# Variations

### From GRwiki

Variations of the Christoffel symbols, Riemann tensor, Ricci tensor, curvature scalar, and inverse metric:

where is the covariant derivative compatible with the metric . The variations are general. They can be replaced by, say, coordinate derivatives or time derivatives. Bryce DeWitt's favorite identity is

where is a matrix and denotes the trace. Other useful relations, where denotes a scalar, include

A term that appears in modified gravity theories is