Hodge star

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This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} be a Riemannian manifold in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} dimensions with metric $g$. The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *: \Omega^i(T^*M) \rightarrow \Omega^{n-i}(T^*M) }

defined as follows: Let $\phi _{1},...,\phi _{n}$ 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

$*\phi _{1}\wedge \cdots \wedge \phi _{i}=\pm \phi _{i+1}\wedge \cdots \wedge \phi _{n}$

where the plus or minus is chosen so that

$\phi _{1}\wedge \cdots \wedge \phi _{i}\wedge *(\phi _{1}\wedge \cdots \wedge \phi _{i})$

is the volume form on $M$. To define the Hodge star operator for general forms, we simply extend the above definition by linearity.

Example

Give $R^{2}$ the standard metric so that $dx,dy$ is a coframe. Then the volume form is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx\wedge dy} . Thus,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle *dx = dy }

and

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and in general

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle * fdx + gdy = fdy - gdx }