Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 4 / c

For forms of the same degree, define the Hodge inner product byOn a $p$-form $\beta$, define the codifferentialequivalently $\delta =(-1{)}^{n(p+1)+1}\ast d\ast$ in dimension $n$. If $\alpha$ has degree $p-1$, Stokes theorem on the compact boundaryless manifold givesRearranging and using the definition of $\delta$ givesThus $\delta$ is the formal adjoint of $d$.