Hodge star operator

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Hodge star operator or Hodge star is a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> defined on the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> of a <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Orientation_(vector_space)/lGXs42Nw">oriented</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> endowed with a <a href="/facts/Degenerate_bilinear_form/9TLGBvSH">nondegenerate</a> <a href="/facts/Symmetric_bilinear_form/UI9b6RK7">symmetric bilinear form</a>. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by <a href="/facts/W._V._D._Hodge/5XsdVugb">W. V. D. Hodge</a>.
For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a> of two basis vectors, and its Hodge dual is the <a href="/facts/Normal_vector/jhtLIhcu">normal vector</a> given by their <a href="/facts/Cross_product/uRBJzkcW">cross product</a>; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-vectors; the dimensions of these spaces are the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficients</a> 
 
 
 
 
 
 
 
 (
 
 
 n
 k
 
 
 )
 
 
 
 
 =
 
 
 
 
 (
 
 
 n
 
 n
 −
 k
 
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\tbinom {n}{k}}={\tbinom {n}{n-k}}}
 
.
The <a href="/facts/Natural_transformation/xwPumTHt">naturalness</a> of the star operator means it can play a role in differential geometry when applied to the cotangent <a href="/facts/Vector_bundle/cXACAa1I">bundle</a> of a <a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo-Riemannian manifold</a>, and hence to <a href="/facts/Differential_form/T9gyHtMv">differential k-forms</a>. This allows the definition of the codifferential as the Hodge adjoint of the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a>, leading to the <a href="/facts/Laplace%25E2%2580%2593Beltrami_operator/AJmyUpWz">Laplace–de Rham operator</a>. This generalizes the case of 3-dimensional Euclidean space, in which <a href="/facts/Divergence/zpwwkFoU">divergence</a> of a vector field may be realized as the codifferential opposite to the <a href="/facts/Gradient/TsR7uukj">gradient</a> operator, and the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> on a function is the divergence of its gradient. An important application is the <a href="/facts/Hodge_decomposition/b6Mju74J">Hodge decomposition</a> of differential forms on a <a href="/facts/Closed_manifold/jMG6G6Bl">closed</a> Riemannian manifold.

Hodge star operator open-in-new

Hodge star operator