Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 3 / b

For a mixed tensor,Write $T=\bar{T}+\delta T$ and $\Gamma =\bar{\Gamma }+\delta \Gamma$. The homogeneous background already satisfies ${\bar{\nabla }}_{\mu }{\bar{T}}^{\mu }{}_{\nu }=0$, and discarding products of two perturbations leavesThis is simply the first variation of the tensor's covariant derivative.