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\begin{document} \Large

\begin{center}
{Mathematical Physics --- PHZ 3113}
{Final (May 2, 2013)}  % \\
\end{center}
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\bigskip

Einstein's convention is used in the following problems and
$\epsilon_{ijk}$ is the Levi-Civita tensor. 

\begin{enumerate}

\item Calculate
\begin{eqnarray} \nonumber 
  \epsilon_{12k}\,\epsilon_{21k} &=& ~~~~~~~~~~~~~~\\ \nonumber
  \epsilon_{ij3}\,\epsilon_{ij3} &=& \\ \nonumber
  \epsilon_{ijk}\,\epsilon_{ijk} &=& 
\end{eqnarray} 

\item Rewrite the expression
\begin{eqnarray} \nonumber
  &\epsilon_{ijk}\,\epsilon_{klm}\,\hat{x}_i\,\partial_j\,(A_l\,B_m)\,,&
  \\ \nonumber &{\rm where\ the}\ \hat{x}_i\ {\rm are\ Cartesian\ unit\
  vectors\ and}\ \partial_j=\frac{\partial~}{\partial x_j}\,,\ \rm into&
\end{eqnarray} 

(A) A vector product. \medskip

(B) A sum of scalar products. 

\item Consider spherical coordinates. Calculate the time derivative
$$ \dot{\hat{\theta}}\ =\ \frac{d\,\hat{\theta}}{d\,t}$$
of the unit vector $\hat{\theta}$ and express the result in spherical 
coordinates. 

\item Find the eigenvalues and (normalized) eigenvectors of the matrix
\begin{eqnarray} \nonumber
  \left( \begin{matrix} +1  &  -1  &   0  \cr
                  -1  &  +2  &  -1  \cr
                   0  &  -1  &  +1  \end{matrix} \right)\ .
\end{eqnarray} 
\end{enumerate} 

\end{document}
