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Line 1: − + − − 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]]
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