Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 4 / b / i / Solution

Varying $\Phi$ and integrating by parts givesSince $\delta F_{ab}=2{\nabla }_{[a}\delta A_{b]}$, variation of the potential and another integration by parts giveThese are the scalar and Maxwell equations of Einstein-Maxwell-dilaton theory.