\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}

\pagestyle{empty}

%\setlength{\parindent}{0pt}\setlength{\parskip}{\baselineskip}

%\renewcommand{\baselinestretch}{2}

%\setlength{\jot}{9pt}

\usepackage{html}
\usepackage{epsf}
%\usepackage{cite}

%\bibliographystyle{plain}

\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 11/8/06}\\
{\it Closed book \hfill Van Dommelen \hfill 11:50-12:40 pm}
\end{center}

Solutions should be fully {\em derived} showing all intermediate
results, using class procedures.  Show all reasoning.  Bare answers are
absolutely not acceptable, because I will assume they come from your
calculator (or the math handbook, sometimes,) instead of from you.
You must state what result answers what part of the question if there
is any ambiguity.  Answer exactly what is asked; you do not get any
credit for making up your own questions and answering those.

You {\em must} use the systematic procedures described in class, not
mess around randomly until you get some answer.  Do not take
shortcuts.  Some reminders: You need to reduce matrices {\em
  completely} to echelon form where appropriate to the question, (but
{\em not} to row canonical if it would be inefficient for large
matrices), using the procedure given in class, find the basis of the
null spaces and orthonormalize eigenvectors using modified
Gram-Schmidt where appropriate, etcetera.  Do not throw away zero rows
in Gaussian elimination.  Coordinate transformations must be found
using transformation matrices only.  Coordinate transformations for
symmetric matrices must be orthonormal.  Inverses of unitary matrices
must be found by transposing.  Principal axes and principal values
must be found as eigenvalue problems using class procedures.
Polynomials of matrices above degree 2 must be found using
diagonalization.  Quadratic forms must be reduced using class
procedures.

One book of mathematical tables, such as Schaum's Mathematical
Handbook, may be used, as well as a calculator, and a handwritten
letter-size formula sheet.

\begin{enumerate}
  
\item {\bf Background:} Statically undetermined structures can have
  internal stress fields of varying nature.  One way to see the
  different possible fields is to write finite element equations for
  the structure and then examine the basis of the null space.

  {\bf Question:} Find a basis of the null space of the matrix
  \begin{displaymath}
    \left(
      \begin{array}{rrrrr}
         3 &  6 &  9 & 12 & 15 \\
         2 &  4 &  4 &  4 &  4 \\
        -2 & -4 & -3 & -2 & -1
      \end{array}
    \right)
  \end{displaymath}

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} The eigenvalues of the $4\times4$ Dirac
  matrix
  \begin{displaymath}
    \left(
      \begin{array}{rrrr}
        1 &  0 &  0 &  p \\
        0 &  1 &  p &  0 \\
        0 &  p & -1 &  0 \\
        p &  0 &  0 & -1
      \end{array}
    \right)
  \end{displaymath}
  describe the possible energies of an electron moving in free space,
  where $p$ is the scaled linear momentum of the electron.
  
  {\bf Question:} Find the eigenvalues of the matrix above.  Do not
  use any Gaussian elimination.  Without solving for the eigenvectors,
  what can you say about the diagonalizability of the Dirac matrix and
  why?

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item
  {\bf Background:} The potential energy of a system of two masses
  aligned in an array using springs is
  \begin{displaymath}
    q = 7 x^2 -12 xy + 23 y^2
  \end{displaymath}
  where $x$ is the displacement of the first mass, and $y$ the one
  of the second.
  
  {\bf Question:} Find the approriate new unknowns $x'$ and $y'$ in
  terms of which $q$ above can be written without an $x'y'$ term.
  Graphically show the relationship between the coordinates $x,y$ and
  $x',y'$.

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

\end{enumerate}

\end{document}

