Hodge star operator

Let V be a $n$-dimensional ($n$ finite) vector space with inner product $g$. The Hodge star operator (denoted by $\ast$) isa linear operator mapping http://planetmath.org/node/3050$\mathrm{p}$-forms on $V$ to $(n-p)$-forms, i.e.,

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

In terms of a basis $\{e^1,\mathrm{\dots },e^n\}$ for $V$ and the corresponding dual basis $\{e_1,\mathrm{\dots },e_n\}$ 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 $\ast$-operatoris defined as the linear operator that maps the basis elements of ${\mathrm{\Omega }}^p(V)$ as

$\ast (e^{i_1}\wedge \mathrm{\cdots }\wedge e^{i_p})$$=$${\displaystyle \frac{\sqrt{|g|}}{(n-p)!}}{g}^{i_1{l}_1}\mathrm{\cdots }{g}^{i_p{l}_p}{\epsilon }_{l_1\mathrm{\cdots }{l}_p{l}_{p+1}\mathrm{\cdots }{l}_n}{e}^{l_{p+1}}\wedge \mathrm{\cdots }\wedge e^{l_n}.$

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

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

$$u\wedge *v=g(u,v)𝐕𝐨𝐥(g)$$

where the notation $g(u,v)$ denotes the inner product on $p$-forms (in coordinates, $g(u,v)=g_{i_1{j}_1}\mathrm{\cdots }{g}_{i_p{j}_p}{u}^{i_1\mathrm{\dots }{i}_p}{v}^{j_1\mathrm{\dots }{j}_p}$) and $𝐕𝐨𝐥(g)$ is the unit volume form associated to the metric. (in coordinates, $𝐕𝐨𝐥(g)=\sqrt{\det (g)}{e}^1\wedge \mathrm{\cdots }\wedge e^n$)

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

$\ast dx=dy\wedge dz,\ast dy=dz\wedge dx,\ast dz=dx\wedge dy.$

The Hodge star operation occurs most frequently in differential geometry in the case where $M^n$ is a $n$-dimensional orientable manifold witha 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.