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\hskip36pc \= Math.\ 302-01\\
\>Spring, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Sixth Examination \\
\it Friday, April 15, 2005
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\noindent {\small {\bf Instructions:} This exam should be done on
your own paper. Your name should be on each sheet and on the back
of the last sheet; the answers should appear written carefully and
in order. If in doubt, show intermediate steps: Full credit may
not be given, even for correct answers, unless work is arranged
clearly and explained. This exam is closed book. You may leave
after handing in your exam paper, but be sure to check your
answers carefully. You may keep this exam sheet.  Each problem is
worth 25 points.}

\Large
\begin{enumerate}
   \item Write down a parametrization of the line through
         $(1,1,1)$ and $(2,3,4)$.
   \item Write down a parametrization of the plane through
         $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
   \item  Suppose
          \begin{equation} \label{eq:vector-field}
             \vec F(x,y) =\left[\begin{array}{r}
                            y \\ -x \end{array}\right],
          \end{equation}
          and suppose $\mathcal{C}$ is the curve $y=x^2$, $0\le
          x\le 1$.  Then compute the line integral
          $$\int_{\mathcal{C}} \vec F\cdot d\vec{r}.$$
   \item Is the vector field in Equation~\ref{eq:vector-field}
         conservative?  State why or why not.  If it is
         conservative, then write down a potential function.
\end{enumerate}
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