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\begin{document}

\begin{center}
{\bf EML 4930/5061 \hfill Analysis in M.E. II \hfill 03/02/11}\\
{\it Closed book \hfill Van Dommelen \hfill 9:35-10:25 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 A curve is described by
  \begin{displaymath}
    x = t \qquad y = 1 + t \qquad z = 1+t^2
  \end{displaymath}
  Find the unit vector in the direction of the curve, the curvature,
  the normal to the curve, the binormal, and the torsion.  Why are you
  not surprised by the value of the torsion?  Note, you should find
  that the normal is
  \begin{displaymath}
    \frac{-t\ihat-t\jhat+\khat}{\sqrt{1+2t^2}}
  \end{displaymath}

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

\item Suppose that an electric field has the form
  \begin{displaymath}
    \vec E = (2x+yz)\ihat+(x + y - \sin z)\jhat + (x^3+y^3) \khat
  \end{displaymath}
  Find the flux of this vector field through the spherical surface
  of radius 2 centered around the point (1,1,1).

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

\item Transform the partial differential equation
  \begin{displaymath}
    u_{yy} + 4 u_{xz} + 3 u_{zz} = 0
  \end{displaymath}
  into the wave equation with unit wave velocity by rotating and
  stretching coordinates.  Give the new coordinates in terms of $x$,
  $y$, and $z$.

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

\end{enumerate}

\end{document}
