= Hodge integration by parts in arbitrary degree
{c}
{title2=$\langle d\alpha,\beta\rangle_{L^2}=\langle\alpha,\delta\beta\rangle_{L^2}$}
On a closed oriented <Riemannian manifold>, let $\alpha$ have degree $r-1$ and $\beta$ degree $r$. The <graded Leibniz rule> and <Stokes theorem> give $\int d\alpha\wedge*\beta=(-1)^r\int\alpha\wedge d*\beta$. The <codifferential> convention $\delta\beta=(-1)^{n(r+1)+1}*d*\beta$, combined with the square of the <Hodge star>, gives $*\delta\beta=(-1)^rd*\beta$. This proves that the <codifferential> is the <formal adjoint> of the <exterior derivative>. A boundary requires the corresponding boundary term or specified boundary conditions.
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