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\begin{center}{ \Large{Worksheets}  \\\large{11/16/2017} }\end{center}
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\begin{enumerate}
\item Consider a surface given by $x=2u$, $y=u^2+v$, $z=v^2$.
\begin{enumerate}
\item Find two tangent vectors to the surface at point A(0,1,1).\vspace{2in}
\item Find a normal vector to the surface at point A.\vspace{1.5in}
\item Write \emph{cartesian} and \emph{parametric} equations for the tangent plane at point A.\vspace{2in}
\end{enumerate}
\item Give a parametrization for the following surfaces
\begin{enumerate}
\item The single cone $z^2=x^2+y^2$, $z\ge 0$\vspace{2in}
\item The elliptic cone $z^2=4x^2+9y^2$, $z\ge 0$\vspace{2in}
\item The unit sphere $x^2+y^2+z^2=1$\vspace{2in}
\item The ellipsoid $x^2 +4y^2+\dfrac{(z-1)^2}{9}=1$\vspace{2in}
\end{enumerate}
\item Let $S$ be the ellipsoid $x^2 +4y^2+\dfrac{(z-1)^2}{9}=1$.\begin{enumerate}
\item Apply the principle ``Level sets of a function are perpendicular to its gradient vectors'' to find a normal vector to $S$ at point A$\left(\frac{1}{3},\frac{1}{3},3\right)$.\vspace{2in}
\item Write the cartesian equation for the tangent plane at point A.\vspace{1.5in}
\item Compared to Problem 1, what makes it (a little) easier for us to find the tangent plane in this problem.\vspace{1in}
\end{enumerate}
\item Consider a surface $S$ given by $x=u^2$, $y=v^2$, $z=uv$, $0\le u,v\le 1$.\begin{enumerate}
\item Find the tangent vectors $T_u$ and $T_v$\vspace{2in}
\item Calculate $\|T_u\times T_v\|$\vspace{2.5in}
\item Find the area of $S$\vspace{2in}
\end{enumerate}
\item A surface $S$ is given by parametric equations $x=u-v$, $y=u+v$, $z=uv$, where $u^2+v^2\le 1$.\begin{enumerate}
\item Find the area of the surface\vspace{5.3in}
\item Compute the integral \[\iint_{S}(x+y)dS\]
\end{enumerate}
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\end{enumerate}
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