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\lhead{Math 4441 Activity 9}
\rhead{<your name(s)>}

\begin{document}
\title{Math 4441 Activity 9}
\author{<your name(s)>}
\date{November 2, 2022}
\maketitle
\vspace{5em}

\setcounter{section}{9}


\begin{exercise}\label{activity-9:christoffel-exercise}
Show that the Christoffel symbols of a surface of revolution satisfy
\[
\Gamma_{22}^1 = -r\,\dot{r}
\quad\text{and}\quad
\Gamma_{12}^2=\Gamma_{21}^2 = \frac{\dot{r}}{r},
\]
with all others being 0.
\end{exercise}
\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Let $\vec{\alpha}(s)=\vec{x}(t(s),\theta(s))$ be a unit-speed surface curve as above.  Show that $\vec{\alpha}$ is a geodesic if and only if
\begin{equation}\label{activity-9:geodesic-ode-exercise}
t'' - (\theta')^2\,r(t)\,\dot{r}(t) = 0
\quad\text{and}\quad
\theta'' + 2t'\,\theta'\,\frac{\dot{r}(t)}{r(t)} = 0,
\end{equation}
where the prime denotes derivative with respect to $s$ and the dot denotes derivative with respect to $t$.
\end{exercise}

\begin{exercise}\label{activity-9:meridians-and-latitudes}
Use Equation~\ref{activity-9:geodesic-ode-exercise} to show that
\begin{enumerate}[label=(\alph*)]
	\item every meridian $\theta=\theta_0$ is a geodesic;
	
	\item a latitude $t=t_0$ is a geodesic if and only if $\dot{r}(t_0)=0$.
\end{enumerate}
\end{exercise}

\begin{exercise}
Based on Exercise~\ref{activity-9:meridians-and-latitudes}, sketch a surface of revolution for which exactly three latitudes are geodesics.  Sketch these three geodesic latitudes, plus at least two more geodesics.
\end{exercise}\begin{exercise}\label{activity-9:onb}
Let $\vec{\alpha}(s)=\vec{x}(t(s),\theta(s))$ be a unit-speed surface curve as above.  Explain why, for each $s\in(c,d)$, there must be some angle $\psi(s)$ such that
\[
\vec{\alpha}'(s) = \sin\psi(s)\,\vec{x}_t + \cos\psi(s)\,\left(\frac{1}{r(t(s))}\,\vec{x}_\theta\right).
\]
\end{exercise}

\begin{exercise}\label{activity-9:forward}
Let $\vec{\alpha}=\vec{x}(t(s),\theta(s))$ be as above, and let $\psi(s)$ be as defined in Exercise~\ref{activity-9:onb}.  Prove that if $\vec{\alpha}$ is a geodesic, then the quantity $r\,\cos\psi$ is constant.
\end{exercise}

\begin{exercise}
Show that, for any surface curve $\vec{\alpha}(s)=\vec{x}(t(s),\theta(s))$,
\begin{equation}\label{activity-9:converse-equation}
(t')^2 = 1 - (r\,\theta')^2 = 1 - \left(\dfrac{r\,\cos\psi}{r}\right)^2,
\end{equation}
where $\psi$ is the angle described above.
\end{exercise}

\begin{exercise}\label{activity-9:converse}
Differentiate both sides of $(t')^2=1-(K/r)^2$ with respect to $s$ to show that
\[
t'\left(t''-\dfrac{K^2}{r^3}\dot{r}\right) = 0.
\]
\end{exercise}

\begin{exercise}
Use Clairaut's relation to identify all geodesics on the right circular cylinder $x^2+y^2=1$.  You should find three infinite families of geodesics, and support your claim with some complete sentences.
\end{exercise}

\begin{exercise}
(\emph{Optional}) Suppose a surface of revolution is asymptotic to its axis of revolution.  Prove that the meridians are the only geodesics on this surface which are asymptotic to the axis of revolution.
\end{exercise}

\end{document}