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{\bf Math 401 HW\#2, due Wednesday 9/11/24} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\
%outlier malcom

\textbf{For full credit, pick one of 1 (theory) or 2 (computation) to upload to gradescope.  \\ \indent Do not turn in \textcolor{blue}{$\star$ in class} problems.  But please finish and understand them!}

 \textcolor{red}{\textbf{UPDATE 8/29: New torsion/curvature problem. Two exercises were moved to HW 3.}}

\begin{enumerate}


%\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up}} \\
%Let $C$ be the circle formed by intersecting the plane $x+z=1$ with the sphere $x^2+y^2+z^2=1$.
%\begin{enumerate}
%\item Find a parametrization of $C$.
%\item Find parametric equations for the tangent line to $C$ at the point $\left( \frac{1}{2}, -\frac{1}{\sqrt{2}} , \frac{1}{2} \right)$
%\end{enumerate}


\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up:}}  Find an equation for the tangent plane to the surface at the point $(1, -2, 1)$:
\[
S= \left\{ \begin{array}{lcl}
x &=& e^{u-v} \\
y &=& u-3v\\
z &=& \frac{1}{2}(u^2+v^2) \\
\end{array} \right.
\]


\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up}}  \\
Find a parametrization for each of the following surfaces (perhaps involving an angular variable that is defined only up to multiples of $2\pi$).
\begin{enumerate}
\item The surface obtained by revolving the curve $z=f(x)$, $a<x<b$ in the $xz$-plane around the $z$-axis, where $a>0$. 
\item The surface obtained by revolving the curve $z=f(x)$, $a<x<b$ in the $xz$-plane around the $x$-axis, where $ 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 \textbf{Exercise 1.9, Theory, in lieu of (2)} \\
 Let $\alpha : I \to \R^3$  be a curve and $[a,b]\subset I.$ Prove that \[ |\alpha(a) - \alpha(b)| \leq L_b^a(\alpha).\] In other words, straight lines are the shortest curves joining two given points. 

\item  \textbf{Computation, in lieu of (1)} \\
For the following curve $\alpha: \R \to \R$, compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, and $k(t)$.  
\[
\alpha(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
\]

\item  \textbf{Exercise 1.37}
 Let $\alpha : I \to \R^3$  be a curve p.b.a.l. with curvature $k(s) >0$ for every $s \in I$.  Prove that $\alpha$ is an arc of a circular helix or an arc of a circle if and only if both the curvature and the torsion of $\alpha$ are constant.  (See example 1.5 for their parametrizations.)

\item  \textbf{Remembering Multi:} For each of the following maps $f: \R^2 \to \R^3$, 
\begin{itemize}
\itemsep-.25em
\item Describe or sketch the (possibly singular) surface $S = f(\R^2)$ 
\item Find a description of $S$ as the locus of an equation $F(x,y,z)=0$. 
\item 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.
\end{itemize}
\begin{enumerate}
\itemsep-.25em
\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) $
%\item[Note:] \textbf{ There are 2 more problems on the next page!}

\end{enumerate}

%\item  Exercise 1.31 \\
%Let $\alpha : I \to \R^3$  be a curve p.b.a.l. Prove that $\alpha$ is a segment of a straight line if and only if the curvature of $\alpha$ vanishes everywhere.




\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find to provide a worthwhile learning experience?

\vfill
\eject

\end{enumerate}

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