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Why using diff. forms in electromagnetism?

  1. Sep 11, 2013 #1
    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. jcsd
  3. Sep 11, 2013 #2

    WannabeNewton

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    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.
     
  4. Sep 12, 2013 #3

    dextercioby

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    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.
     
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