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\lhead{Math 4441 Activity 8}
\rhead{<your name(s)>}

\begin{document}
\title{Math 4441 Activity 8}
\author{<your name(s)>}
\date{October 26, 2022}
\maketitle
\vspace{5em}

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\begin{exercise}
Recall that $\{\vec{x}_1(u^1,u^2),\vec{x}_2(u^1,u^2)\}$ gives a basis for $T_{\vec{x}(u^1,u^2)}\vec{x}$.  The purpose of this problem is to produce an \emph{orthonormal} basis.
\begin{enumerate}
	\item Check that $\vec{x}_1(u^1,u^2)$ and $\vec{x}_2(u^1,u^2)$ are perpendicular, for all $(u^1,u^2)$.  Thus we can obtain an orthonormal basis by scaling $\vec{x}_1$ and $\vec{x}_2$ to have unit length.\label{activity-8:perpendicular}
	\begin{solution}
	<your solution>
	\end{solution}
	
	\item For $i=1,2$, let $\vec{E}_i := \vec{x}_i/\|\vec{x}_i\|$.  Check that
	\[
	\vec{E}_1(u^1,u^2) = (-\sin u^1,\cos u^1, 0)
	\quad\text{and}\quad
	\vec{E}_2(u^1,u^2) = (-\cos u^1\,\sin u^2, -\sin u^1\,\sin u^2, \cos u^2)
	\]
	for all $(u^1,u^2)$.
	
	\item Check that the surface normal vector $\vec{n}$ is given by
	\[
	\vec{n}(u^1,u^2) = (\cos u^1\,\cos u^2,\sin u^1\,\cos u^2,\sin u^2).
	\]
	\emph{Hint: We defined $\vec{n}$ to be a scaled version of $\vec{x}_1\times\vec{x}_2$.  How should this relate to $\vec{E}_1\times\vec{E}_2$?}
\end{enumerate}
\end{exercise}

\begin{exercise}\label{activity-8:circle-of-latitude}
Fix a value $-\pi/2<\phi_0<\pi/2$ and present the circle with latitude $\phi_0$ as a surface curve.  That is, identify a curve $\vec{\alpha}_U$ in $(-\pi,\pi)\times(-\pi/2,\pi/2)$ with the property that $\vec{\alpha}(t):=(\vec{x}\circ\vec{\alpha}_U)(t)$ parametrizes the circle with latitude $\phi_0$.
\end{exercise}

\begin{exercise}\label{activity-8:parallel-hypothesis}
Let $\vec{x}$ and $\vec{\alpha}$ be as above.  We want to construct a function $\vec{V}(t)$ which produces a vector tangent to $\vec{x}$ at $\vec{\alpha}(t)$, for each $t\in(-\pi,\pi)$.  What should it mean to say that $\vec{V}$ doesn't change as we move along $\vec{\alpha}$?\\
\emph{Note: Write down an answer to this in complete sentences \textbf{before} moving on.  You won't get any points for being right, but you will get points for giving a thoughtful hypothesis or explanation.}
\end{exercise}

\begin{exercise}\label{activity-8:coordinate-basis}
Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ be as above, and consider $\vec{E}_1(t)=\vec{E}_1(\vec{\alpha}_U(t))$ and $\vec{E}_2(t)=\vec{E}_2(\vec{\alpha}_U(t))$.  Show that
\[
\vec{E}_1'(t) = \sin\phi_0\,\vec{E}_2(t) - \cos\phi_0\,\vec{n}(t)
\quad\text{and}\quad
\vec{E}_2'(t) = -\sin\phi_0\,\vec{E}_1(t).
\]
For which values of $\phi_0$ are the vector fields $\vec{E}_1(t)$ and $\vec{E}_2(t)$ parallel?  Give an intuitive explanation of why these vector fields are not parallel for most values of $\phi_0$.\\
\emph{Hint: Think about very cold circles of latitude.}
\end{exercise}

\begin{exercise}\label{activity-8:constant-length}
Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ be as above, and suppose $\vec{V}(t)$ is parallel along $\vec{\alpha}$.  Show that $\|\vec{V}(t)\|$ is constant.\\
\emph{Hint: Since $\vec{V}'(t)$ is perpendicular to $T_{\vec{\alpha}(t)}\vec{x}$ and $\vec{V}(t)$ is a tangent vector, $\langle\vec{V}'(t),\vec{V}(t)\rangle\equiv 0$.}
\end{exercise}

\begin{exercise}\label{activity-8:ode}
Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ and $\vec{V}(t)$ be as above.  Show that $\vec{V}(t)$ is parallel along $\alpha$ if and only if $\theta'(t)=-\sin\phi_0$, for all $t\in(-\pi,\pi)$.\\
\emph{Hint: Use the product and chain rules to compute $\vec{V}'(t)$, and then apply the formulas from Exercise~\ref{activity-8:coordinate-basis} to simplify the result.  Remember that $\vec{V}(t)$ is parallel along $\vec{\alpha}$ if and only if $\vec{V}'(t)$ has no tangential component.}
\end{exercise}

\begin{exercise}
Let $\vec{\alpha}=\vec{x}\circ\vec{\alpha}_U$ be as above.  If we parallel transport a vector $\vec{V}_0$ along $\vec{\alpha}$ for time $2\pi$, we will return to our original point with a new vector $\vec{V}(t_0+2\pi)$.  Determine the angle between this new vector and the original, $\vec{V}_0$.\\
\emph{Note: Technically, $\vec{\alpha}(t)$ is only defined for $t\in(-\pi,\pi)$, but let's ignore this for now.}
\end{exercise}

\begin{exercise}
Georgia Tech's campus lies approximately $33.77^\circ$ north of the equator.  If the Institute were to install a Foucault pendulum, approximately what would be its period, in hours?  That is, how many hours would be required for the vector $\vec{V}_0$ identified above to make one full rotation in the tangent plane?
\end{exercise}

\end{document}