Hodge dual

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. It has the following property, which defines it completely: given an oriented orthonormal basis e₁,e₂,...,e_n we have

More abstractly, if α is a k-vector * α can be completely defined by the following identity: for any k-vector ζ 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), and ω is the normalised volume form defined by the inner product and the orientation.

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 the construction above for each tangent space of an n-dimensional oriented Riemannian manifold, and get the Hodge dual n− k-form, of a k-form. More generally, in the not oriented case, one can define the hodge star of a k-form is a n− k- pseudo differential form .

Identities

* * = ( - 1)^{k(n - k) + s}id

on Ω^k(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 or Laplace Beltrami-operator ; it clearly leaves the degree of a form unchanged, since d increments the degree while *d* decrements the degree, both by 1.