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{\bf Math 222 in class problems} \hspace{.125in} \textbf{Week: April 6, 2021} \hspace{.5in}  \textbf{Name:} %outlier malcom
\begin{enumerate}

%DO CHANGE OF COORDINATES IN WEEK 7

%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.

\item  Find a parametrization for each of the following surfaces (hint: use an angular variable).
\begin{enumerate}
\item The surface obtained by revolving the curve $z=f(x)$, $a<x<b$ in the $xz$-plane around the $z$-axis,  $a>0$. 
\item The surface obtained by revolving $z=f(x)$, $a<x<b$ in the $xz$-plane around the $x$-axis,  $ f(x)>0$.
\item The lower sheet of the hyperboloid $z^2-2x^2 - y^2 =1$.
\item The cylinder $x^2+z^2=9$.
\end{enumerate}
  
  \item  For each of the following maps $f: \R^2 \to \R^3$, describe the (possibly singular) surface $S = f(\R^2)$ and find a description of $S$ as the locus of an equation $F(x,y,z)=0$.  Find the points where $\partial_u f$ and $\partial_v f$ are linearly dependent, and describe the singularities of $S$ (if any) at these points.
\begin{enumerate}
\item $f(u,v) = (2u+v, u-v, 3v) $
\item$f(u,v) = (au \cos v, bu \sin v, u) \mbox{ with } a, b>0$
\item$f(u,v) = (u \cos v, u\sin v, u^2) $
\end{enumerate}



\item Find the a surface parametrization of the cap cut from the sphere $x^z+y^2+z^2=4$ by the cone $z=\sqrt{x^2+y^2}$ in terms of two variables.  Give bounds on the two variables.  Compute the surface area of said cap.

  
\item Find the surface area of the part of the paraboloid $z=x^2 + y^2$ inside the cylinder $x^2 + y^2 = a^2$


\item On HW \# 9:
Find the surface area of the ellipsoid $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} =1$.  You should include a computation of why the surface area of a sphere of radius $R$ is $4 \pi R^2$. 


\item Match the parametric equations with the surfaces. 
\end{enumerate}


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