,

Sorry if this is the wrong section to post this.

For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I decided to type it out and present it in what I hope is a logical and obvious manner.

It took me about four days of working in my spare time, but I just finished and thought it might be worth sharing. The Laplacian is something that comes up a lot in textbooks, but never really gets a good explanation of why it is has its final form.

Anyways, here it is. Please excuse any spelling errors. I do think the math is all correct though.

 

[zwierz]

it would be better if you study tensor calculus and get familiar with invariant definition of the Laplace operator: $\Delta=g^{ij}\nabla_i\nabla_j$

 

[LyleJr]

it would be better if you study tensor calculus and get familiar with invariant definition of the Laplace operator: $\Delta=g^{ij}\nabla_i\nabla_j$

I agree. This was just a for fun exercise to pass the time.

 

[LyleJr]

I found a major mistake on page one, of all places. Corrected version is attached.