Hodge star operator
Let V be a -dimensional ( finite) vector space with inner product . The Hodge star operator (denoted by ) is a linear operator mapping http://planetmath.org/node/3050-forms on to -forms, i.e.,
In terms of a basis for and the corresponding dual basis for (the star used to denote the dual space is not to be confused with the Hodge star!), with the inner product being expressed in terms of components as , the -operator is defined as the linear operator that maps the basis elements of as
Here, , and is the Levi-Civita permutation symbol
This operator may be defined in a coordinate-free manner by the condition
where the notation denotes the inner product on -forms (in coordinates, ) and is the unit volume form associated to the metric. (in coordinates, )
Generally , where is the identity operator in . In three dimensions, for all . On with Cartesian coordinates, the metric tensor is , and the Hodge star operator is
The Hodge star operation occurs most frequently in differential geometry in the case where is a -dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor and is a cotangent vector space of . Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.
Title | Hodge star operator |
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Canonical name | HodgeStarOperator |
Date of creation | 2013-03-22 13:31:41 |
Last modified on | 2013-03-22 13:31:41 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 11 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 53B21 |
Synonym | Hodge operator |
Synonym | star operator |
Defines | hodge star operator |