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# Hodge star operator

Let V be a $n$-dimensional ($n$ finite) vector space with inner product $g$. The *Hodge star operator* (denoted by $\ast$) is
a linear operator mapping $p$-forms on $V$ to $(n-p)$-forms, i.e.,

$\ast:\Omega^{p}(V)\to\Omega^{{n-p}}(V).$ |

In terms of a basis $\{e^{1},\ldots,e^{n}\}$ for $V$ and the corresponding dual basis $\{e_{1},\ldots,e_{n}\}$ for $V^{*}$ (the star used to denote the dual space is not to be confused with the Hodge star!), with the inner product being expressed in terms of components as $g=\sum_{{i,j=1}}^{n}g_{{ij}}e^{i}\otimes e^{j}$, the $\ast$-operator is defined as the linear operator that maps the basis elements of $\Omega^{p}(V)$ as

$\displaystyle\ast(e^{{i_{1}}}\wedge\cdots\wedge e^{{i_{p}}})$ | $\displaystyle=$ | $\displaystyle\!\!\!\!\frac{\sqrt{|g|}}{(n-p)!}g^{{i_{1}l_{1}}}\cdots g^{{i_{p}% l_{p}}}\varepsilon_{{l_{1}\cdots l_{p}\,l_{{p+1}}\cdots l_{n}}}e^{{l_{{p+1}}}}% \wedge\cdots\wedge e^{{l_{{n}}}}.$ |

Here, $|g|=\det g_{{ij}}$, and $\varepsilon$ is the Levi-Civita permutation symbol

This operator may be defined in a coordinate-free manner by the condition

$u\wedge*v=g(u,v)\,\mathop{\bf Vol}(g)$ |

where the notation $g(u,v)$ denotes the inner product on $p$-forms (in coordinates, $g(u,v)=g_{{i_{1}j_{1}}}\cdots g_{{i_{p}j_{p}}}u^{{i_{1}\ldots i_{p}}}v^{{j_{1}% \ldots j_{p}}}$) and $\mathop{\bf Vol}(g)$ is the unit volume form associated to the metric. (in coordinates, $\mathop{\bf Vol}(g)=\sqrt{\operatorname{det}(g)}e^{1}\wedge\cdots\wedge e^{n}$)

Generally $\ast\ast=(-1)^{{p(n-p)}}\operatorname{id}$, where $\operatorname{id}$ is the identity operator in $\Omega^{p}(V)$. In three dimensions, $\ast\ast=\operatorname{id}$ for all $p=0,\ldots,3$. On $\mathbb{R}^{3}$ with Cartesian coordinates, the metric tensor is $g=dx\otimes dx+dy\otimes dy+dz\otimes dz$, and the Hodge star operator is

$\displaystyle\ast dx=dy\wedge dz,\ \ \ \ \ \ \ast dy=dz\wedge dx,\ \ \ \ \ \ % \ast dz=dx\wedge dy.$ |

The Hodge star operation occurs most frequently in differential geometry in the case where $M^{n}$ is a $n$-dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor $g$ and $V$ is a cotangent vector space of $M^{n}$. Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.

## Mathematics Subject Classification

53B21*no label found*

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