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\begin{document}

\begin{center}
{\bf EML 4930/5930 \hfill Analysis in M.E. II \hfill 03/03/08}\\
{\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 Consider the following volume in Cartesian coordinates, shaped
  like a square cylinder with an uneven top assuming that $z$ is
  upwards:
  \begin{displaymath}
    0\le x\le \pi
    \qquad
    0\le y\le\pi
    \qquad
    0\le z\le x^2 + y^2 + xy
  \end{displaymath}
  There is a heat flux through this volume given by
  \begin{displaymath}
    \vec q = \sin(x)\cos(y)\ihat -\cos(x)\sin(y)\jhat
  \end{displaymath}
  and we are interested in the net heat flow coming out of the volume,
  as given by
  \begin{displaymath}
    \int_S \vec q\cdot\vec n \dd S
  \end{displaymath}
  where $S$ consists of all 6 outside surfaces of the volume.
  \begin{enumerate}
  \item Show that the heat flow out of the bottom surface
    $z=0$ is zero.
  \item Show that the heat flow out of the side surfaces $x=0$, $x=\pi$,
    $y=0$, and $y=\pi$ is also zero.
  \item That leaves the heat flow through the top surface.  Write an
    explicit expression for $\vec n\dd S$ in terms of $x$ and $y$ for
    this top surface.
  \item Write the integral to be evaluated on the top surface out
    completely, but do not integrate it yet.
  \item Show that $\vec q =\nabla\times\big[\sin(x)\sin(y)\khat\big]$.
  \item Based on that fact, explain what the top integral will be.
  \end{enumerate}

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

\item Boundary layer coordinates $x_1$, $x_2$, and $x_3$ are defined
  by the following expression for the position vector $\vec r=(x,y,z)$:
  \begin{displaymath}
    \vec r = \vec r_0(x_1) + x_2 \vec n(x_1) + x_3 \khat
  \end{displaymath}
  where $(x_1)$ indicates that those vectors are functions of $x_1$.
  In particular,
  \begin{displaymath}
    \frac{\D \vec r_0}{\D x_1} = \vec s(x_1)
    \qquad
    \frac{\D \vec n}{\D x_1} = \kappa(x_1)\vec s + \lambda(x_1)\vec n
  \end{displaymath}
  and $\vec s$, $\vec n$ and $\khat$ are a set of mutually orthogonal
  unit vectors.
  \begin{enumerate}
  \item Explain why $\lambda=0$.
  \item Explain why this is an orthogonal curvilinear coordinate
    system.
  \item Find the expression for the Laplacian of a quantity $w$ in
    terms of these coordinates.
  \item Work the expression out completely.
  \end{enumerate}

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

\item Consider the following partial differential equation
  \begin{displaymath}
    u_{xx} - 6 u_{xy} + 5 u_{yy} = e^x
  \end{displaymath}
  \begin{enumerate}
  \item Find a particular solution.
  \item Classify the equation.
  \item Derive the general solution $u(x,y)$ to the equation,
    (expressed in terms of $x$ and $y$, of course).
  \end{enumerate}

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

\end{enumerate}

\end{document}
