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\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Levi-Civita Tensor 2 Applications} \\ (January 14, 2013) \\
\end{center}
\setcounter{equation}{0}

\rightline{Group \#~~~~~~~~}

\leftline{Participating students (print):} \vskip 60pt

\centerline{------------------------------------------------------}

\bigskip

\begin{enumerate}

\item In a {\bf cyclic permutation} the first element becomes the 
last and the others stay in their order. Backward the last becomes
the first and the others stay in their order. 

Write down the values of the cyclic permutations of $\epsilon_{123}$ 
and then of $\epsilon_{132}$. Do you get all $3D$ values this way? 
Which are positive and which are negative? \bigskip

\item Calculate in $3D$
\begin{eqnarray}
   \sum_{j=1}^3 \sum_{k=1}^3 \epsilon_{1jk}\,a_j\,b_k =
   ~~~~~~~~~~~~~~~~~~~~~~~\\
   \sum_{j=1}^3 \sum_{k=1}^3 \epsilon_{2jk}\,a_j\,b_k =
   ~~~~~~~~~~~~~~~~~~~~~~~\\
   \sum_{j=1}^3 \sum_{k=1}^3 \epsilon_{3jk}\,a_j\,b_k =
   ~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray}

\item Calculate in $3D$ 
\begin{eqnarray}
   \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \epsilon_{ijk}\,
   \hat{x}_i\,\,a_j\,b_k = ~~~~~~~~~~~~~~~
\end{eqnarray}
\noindent and compare with $\vec{a}\times\vec{b}$. 
\vskip 60pt % \vfil \eject

{\bf Definition of the determinant of a $nD$ matrix:} \bigskip

\begin{eqnarray} \nonumber
   \left|\matrix{a_{11} & \dots & a_{1n} \cr
                   .    &       &   .    \cr
                   .    &       &   .    \cr
                   .    &       &   .    \cr
                 a_{n1} & \dots & a_{nn}}\right|\ =
   ~~~~~~~~~~~~~~~~~~~~~~~~~\\ \\ \nonumber 
   \sum_{i_1=1}^n \dots \sum_{i_n=1}^n 
   \epsilon_{i_1\dots i_n}\,a_{1i_1}\dots a_{ni_n}\,.
\end{eqnarray} \bigskip

\item Calculate in $2D$
\begin{eqnarray}
   \sum_{i=1}^2 \sum_{j=1}^2 \epsilon_{ij}\,a_{1i} a_{2j}
   \ = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray} \bigskip

\item Calculate in $3D$
\begin{eqnarray}
   \sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3 \epsilon_{ijk}\,
   a_{1i} a_{2j} a_{3k}\ = ~~~~~~~~~~~~~~~~~~~
\end{eqnarray} 
using cyclic permutations. \bigskip

\item Substitute in the previous expression 
\begin{eqnarray} 
  a_{11}=\hat{x}_1,\ a_{12}=\hat{x}_2,\ a_{13}=\hat{x}_3\,.
\end{eqnarray} \bigskip

\item Substitute in the previous expression 
\begin{eqnarray} 
  &~& a_{21}=a_1\,,\ a_{22}=a_2\,,\ a_{23}=a_3\,,~~~~~\\ 
  &~& a_{31}=b_1\,,\ a_{32}=b_2\,,\ a_{33}=b_3\,.~~~~~
\end{eqnarray} \bigskip

\item In $3D$, write $\vec{a}\times\vec{b}$ as determinant (book p.23).
\bigskip

\begin{eqnarray} 
  \vec{a}\times\vec{b}\ =~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray} \bigskip

\item {\bf Proof the $3D$ identity}
\begin{eqnarray} \label{epsdel}
  \sum_{i=1}^3 \epsilon_{ijk}\epsilon_{ilm} =
  \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}
\end{eqnarray} 
by calculating the expression for the nine possibilites
of values for $jk$, i.e., 11, 12, 13, 21, 22, 23, 31, 
32, 33. \bigskip

\end{enumerate}

\end{document}

\item Use the identity (\ref{epsdel}) to calculate
\begin{eqnarray} 
  (\hat{a}\times\hat{b})\cdot(\hat{a}\times\hat{b})
\end{eqnarray} 
where $\hat{a}$ and $\hat{b}$ are unit vectors. Eliminate all 
$\hat{a}\cdot\hat{b}$ in favor of $\cos\theta$.\bigskip

\item Show $\vec{a}\times\vec{b}=a\,b\,\sin\theta$ (book p.21).
