# Why using diff. forms in electromagnetism?

1. Sep 11, 2013

### 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?

2. Sep 11, 2013

### 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.

3. Sep 12, 2013

### 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.