Inner Product of k-vectors
The Hodge dual induces an inner product on the space of k-vectors, that is, on the exterior algebra of V. Given two k-vectors and, one has
where ω is the normalised n-form (i.e. ω ∧ ∗ω = ω). In the calculus of exterior differential forms on a pseudo-Riemannian manifold of dimension n, the normalised n-form is called the volume form and can be written as
where is the matrix of components of the metric tensor on the manifold in the coordinate basis.
If an inner product is given on, then this equation can be regarded as an alternative definition of the Hodge dual. The wedge products of elements of an orthonormal basis in V form an orthonormal basis of the exterior algebra of V.
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