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

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 \begin{eqnarray*} \ast dx = dy\wedge dz,\ \ \ \ \ \ \ast dy = dz\wedge dx,\ \ \ \ \ \ \ast dz = dx\wedge dy. \end{eqnarray*} 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.