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\begin{document}

\begin{center}
{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 11/02/07}\\
{\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.  Geometry must be done
using vector operations.  Use standard matrix methods to determine
linear independence and simplified bases of vector spaces.  You need to
reduce matrices to echelon form where elimination is called for, using
the basic row operations and following the class procedures.  Do not
take shortcuts.  Do not reduce further if there is no need.
Eigenvalues must be found using minors only.  Eigenvectors must be
found by identifying the basis vectors of the appropriate null space
if there are multiple eigenvalues, using the appropriate procedures.
Eigenvectors to symmetric matrices must be orthonormal.  Higher matrix
powers and polynomials must be found through transformation, not
crunching.  Inverses must be found the quick way, where possible.

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:} In finite volume fluid computations, the flow
  field may be divided into simple volumes, and you then need to find
  expressions for the areas of the surfaces of these volumes.

  {\bf Question:} Assuming that point A has coordinates (1,2,3),
  B (1,1,1), and C (2,4,6), find the area of the parallelogram that
  has BA and BC as sides.  Also find a vector of unit length normal
  to the parallelogram.

\begin{rawhtml}
<A HREF="figures/1.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Finding eigenvalues and eigenvectors of larger
  matrices is important for many mechanical engineering areas, such as
  the motion of systems of point masses.

  {\bf Question:} Find a complete set of eigenvalues and eigenvectors
  to the following matrix, using the required procedures:
  \begin{displaymath}
    \left(
      \begin{array}{cccc}
         1 & 0 & 1 & 0 \\
         0 & 2 & 0 & 1 \\
         0 & 2 & 1 & 2 \\
         0 & 1 & 0 & 2
      \end{array}
    \right)
  \end{displaymath}
  Is the matrix singular or defective?

\begin{rawhtml}
<A HREF="figures/2.jpg">Solution.</A>
\end{rawhtml}
  
\item {\bf Background:} Quadratic forms appear in such applications as
  the kinetic energy of rotating bodies.

  {\bf Question:} Consider the quadratic form
  \begin{displaymath}
    q=10 x_1^2 + 6x_1x_2 + 2 x_2^2
  \end{displaymath}
  Using the class linear algebra procedures, determine over what angle
  we need to rotate the axis system to simplify this quadratic form.
  Use the result to {\em accurately} sketch the locations in the
  $x_1,x_2$-plane where $q=1$.

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

\end{enumerate}

\end{document}
