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In 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
- <math>\alpha\wedge *\alpha = \omega.<math>
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 <math>e_1,e_2,...,e_n<math> we have
- <math>*(e_1\wedge e_2\wedge ... \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge ... \wedge e_n.<math>
More abstractly, if <math>\alpha<math> is a k-vector <math>*\alpha<math> can be completely defined by the following identity, for any k-vector <math>\zeta<math> we have
- <math>\langle\zeta\wedge *\alpha,\omega\rangle = \langle\zeta, \alpha \rangle,<math>
where <math>\langle-,-\rangle <math> 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
- <math>**=(-1)^{k(n-k)+s}id<math>
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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