= Solution
For forms of the same degree, define the Hodge inner product by
$$
\langle\alpha,\beta\rangle_X
=\int_X\alpha\wedge *\beta.
$$
On a $p$-form $\beta$, define the <codifferential>
$$
\delta\beta=(-1)^p*^{-1}d*\beta,
$$
equivalently $\delta=(-1)^{n(p+1)+1}*d*$ in dimension $n$. If $\alpha$ has degree $p-1$, <Stokes theorem> on the compact boundaryless manifold gives
$$
0=\int_Xd(\alpha\wedge *\beta)
=\int_Xd\alpha\wedge *\beta+(-1)^{p-1}\int_X\alpha\wedge d*\beta.
$$
Rearranging and using the definition of $\delta$ gives
$$
\langle d\alpha,\beta\rangle_X
=\langle\alpha,\delta\beta\rangle_X.
$$
Thus $\delta$ is the formal adjoint of $d$.
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