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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 12}\\%
\vspace {3mm} (April 18, 2011) \\
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\item{(38)} Use $F_{\gamma\delta}=\partial_{\gamma}A_{\delta}-
\partial_{\delta} A_{\gamma}$ to show 
$\partial_{\alpha}\,^*F^{\alpha\beta}=0$, where
$$^*F^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\gamma\delta}
                   F_{\gamma\delta} $$
is the dual tensor. Here $\epsilon^{\alpha\beta\gamma\delta}$ is 
the completely antisymmetric Levi-Cevita tensor: $+1$ for
$(\alpha,\beta,\gamma,\delta)$ an even permutation of $(0,1,2,3)$,
$-1$ for $(\alpha,\beta,\gamma,\delta)$ an odd permutation of 
$(0,1,2,3)$. Not assigned.
\medskip

\item{(39)} Under a gauge transformation the vector potential
transform according to 
$$A^{\alpha}\to A'^{\,\alpha}=A^{\alpha}+\partial^{\,\alpha}\Lambda\ .$$
Calculate the corresponding transformation of the electromagnetic
field tensor
$$F^{\alpha\beta}\to F'^{\,\alpha\beta}\ .$$
Due April~22 before class (4~points).

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