Hodge Star

On an oriented $n$-D Riemannian Manifold, the Hodge star is a linear Function which converts alternating Differential k-Form to alternating $(n-k)$-forms. If $w$ is an alternating k-Form, its Hodge star is given by

$$w(v_1, \ldots, v_k) = (^*w)(v_{k+1}, \ldots, v_n)$$

when $v_1$, ..., $v_n$ is an oriented orthonormal basis.

See also Stokes' Theorem