
\noindent
(A) Compare $\partial_{\alpha}F^{\alpha\beta}=(4\pi/c)\,J^{\beta}$ 
for $\beta=0$ with the inhomogeneous Maxwell equations 
$$ \nabla\cdot\vec{E} = 4\pi \rho = \frac{4\pi}{c}\,J^0
   ,~~~\nabla \times \vec{B} - {1\over c}{\partial\vec{E}
   \over\partial t}={4\pi\over c}{\vec J} $$
to identify $F^{00}$, $F^{10}$, $F^{20}$ and $F^{30}$ in terms of
$\vec{E}$ and $\vec{B}$. Continue with $\beta=1$, then $\beta=2$. 
Do you need $\beta=3$ too?

\noindent (B) The result of (A) translates into (see the lecture notes) 
$$ ({^*\!F}^{\alpha\beta}) = \left(\begin{matrix} 
         0      & ^*F^{01} & ^*F^{02} & ^*F^{03} \\
       ^*F^{10} &  0       & ^*F^{12} & ^*F^{13} \\
       ^*F^{20} & ^*F^{21} &  0       & ^*F^{23} \\
       ^*F^{30} & ^*F^{31} & ^*F^{32} &  0 \end{matrix}\right)\
    =\ \left(\begin{matrix} 
         0   & -B^x & -B^y & -B^z \\
         B^x &  0   &  E^z & -E^y \\
         B^y & -E^z &  0   &  E^x \\
         B^z &  E^y & -E^x &  0   \end{matrix}\right)\ . $$
Show that this form and $\partial_{\alpha}\,^*F^{\alpha\beta}=0$ 
imply the homogeneous Maxwell equations 
$$ \nabla\cdot \vec{B} = 0\,,~~~\nabla \times \vec{E} +
   {1\over c} {\partial \vec{B}\over \partial t} = 0\,. $$
Fix $\beta$ to 0, 1, 2, 3 and discuss each case separately.
\bigskip
\end{enumerate}

