Hodge integration by parts for one-forms (${\int }_M⟨df,\alpha {⟩}_g{\omega }_g=-{\int }_{M}f\ast d\ast \alpha {\omega }_g$)

On an oriented boundaryless Riemannian manifold, apply Stokes theorem to the compactly supported form $f\ast \alpha$. The graded Leibniz rule gives $\int df\wedge \ast \alpha =-\int fd\ast \alpha$. The Hodge star operator identifies these top forms with the scalar integrands displayed above. This proves that the codifferential on one-forms is $\delta =-\ast d\ast$, the formal adjoint of the exterior derivative. Compact support of $f$ suffices.