Difference between revisions of "Hodge star"

Revision as of 02:35, January 25, 2010

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-{{math-a}}
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-Let <math>M</math> be a [[Riemannian manifold]] in <math>n</math> dimensions with [[metric]] <math>g</math>. The '''Hodge star''' operator is a [[linear]] [[operator]] from i-[[differential form]]s to (n-i)-differential forms
-<math>
-*: \Omega^i(T^*M) \rightarrow \Omega^{n-i}(T^*M)
-</math>
-defined as follows: Let <math>\phi_1,...,\phi_n</math> be a local orthonormal [[coframe]] (i.e., a collection of locally defined 1-forms which are [[orthonormal]] with respect to the induced metric on the <!--Don't wikilink this unless you know what you are doing-->cotangent space). Then we define
-<math>
-*\phi_1\wedge\cdots\wedge\phi_i = \pm \phi_{i+1}\wedge\cdots\wedge\phi_n
-</math>
-where the plus or minus is chosen so that
-<math>\phi_1\wedge\cdots\wedge\phi_i\wedge *(\phi_1\wedge\cdots\wedge\phi_i)</math>
-is the [[volume]] form on <math>M</math>. To define the Hodge star operator for general forms, we simply extend the above definition by [[linearity]].
-==Example==
-Give <math>R^2</math> the standard metric so that <math>dx, dy</math> is a coframe. Then the volume form is <math>dx\wedge dy</math>. Thus,
-<math>
-*dx = dy
-</math>
-and
-<math>
-*dy = -dx
-</math>
-and in general
-<math>
-* fdx + gdy = fdy - gdx
-</math>
-[[Category:Calculus]]