Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 17; 8.

Here’s another example of calculating the covariant derivative in the Schwarzschild (S) metric. We’re given a vector with coordinates in the S metric of:

The covariant derivative is given by

Since the only non-zero component of is and it depends only on , most of the terms are zero.

The Christoffel symbols for the S metric are:

The one non-zero derivative is

and the values of the second term in 2 are

with all other terms being zero.

The covariant derivative is then (with the row index and the column index):