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\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
Curl; Vector Integration Homework (February 7, 2013) \\
\end{center}
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The dual of the Euclidean electromagnetic field tensor
is defined by
\begin{eqnarray} \label{dFmn}
   ^*F_{\alpha\beta} = \frac{1}{2}\,\epsilon_{\alpha\beta\mu\nu}
     F_{\mu\nu}
\end{eqnarray} 
where the indices run from 1 to 4, the Einstein convention is used 
and $F_{\mu\nu}$ is an anti-symmetric (i.e., $F_{\mu\nu}=-F_{\nu\mu}$) 
rank two tensor
\begin{eqnarray} \nonumber
  \left(F_{\mu\nu}\right)\ =\ \left( \matrix{ 
           0   & F_{12} & F_{13} & F_{14} \cr
        -F_{12}&   0    & F_{23} & F_{24} \cr
        -F_{13}& -F_{23}&   0    & F_{34} \cr
        -F_{14}& -F_{24}& -F_{34}& 0    } \right)\ .
\end{eqnarray} 

\begin{enumerate}

\item Calculate the elements of $^*F_{\alpha\beta}$ in terms of the 
independent elements of $F_{\mu\nu}$ and write $^*F_{\alpha\beta}$
as a matrix.

\item The electromagnetic field tensor can be written in form of
derivatives of a 4-potential 
\begin{eqnarray} \label{Fmn}
  F_{\mu\nu}\ =\ \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}\,.
\end{eqnarray} 
Use this expression to proof the homogeneous Maxwell equations 
in their form
\begin{eqnarray} \label{ddFmn}
  \partial_{\alpha}\ ^*F_{\alpha\beta}\ =\ 0\,.
\end{eqnarray} 

Remark: Therefore (\ref{Fmn}) implies that no magnetic monopoles 
exist, because (\ref{ddFmn}) includes (upon identification of the
$\vec{E}$ and $\vec{B}$ fields) the relation $\nabla\cdot\vec{B}
=0$.

\end{enumerate}

\end{document}
