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

Note that the expression found in this exercise is only the Laplacian as applied to scalar functions. The full expression for applied to arbitrary tensors in terms of 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 "" terms are ambiguous. It would be better to write them as or instead, so that it's clear the covariant derivative acts only on , and not on .