Codifferential ($\delta$)

The codifferential is the formal $L^2$ adjoint of the exterior derivative. On $p$-forms in dimension $n$, one convention is

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

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.

A differential form is coclosed when its codifferential vanishes. On a closed Riemannian manifold, a form is a harmonic differential form exactly when it is both a closed differential form and coclosed: the identity $⟨\Delta \alpha ,\alpha ⟩=\Vert d\alpha {\Vert }^2+\Vert \delta \alpha {\Vert }^2$ proves the converse as well as necessity.