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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Levi-Civita Tensor 1}\\ (January 11, 2013) \\
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\rightline{Group \#~~~~~~~~}

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

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

\bigskip

\begin{enumerate}
\item Use binary numbers 0, 1 and write down the numbers 0 to 3.

0 \\ 1 \\ 2 \\ 3

Add one more column in which you substitute $0\to 1$, $1\to 2$ and 
one last column in which you count the decimal numbers from 1 to 4.
\item Use numbers with base 3 and symbols 0, 1, 2 to write down 
the numbers 0 to 26. \eject
 0\\  1\\  2\\  3\\  4\\  5\\  6\\  7\\  8\\  9\\ 10\\ 11\\ 12\\ 13\\
14\\ 15\\ 16\\ 17\\ 18\\ 19\\ 20\\ 21\\ 22\\ 23\\ 24\\ 25\\ 26\\
Add one more column in which you substitute $n\to n+1$ for 
$n=0,\,1,\,2$ and one last column in which you count the
decimal numbers from 1 to 27.

\item Write down the {\bf permutations} of $1\,2$. 
How many permutations of $1\,2$ are there? % \bigskip

\item Write down the transposition of $1\,2$. % \bigskip

\item Use the permutations of $1\,2$ to write down the 
permutations of $1\,2\,3$ by starting in each case with 3 on 
the right and transposing the number 3 with its left neighbor
until this is no longer possible. How many permutations of 
$1\,2\,3$ are there? % \bigskip % \vfil \eject

\item Along the same lines: Use the permutations of $1\,2\,3$ to 
write down the permutations of $1\,2\,3\,4$. How many permutations 
of $1\,2\,3\,4$ are there? % \bigskip 

\item Proof that there are $n!$ permutions $\pi_1,\dots,\pi_n$
of $1,\dots,n$. % \bigskip

\item With $i_1=1,\dots,n,\ i_2=1,\dots,n,\ \dots\ ,\ i_n=1,\dots,n$ 
the definition of the Levi-Civita tensor is
$$  \epsilon_{i_1,\dots,i_n}\ =\ \cases{
  +1~{\rm for}~i_1,\dots,i_n~{\rm even~permutation,} \cr
  -1~{\rm for}~i_1,\dots,i_n~{\rm odd~permutation,} \cr
   0~{\rm for}~i_1,\dots,i_n~{\rm no~permutation.} } $$
A permutation $i_1,\dots,i_n$ is even, when it is generated
by an even number of transpositions of $1,\dots,n$ and odd,
when it is generated by an odd number of transpositions of 
$1,\dots,n$.  \smallskip

How many non-zero elements are there?  \smallskip

How many non-zero positive elements are there?  \smallskip % \eject

\item Write down all elements of the Levi-Civita tensor for $n=2$.
% \bigskip

\item Write down all elements of the Levi-Civita tensor for $n=3$.
% \bigskip

\item Write down all non-zero elements of the Levi-Civita tensor for 
$n=4$. % \bigskip

\end{enumerate}

\end{document}

