\documentclass{article}

% paper size. [11in]
%\setlength{\paperheight}{11in}
% top margin (1 inch is added automatically) [17pt]
\setlength{\topmargin}{-24pt}
% height of the header line [12pt]
%\setlength{\headheight}{12pt}
% separation between header and body [25pt]
\setlength{\headsep}{12pt}
% from start of body to the first base line [12pt]
%\setlength{\topskip}{12pt}
% size of the body of the page (no headers or footers) [7.618in]
\setlength{\textheight}{9truein}
% bottom of body to base of footer [30pt]
\setlength{\footskip}{24pt}

% paper size. [8.5in]
%\setlength{\paperwidth}{8.5in}
% left margin (1 inch is added automatically) [39pt]
\setlength{\oddsidemargin}{0truein}
% the same for left-hand pages [40pt]
\setlength{\evensidemargin}{0truein}
% size of the body of the page [5.417in]
\setlength{\textwidth}{6.5truein}
% width and separation of marginal notes
\setlength{\marginparwidth}{.8truein}\setlength{\marginparsep}{.1truein}

\setlength{\parskip}{\baselineskip}

\pagestyle{empty}

\usepackage{html}
\usepackage{epsf}

\newcommand{\ihat}{\hat\imath}
\newcommand{\jhat}{\hat\jmath}
\newcommand{\khat}{\hat k}
\newcommand{\tfrac}[2]{{\textstyle\frac{#1}{#2}}}
\newcommand{\D}{{\rm d}}
\newcommand{\dd}{\,{\rm d}}

\begin{document}

\begin{center}
{\bf EML 4930/5930 \hfill Analysis in M.E. II \hfill 02/25/09}\\
{\it Closed book \hfill Van Dommelen \hfill 9:40-10:30 am}
\end{center}

Show all reasoning and intermediate results leading to your answer, or
credit will be lost. One book of mathematical tables, such as Schaum's
Mathematical Handbook, may be used, as well as a calculator and one
handwritten letter-size single formula sheet.

\begin{enumerate}

\item Parallelogram $ABCD$ has its corner point $A$ at the origin.
  The diagonally opposite point is C.
  \begin{enumerate}
  \item Write the area of this parallelogram in terms of the vectors
    $\vec{r}_B$ and $\vec{r}_C$.  Clean up.
  \item Parallelepiped $ABCDEFGH$ has point $G$ diagonally opposite to
    $A$ and has $E$ on the same edge as $A$.  Write diagonal $AG$ in
    terms of $\vec{r}_B$, $\vec{r}_C$, and $\vec{r}_E$.
  \item Write the volume of the parallepiped $ABCDEFGH$ in terms of
    $\vec{r}_B$, $\vec{r}_C$, and $\vec{r}_E$.  Clean up.
  \item Evaluate all of the above if $\vec{r}_B=(3,-1,1)$,
    $\vec{r}_C=(4,4,0)$, and $\vec{r}_E=(0,0,1)$.  Clean up.
  \end{enumerate}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}

\item A screw thread can be modeled as the spiral
  \begin{displaymath}
    x = \cos t \qquad y = \sin t \qquad z = C t
  \end{displaymath}
  where $C$ is a constant.  This spiral is on the surface of a
  circular cylinder of unit radius around the $z$-axis.
  \begin{enumerate}
  \item Give the unit vector tangential to the curve at a given point.
  \item Find the curvature and torsion of the spiral.
  \item Find the unit normal to the plane of the osculating circle and
    its radius.  Is the center of the circle on the axis of the
    cylinder?

  \end{enumerate}


\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}

\item Reduce the PDE
  \begin{displaymath}
    5 u_{xx} + 4 u_{xy} + 2 u_{yy} + u_x = 0
  \end{displaymath}
  to canonical form by rotating the coordinate system.  Give the
  rotated coordinates in terms of the original ones and the canonical
  PDE.

\begin{rawhtml}
<A HREF="figures/3.jpg">Solution.</A>
\end{rawhtml}

\end{enumerate}

\end{document}
