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\lhead{Math 4441 Activity 12}
\rhead{<your name(s)>}

\begin{document}
\title{Math 4441 Activity 12}
\author{<your name(s)>}
\date{December 2, 2022}
\maketitle
\vspace{5em}

\setcounter{section}{12}

\begin{exercise}
Suppose $\vec{x}\colon U\to\mathbb{R}^3$ is a simple surface whose image\footnote{This isn't technically possible: in order to be \emph{simple}, $\vec{x}$ will have to miss at least one point of the sphere.  But let's ignore this.} is the sphere of radius $R>0$ centered at the origin, with outward-pointing normal vector.
\begin{enumerate}[label=(\alph*)]
	\item Show that
	\[
	(L^i_j) = \left(\begin{matrix}
	-1/R & 0\\
	0 & -1/R
	\end{matrix}\right).
	\]
	\begin{solution}
	<solution>
	\end{solution}
	
	\item Identify the principal directions, principal curvatures, and Gaussian curvature of $\vec{x}$.
	
	\item Compute the surface integral $\iint_{\vec{x}}K\,dA$, where $K\colon\mathrm{im}(\vec{x})\to\mathbb{R}$ is the Gaussian curvature of $\vec{x}$.
	
	\item Does $K$ change as you vary the value of $R$?  Does $\iint_{\vec{x}}K\,dA$ change?
\end{enumerate}
\end{exercise}

\begin{exercise}\label{activity-12:torus-exercise}
Consider the simple surface $\vec{x}\colon(-\pi,\pi)\times(-\pi,\pi)\to\mathbb{R}^3$ defined by
\[
\vec{x}(u^1,u^2) = (\cos u^1\,(R-r\,\sin u^2),\sin u^1\,(R-r\,\sin u^2),r\,\cos u^2),
\]
where $R>r>0$ are arbitrary real numbers.  The first and second fundamental forms of this surface are given by
\[
(g_{ij}) = \left(\begin{matrix}
(R-r\,\sin u^2)^2 & 0\\
0 & r^2
\end{matrix}\right)
\quad\text{and}\quad
(L_{ij}) = \left(\begin{matrix}
\sin u^2\,(R-r\,\sin u^2) & 0\\
0 & -r
\end{matrix}\right).
\]
\begin{enumerate}[label=(\alph*)]
	\item Identify the principal directions of $\vec{x}$.  Specifically, express them in terms of $\vec{x}_1$ and $\vec{x}_2$, and also describe them in terms of a figure, either by drawing a picture or writing some sentence(s).
	
	\item Identify the principal curvatures $\kappa_1$ and $\kappa_2$ of $\vec{x}$.
	
	\item Compute the surface integral $\iint_{\vec{x}}K\,dA$, where $K\colon\mathrm{im}(\vec{x})\to\mathbb{R}$ is the Gaussian curvature of $\vec{x}$.
	
	\item Does $K$ change as you vary the values of $R$ and $r$?  Does $\iint_{\vec{x}}K\,dA$ change?
\end{enumerate}
\end{exercise}

\begin{exercise}\label{activity-12:revolution-formulas}
Let $\vec{\alpha}(t)=(r(t),z(t))$, $t\in(a,b)$, be a unit speed, $C^\infty$, injective curve in the $rz$-plane, with $r(t)>0$.  Consider the surface of revolution
\[
\vec{x}(t,\theta) := (r(t)\,\cos\theta,r(t)\,\sin\theta,z(t)),
\quad
t\in(a,b)~
\theta\in(-\pi,\pi).
\]
We have previously computed the first and second fundamental forms of this surface to be given by
\[
(g_{ij}) = \left(\begin{matrix}
1 & 0\\
0 & r^2
\end{matrix}\right)
\quad\text{and}\quad
(L_{ij}) = \left(\begin{matrix}
\dot{r}\,\ddot{z}-\ddot{r}\,\dot{z} & 0\\
0 & r\,\dot{z}
\end{matrix}\right).
\]
Show that $K\,\sqrt{g}=-\ddot{r}$, and conclude that $\iint_{\vec{x}}K\,dA=-2\pi\,\dot{r}(t)\vert_a^b$.
\end{exercise}
\begin{solution}
<solution>
\end{solution}

\begin{exercise}
This exercise is a continuation of Exercise~\ref{activity-12:revolution-formulas}, where we'll compute the integral $\iint_{\vec{x}}K\, dA$ for some particular surfaces of revolution.
\begin{enumerate}[label=(\alph*)]
	\item Suppose that the closed curve $\vec{\alpha}$ depicted in Figure~12.2(a) satisfies the conditions given in Exercise~\ref{activity-12:revolution-formulas}.  Describe the resulting surface $\vec{x}$.  What is the value of $\iint_{\vec{x}}K\,dA$?
	
	\item Suppose that the curve $\vec{\alpha}$ depicted in Figure~12.2(b) satisfies the conditions given in Exercise~\ref{activity-12:revolution-formulas}.  Describe the resulting surface $\vec{x}$.  What is the value of $\iint_{\vec{x}}K\,dA$?
	
	\item Suppose that the curve $\vec{\alpha}$ depicted in Figure~12.2(c) satisfies the conditions given in Exercise~\ref{activity-12:revolution-formulas}.  Describe the resulting surface $\vec{x}$.  What is the value of $\iint_{\vec{x}}K\,dA$?
\end{enumerate}
\end{exercise}

\begin{exercise}
Consider the closed surface depicted in Figure~12.3.  Make a conjecture about the value of $\iint_{\vec{x}}K\,dA$ for this surface.  It's okay if you're unsure of a precise numerical value, but explain what the previous exercises seem to indicate about the value of this integral.
\end{exercise}

\end{document}