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

.