= Solution
An <orientation> selects the positive ordered bases in each tangent space. On an oriented $n$-dimensional <Riemannian manifold>, the <Riemannian volume form> $\omega_g$ is the unique smooth $n$-form satisfying
$$
\omega_g(e_1,\ldots,e_n)=1
$$
for every positively oriented orthonormal frame. In positively oriented local coordinates,
$$
\omega_g=\sqrt{\det(g_{ij})}\,dx^1\wedge\cdots\wedge dx^n.
$$
The metric induces an <inner product> on the bundle $\Lambda^pT^*M$ of $p$-forms. The <Hodge star operator> is the unique linear map
$$
*:\Omega^p(M)\longrightarrow\Omega^{n-p}(M)
$$
such that
$$
\alpha\wedge *\beta=\langle\alpha,\beta\rangle_g\,\omega_g
$$
for all $p$-forms $\alpha,\beta$. With the <codifferential> $\delta$, the <Laplace-Beltrami operator> on differential forms is
$$
\Delta=d\delta+\delta d.
$$
The <Hodge decomposition theorem> says that on a compact oriented Riemannian manifold,
$$
\Omega^p(M)
=\mathcal H^p(M)\mathbin\oplus d\Omega^{p-1}(M)
\mathbin\oplus\delta\Omega^{p+1}(M),
$$
an $L^2$-orthogonal direct sum, where $\mathcal H^p(M)=\ker\Delta$ is the finite-dimensional space of <harmonic differential form>[harmonic $p$-forms]. Every <de Rham cohomology> class has exactly one harmonic representative.
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