Hodge star

From Conservapedia

$\pi_1(S^1)=?\,$

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

Let $M$ be a Riemannian manifold in $n$ dimensions with metric $g$ . The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

$*: \Omega^i(T^*M) \rightarrow \Omega^{n-i}(T^*M)$

defined as follows: Let $φ 1,...,φ n$ 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 $d x, d y$ is a coframe. Then the volume form is $dx\wedge dy$ . Thus,

$* d x = d y$

and

$* d y = - d x$

and in general

$* f d x + g d y = f d y - g d x$

Hodge star

From Conservapedia

Example

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