= Hodge integration by parts for one-forms
{c}
{title2=$\int_M\langle df,\alpha\rangle_g\omega_g=-\int_M f*d*\alpha\,\omega_g$}
On an oriented boundaryless <Riemannian manifold>, apply <Stokes theorem> to the compactly supported form $f*\alpha$. The <graded Leibniz rule> gives $\int df\wedge*\alpha=-\int f\,d*\alpha$. The <Hodge star operator> identifies these top forms with the scalar integrands displayed above. This proves that the <codifferential> on one-forms is $\delta=-*d*$, the <formal adjoint> of the <exterior derivative>. Compact support of $f$ suffices.
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