Solution
= Solution
The metric induces an inner product on decomposable $r$-forms by
$$
\langle\alpha_1\wedge\cdots\wedge\alpha_r,
\beta_1\wedge\cdots\wedge\beta_r\rangle
=\det(\langle\alpha_i,\beta_j\rangle),
$$
extended bilinearly. On an oriented Riemannian manifold, the <Hodge star operator> is characterized by
$$
\alpha\wedge *\beta=\langle\alpha,\beta\rangle\,d\operatorname{vol}_g.
$$
On $r$-forms in dimension $n$,
$$
\boxed{*^2=(-1)^{r(n-r)}.}
$$