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Maxwell equations from tensor notation to component notation

  1. Apr 30, 2016 #1
    1. The problem statement, all variables and given/known data

    Verify that ##\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}## is equivalent to ##\partial_{[\mu}F_{\nu\lambda]}=0##,

    and that they are both equivalent to ##\tilde{\epsilon}^{ijk}\partial_{j}E_{k}+\partial_{0}B^{i}=0## and ##\partial_{i}B^{i}=0##.

    2. Relevant equations

    3. The attempt at a solution

    ##\partial_{[\mu}F_{\nu\lambda]}=0##

    ##\implies \frac{1}{3!}(\partial_{\mu}F_{\nu\lambda}-\partial_{\mu}F_{\lambda\nu}-\partial_{\nu}F_{\mu\lambda}+\partial_{\nu}F_{\lambda\mu}-\partial_{\lambda}F_{\nu\mu}+\partial_{\lambda}F_{\mu\nu})=0##, using the definition of antisymmetrization of a tensor

    ##\implies \partial_{\mu}F_{\nu\lambda}-\partial_{\mu}F_{\lambda\nu}-\partial_{\nu}F_{\mu\lambda}+\partial_{\nu}F_{\lambda\mu}-\partial_{\lambda}F_{\nu\mu}+\partial_{\lambda}F_{\mu\nu}=0##

    ##\implies \partial_{\mu}F_{\nu\lambda}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}+\partial_{\lambda}F_{\mu\nu}=0##

    ##\implies 2(\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu})=0##

    ##\implies \partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}=0##

    Is my working so far correct?
     
  2. jcsd
  3. May 4, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, I believe so.
     
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