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\begin{document}

\begin{center}
{\bf EML 4930/5061 \hfill Analysis in M.E. II \hfill 03/03/10}\\
{\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 vector field
  \begin{displaymath}
    \vec A = (2x+2y+yz, 2x+3y^2+xz,2z+xy)
  \end{displaymath}
  Find the curl of this vector field.  Show that $\vec A$ must be the
  gradient of some function $\phi$. Then rigorously derive the most
  general form of this function.  Now consider the line integral
  $\int\vec A\D\vec r$ starting from the origin to the point (1,1,1)
  through the path
  \begin{displaymath}
    y=\left(\frac{\sinh x}{\sinh 1}\right)^3
    \qquad
    z=\left(\frac{\sin x}{\sin 1}\right)^4
  \end{displaymath}
  and then back again to the origin over the path
  \begin{displaymath}
    x=\left(\frac{\sinh z}{\sinh 1}\right)^3
    \qquad
    y=\left(\frac{\sin z}{\sin 1}\right)^2
  \end{displaymath}
  Give the value of this integral.  Also give the value of the two
  individual pieces that the integral consists of.

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

\item Consider the curvilinear coordinates $\xi$, $\eta$, and $\theta$
  given by
  \begin{displaymath}
    x=\xi^3 - 3\xi\eta^2 \qquad y = 3\xi^2\eta-\eta^3 \qquad z=\theta
  \end{displaymath}
  Show that this curvilinear coordinate system is orthogonal.  Find
  the metric indices $h_\xi$, $h_\eta$, and $h_\theta$.  Using these
  metric indices {\em only}, find the derivative of the unit vector
  $\ihat_\xi$ with respect to $\eta$.

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

\item Classify and solve the PDE
  \begin{displaymath}
    u_{tt} + u_t + t = u_{xx} + u_x + x
  \end{displaymath}
  Suppose you wanted to solve this PDE in a rectangle in the
  $xt$-plane, what would be appropriate BC/IC to add?

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

\end{enumerate}

\end{document}
