Editing Hodge star operator (section)

=== Computation in index notation ===
We compute in terms of [[tensor index notation]] with respect to a (not necessarily orthonormal) basis <math display="inline">\left\{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right\}</math> in a tangent space <math>V = T_p M</math> and its dual basis <math>\{dx_1,\ldots,dx_n\}</math> in <math>V^* = T^*_p M</math>, having the metric matrix <math display="inline">(g_{ij}) = \left(\left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right\rangle\right)</math> and its inverse matrix <math>(g^{ij}) = (\langle dx^i, dx^j\rangle)</math>. The Hodge dual of a decomposable ''k''-form is:
<math display="block">
{\star}\left(dx^{i_1} \wedge \dots \wedge dx^{i_k}\right) 
\ =\  
\frac{\sqrt{\left|\det [g_{ij}]\right|}}{(n-k)!} g^{i_1 j_1} \cdots g^{i_k j_k} \varepsilon_{j_1 \dots j_n} dx^{j_{k+1}} \wedge \dots \wedge dx^{j_n}.
</math>

Here <math>\varepsilon_{j_1 \dots j_n}</math> is the [[Levi-Civita symbol]] with <math>\varepsilon_{1 \dots n} = 1</math>, and we [[Einstein notation|implicitly take the sum]] over all values of the repeated indices <math> j_1,\ldots,j_n</math>. The factorial <math>(n-k)!</math> accounts for double counting, and is not present if the summation indices are restricted so that <math>j_{k+1} < \dots < j_n</math>. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to [[Pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian manifold]]s.

An arbitrary differential form can be written as follows:
<math display="block">
\alpha \ =\  \frac{1}{k!}\alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k} 
\ =\ \sum_{i_1 < \dots < i_k} \alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k}.
</math>

The factorial <math>k!</math> is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component <math>\alpha_{i_1, \dots, i_k}</math> so that the Hodge dual of the form is given by
<math display="block">
{\star}\alpha = \frac{1}{(n-k)!}({\star} \alpha)_{i_{k+1}, \dots, i_n} dx^{i_{k+1}} \wedge \dots \wedge dx^{i_n}.
</math>

Using the above expression for the Hodge dual of <math>dx^{i_1} \wedge \dots \wedge dx^{i_k}</math>, we find:<ref>{{cite book| last=Frankel|first=T.| title=The Geometry of Physics| publisher=Cambridge University Press| year=2012| isbn=978-1-107-60260-1| edition=3rd}}</ref>
<math display="block">
({\star} \alpha)_{j_{k+1}, \dots, j_n} = \frac{\sqrt{\left|\det [g_{ab}]\right|}}{k!} \alpha_{i_1, \dots, i_k}\,g^{i_1 j_1}\cdots g^{i_k j_k} \,\varepsilon_{j_1, \dots, j_n}\, .
</math>

Although one can apply this expression to any tensor <math>\alpha</math>, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor.  It is thus equivalent to antisymmetrization followed by applying the Hodge star.

The unit volume form <math display="inline">\omega = {\star} 1\in \bigwedge^n V^*</math> is given by:
<math display="block">\omega = \sqrt{ \left| \det [g_{ij}] \right| }\;dx^1 \wedge \cdots \wedge dx^n .</math>