[ 2 posts ] 
 Exercise [22.32] 
Author Message

Joined: 26 Mar 2010, 04:39
Posts: 109
Post Exercise [22.32]
Here is exercise 22.32 in which we are asked to find the expression for the Laplacian operator in spherical polars.

This problem can be solved by working out the change of coordinates from the usual Cartesian coordinates, but I solved it by working out the connection coefficients using the result from Exercise 14.26

File comment: Exercise 22.32
Exercise22.32.pdf [66.6 KiB]
Downloaded 130 times
17 Aug 2010, 05:05

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Exercise [22.32]
Note that the expression found in this exercise is only the Laplacian as applied to scalar functions. The full expression for \nabla^2 \equiv g^{ab}\nabla_a\nabla_b applied to arbitrary tensors T^{c\ldots d}_{e\ldots f} in terms of \left(\phi,\theta\right) coordinates can be found in a similar manner, but it's a far more complicated expression, and requires considerably lengthier calculations. (So I shan't post it here...!!!).

Also, it's a minor quibble, but note that in the posted solution, on the two lines immediately following "Taking the covariant derivatives of these expressions, we find:", the "\nabla f\Gamma" terms are ambiguous. It would be better to write them as \left(\nabla f\right)\Gamma or \Gamma\nabla f instead, so that it's clear the covariant derivative acts only on f, and not on \left(f\Gamma\right).

03 Mar 2013, 16:01
   [ 2 posts ]