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):