\documentclass[12pt]{article} % BB Jan 11 2013.
\usepackage{epsfig}
\textheight 9.0in
\textwidth  6.0in
\topmargin -1.0in  
\headsep    0.4in  
\oddsidemargin  0.25in  
\evensidemargin 0.25in  

\def\sqr#1#2{{\vcenter{\hrule height.#2pt
   \hbox{\vrule width.#2pt height#1pt \kern#1pt
      \vrule width.#2pt}
   \hrule height.#2pt}}}
\def\square{{\mathchoice\sqr64\sqr64\sqr{4.2}3\sqr{3.0}3}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
% \pagestyle{empty}

\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Einstein Convention} \\

 Homework -- January 23, 2013 \\
\end{center}
\setcounter{equation}{0}

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

\bigskip

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.3]{Einstein1921}
\includegraphics[scale=0.73]{Levi-Civita}
% \caption{Einstein (left), Levi-Civita (right).}
\end{center}
\end{figure}

\centerline{Einstein (left), Levi-Civita (right).} \bigskip

\noindent When asked what he liked best about Italy, Einstein 
said ``Spaghetti and Levi-Civita"

% \begin{enumerate} \item \end{enumerate}

Einstein was a lazy person, so he got quickly tired of writing all
these sums we have seen and invented the convention (known as 
Einstein convention): {\bf Two identical indices are automatically 
summed.}

\bigskip

Sometimes (but not always) one indicates indices, which one sums in 
the next step by {\bf contractions}, i.e., connects them by a lines. 
Examples:
\begin{eqnarray} \nonumber
  \epsilon_{ijk}\epsilon_{ilm} &=&
  \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl} \\ \nonumber
  \\ \nonumber
  \delta_{ij}\,x_i\,y_j &=& x_i\,y_i\ =\ \vec{x}\cdot\vec{y}
  \\ \nonumber \\ \nonumber
  \delta_{ij}\,x_i\,y_j &=& x_j\,y_j\ =\ \vec{x}\cdot\vec{y}
\end{eqnarray}  \bigskip

{\bf Homework}: \medskip

Re-write homeworks 2 and 3 using the Einstein convention and
draw the contractions.

\end{document}
