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

\delta \Gamma^a_{bc} & = & \frac{1}{2}g^{ad} \Bigl( \nabla_b \delta g_{dc} + \nabla_c \delta g_{bd} 
     - \nabla_d \delta g_{bc} \Bigr) \\
\delta R^a{}_{bcd} & = & \nabla_c \delta\Gamma^a_{bd} - \nabla_d \delta\Gamma^a_{bc} \\
\delta R_{bd} & = & \nabla_a \delta\Gamma^a_{bd} - \nabla_d \delta\Gamma^a_{ba} \\
\delta R & = & \nabla^a \nabla^b \delta g_{ab} - \nabla^a \nabla_a \delta (\ln |g|) - R^{ab}\delta g_{ab} \\
\delta g^{ab} & = & -g^{ac} \delta g_{cd}\,  g^{db} 

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

\delta(\ln(\det M)) = Tr(M^{-1} \delta M)

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

\delta \nabla_a \nabla_b S & = & \nabla_a \nabla_b \delta S - \frac{1}{2} \nabla^c S \Bigl( \nabla_a\delta g_{bc} + \nabla_b\delta g_{ac} - \nabla_c\delta g_{ab}
    \Bigr) \\
\delta \nabla^2 S & = & \nabla^2 \delta S - (\nabla^a \nabla^b S) \delta g_{ab} - (\nabla^a S) \nabla^b\delta g_{ab} + (\nabla^a S) \nabla_a \delta (\ln\sqrt{|g|}) 

A term that appears in modified gravity theories is

 g^{\mu\nu} \delta \Gamma^\sigma_{\mu\nu} - g^{\mu\sigma} \delta \Gamma^\rho_{\rho\mu} =  g^{\mu\nu} \nabla^\sigma(\delta g^{\mu
\nu}) - \nabla_\rho (\delta g^{\sigma\rho}).
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