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{{math-a}}

{{math-a}}

Let <math>M</math> be a Riemannian n-manifold with metric <math>g</math>. The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms  

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>

<math>

</math>

</math>

defined as follows: Let <math>\phi_1,...,\phi_n</math> be a local orthonormal co-frame (i.e., a collection of locally defined 1-forms which are orthonormal with respect to the induced metric on the cotangent space). Then we define

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 [[cotangent]] space). Then we define

<math>

<math>

<math>\phi_1\wedge\cdots\wedge\phi_i\wedge *(\phi_1\wedge\cdots\wedge\phi_i)</math>

<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.

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==

==Example==

* fdx + gdy = fdy - gdx

* fdx + gdy = fdy - gdx

</math>

</math>