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{\bf Math 401 HW \#5, due Wednesday 10/2/24} \hspace{.5in} \textbf{NAME:} \hspace{4in}

\textbf{UPDATED:   \# 3 is about orientability, not the hyperbolic paraboloid}

\begin{enumerate}
\item Guided Jones 11-10, \Homer d'oh \\
Suppose $0 \leq a \leq b$.  Find the surface area of the torus  obtained by revolving the circle $(x-b)^2 + z^2 = a^2$ in the $xz$-plane about the $z$-axis.  \\ Suggestion: Show that the torus admits the parametrization $ 0 \leq \varphi, \theta \leq 2\pi$ by 
\[
\begin{array}{lcl}
x&=& (b + a \cos \varphi) \cos \theta \\
y&=& (b + a \cos \varphi) \sin \theta \\
z&=& a \sin \varphi \\
\end{array}
\]
\vspace{-.25in}
\begin{figure}[h]
\begin{centering}
  \includegraphics[width=2.5in]{HW5_donut.png}
  \vspace{-.125in}
  \caption{The hollow blue donut}
  \label{fig:boat1}
\end{centering}
\end{figure}





\item Exercise 3.17 \\
If $S$ is an oriented surface, $N$ is the corresponding unit normal field, and $p\in S$, then we say that a basis $\{ a,b\}$ of the tangent plane $T_pS$ is \emph{positively oriented} when $\mbox{det}(a,b,N(p)) >0$.  Otherwise we say that it is \emph{negatively oriented}.  If $S_1$ and $S_2$ are two oriented surfaces, we say that a local diffeomorphism $f:S_1 \to S_2$ \emph{preserves orientation} if its differential at each point of $S_1$ takes positively oriented bases on $S_1$ into positively oriented bases on $S_2$.  We define a function 
\[
\mbox{Jac} f:S_1 \to \R
\]
that is called the \emph{Jacobian} of $f$ by the equation
\[
(\mbox{Jac} f)(p)=\mbox{det}((df)_p(e_1),(df)_p(e_2),N_2(f(p))), 
\]
where $\{ e_1, e_2 \}$ is a positively oriented orthonormal basis of $T_pS_1$.  \\\textbf{Prove that, if $S_1$ and $S_2$ are connected, $f$ preserves orientation if and only if its Jacobian is positive everywhere.}



\item Exercise 3.16 \\
Consider two diffeomorphic surfaces $S_1$ and $S_2$.  Show that $S_1$ is orientable if and only if $S_2$ is orientable.

 

%\item Exercise 3.16 \\
%Consider two diffeomorphic surfaces $S_1$ and $S_2$.  Show that $S_1$ is orientable if and only if $S_2$ is orientable.



\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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