Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 4 / c / Solution

The first vacuum Maxwell equations equation is preserved for every vector field because part a givesOn two-forms in four dimensions, the mixed volume tensor $T_{ab}{}^{cd}$ represents twice the Hodge star operator. Part b therefore says that a conformal Killing field commutes with the Hodge star:It follows thatHence $\tilde{F}={\mathcal{L}}_{X}F$ satisfies both vacuum Maxwell equations whenever $F$ does.