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{\bf Math 401 HW\#4, due Wednesday 9/25/24} \hspace{.5in} \textbf{NAME:} \hspace{4in}\\
%outlier malcom

\textbf{UPDATED:   For full credit turn in solutions to 3 out of 4 problems}
\begin{enumerate}


\item \textbf{Exercise \S 2 (8)}  \\
\vspace{-.25in}
\begin{figure}[h]
\begin{centering}
  \includegraphics[width=.75\linewidth]{hw4_triply-periodic.png}
 \caption{Hint: Appeal to trig identities to get linearly independent translations of $\R^3.$ \\ {\tiny \url{https://math.stackexchange.com/questions/3589185/a-rigid-motion-that-interchanges-the-the-two-regions-determined-by-the-gyroid}}}
  \label{fig:boat1}
\end{centering}
\end{figure}
\item \textbf{Exercise 2.43} \\
If $X:U\to S$ is a parametrization of a surface, show that $X: U \to X(U)$ is a diffeomorphism.



\item \textbf{Exercise \S 2 (9)} \\
 Let $\phi: S \to \R^3$ be a differentiable map defined on a surface that satisfies
 \begin{itemize}
\item $(d\phi)_p:T_pS \to \R^3$ is injective for all $p\in S$.
\item $\phi:S \to \phi(S)$ is a homeomorphism.
\end{itemize}
Prove that $\phi(S)$ is a surface and that $\phi:S \to \phi(S)$ is a diffeomorphism.  If $S$ is compact, show that the second requirement is equivalent to requiring that $\phi$ be injective.  



\item \textbf{Exercise 2.55} \\
Let 
\[
S = \{ p \in \R^3 \ | \ |p|^2-\langle p,a \rangle^2 = r^2 \},
\]
with $|a|=1$ and $r >0$, be a right cylinder of radius $r$ whose axis is the line passing through the origin with direction $a$. Prove that
\[
T_pS = \{ v \in \R^3 \ | \  \langle p,v \rangle - \langle p,a \rangle \langle a,v \rangle =0 \}.
\]
Conclude that all the normal lines of $S$ cut the axis perpendicularly.


 

\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you like?

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\end{enumerate}

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