Let $(M,g)$ be a compact oriented Riemannian manifold without boundary, and let $\alpha\in A^k(M;\mathbb C)$ be a smooth complex-valued $k$-form. Let $d^*$ denote the formal adjoint of the [exterior derivative](/theorems/1525) with respect to the $L^2$ [inner product](/page/Inner%20Product) induced by $g$, and let