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Hodge DualIn mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. In rough terms it is defined by dividing into a volume element ω, thought of as n vectors in an orthonormal basis wedged together, so that -
This works for pure k-vectors α from a standard basis giving volume 1. It has the following property, which defines it completely: given an oriented orthonormal basis we have -
More abstractly, if is a k-vector can be completely defined by the following identity, for any k-vector we have -
where denoted the inner product on the exterior algebra of V induced from the inner product on V (i.e. the all wedge products of elements of orthonormal basis in V form an orthonormal basis of exterior algebra). A common example of the star operator is the case n = 3, when it can be taken as the correspondence between the vectors and the skew-symmetric matrices of that size. This is used implicitly in vector calculus, for example to create the cross product vector from the wedge product of two vectors. In case n = 4 the Hodge dual acts an endomorphism of the second exterior power, of dimension 6; it splits it into self-dual and anti-self-dual subspaces, on which it acts respectively as +1 and −1. One can repeat one each tangent space of an n-dimensional Riemannian manifold, and get a Hodge dual n−k-form, from a k-form. Identities -
on Ωk(M), where s is the signature of pseudo-Riemannian manifold M. The combination of * and the exterior derivative d generates the classical operators div, grad and curl, in three dimensions. This works out as follows: d can take a 0-form (function) to a 1-form, a 1-form to a 2-form, or a 2-form to a 3-form (applied to a 3-form it just gives zero). The first case written out in components is identifiable as the grad operator. The second followed by * is an operator on 1-forms that in components is curl. The final case prefaced and followed by *, so *d*, takes a 1-form to a 0-form (function); written out in components it is div. One advantage of this expression is that the identity d2 = 0, which is true in all cases, sums up two others, namely curl of a grad and div of a curl are identically zero. The symmetrised form *d*d + d*d* is a definition of the Hodge Laplacian; it clearly leaves the degree of a form unchanged, since d increments the degree while *d* decrements the degree, both by 1.
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