In d dimensions, the Hodge star sets up a duality between p-forms at (d-p)-forms, via
\alpha \wedge \beta = ({}^\star\alpha , \beta) \, \omega
where \omega is the volume form (a basis form in the one dimensional space of d-forms), and where \alpha is a p-form, and where \beta, \; {}^\star\alpha are (d-p)-forms.
So for example, when d=4, we can describe a current using a one-form J or a dual three-form *J, and given a two-form describing the EM field, F, we can take its dual to find another two-form, *F. Half of Maxwell's equations are then seen to be trivial: dF = 0 automatically since F=dA. The other half are then d {}^\star F = 4 \pi \, {}^\star J.
The book Gravitation by Misner, Thorne, and Wheeler contains some nice discussion. For example, from the simple two-form F = -q/r^2 \, dt \wedge dr (which describes a Coloumb-type purely radial electrostatic field), we obtain the simple two-form {}^\star F= q \sin(\theta) d\theta \wedge d\phi, which is suitable for integration over a sphere. The duality becomes even more vivid if you use the coframe
\sigma^{\hat{0}} = -dt, \; \sigma^{\hat{1}} = dr, \; \sigma^{\hat{2}} = r \, d\theta, \; \sigma^{\hat{3}} = r \sin(\theta) \, d\phi[/tex]<br />
Then we obtain<br />
F = q/r^2 \, \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} , \; {}^\star F = q/r^2 \, \sigma^{\hat{2}} \wedge \sigma^{\hat{3}}<br />
which we could have obtained even faster using <br />
{}^\star \left( \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} \right) =\sigma^{\hat{2}} \wedge \sigma^{\hat{3}}<br />
However, the "coordinate coframe" is usually more convenient when we are actually integrating. Note that I am talking about Minkowski spacetime here, not a curved spacetime. The volume form is simply the wedge product of the four covectors making up the coframe. However, this can be readily generalized to work on curved spacetimes.<br />
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(This example is potentially misleading, since in general, F will be a "rank two" two-form, meaning that it contains both dt \wedge dr and d\theta \wedge d\phi terms. A rank one exterior form is often called a "simple" form. Don't confuse this "rank" with "second rank tensor"! In our example, our "rank one" or "simple" two-form F corresponds to an antisymmetric "second rank" tensor.)<br />
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Another good book is Burke, Applied Differential Geometry.
