I'm unable to reproduce the Christoffel element 2 3 3. It seems to me that it must vanish because the metric is diagonal and derivatives of the second coordinate (phi) vanishes. Have I missed something ?
The elements 3 2 3 and 3 3 2 also vanishes, for me. I get element 3 3 3 to cot(theta).
Thank you for a very instructive article.