Why using diff. forms in electromagnetism?

[christianpoved]

In electromagnetism we introduce the following differential form
\begin{array}{c}
\mathbb{F}=F_{\mu \nu}dx^{\mu}\wedge dx^{\nu}
\end{array}
Then the homogeneus Maxwell equations are equivalent to:
\begin{array}{c}
d\mathbb{F} = 0
\end{array}
And is nice, but what purpose does this have?, there is something interesting in saying that F is a closed form?

 

[WannabeNewton]

By the Poincare lemma, if $dF = 0$ then (at least locally) $F = dA$ for some 1-form $A$; $A$ is of course none other than the 4-potential. This is why we can describe electromagnetism using the 4-potential.

Also, writing down Maxwell's equations as $dF = 0$ and $d{\star}F = {\star}j$ allows us to easily use Stokes' theorem $\int_{\Omega}d\omega = \int _{\partial \Omega} \omega$ when needed. There are other uses of course of writing down Maxwell's equations as $dF = 0$ and $d{\star}F = {\star}j$ (one of them being pure elegance!) and you will see the above form a lot in gauge theoretic treatments.

 

[dextercioby]

Differential forms are at the heart of modern physics, expecially gauge field theory (for which vacuum classical electrodynamics is the simplest case). And GR looks spectacular in terms of forms.