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\title{Math 147A: Homework 6}
\date{}


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\begin{document}
\maketitle
\textbf{Suggested due date Thursday, May 26.}

\begin{enumerate}
\item Provide details to example 6.3.5 in text, i.e. show that the stereographic projection is a conformal map between the sphere and the plane.
\item (6.4.1) Determine the area of the part of the paraboloid $z=x^2 + y^2$ with $z \leq 1$ and compare with the area of the hemisphere $x^2+y^2+z^2 = 1$, $z\leq 0$.
\item (6.4.2) A sailor circumnavigates Australia by a route consisting of a triangle whose sides are arcs of great circles.  Prove that at least one interior angle of the triangle is greater than or equal to $\frac{\pi}{3}+\frac{10}{169}$ radians.  Take the Earth to be a sphere of radius 6,500 km and assume that the area of Australia is 7.5 million square kilometers.
\item (6.4.3) A spherical polygon on $S^2$ is the region formed by intersection of $n$ hemispheres of $S^2$, where $n$ is an integer greater than or equal to 3.  Show that if $\alpha_1,\ldots,\alpha_n$ are the interior angles of such a polygon, its area is equal to 
\begin{equation*}
\sum_{i=1}^n \alpha_i - (n-2)\pi.
\end{equation*}
\item (6.4.4) Suppose that $S^2$ is covered by spherical polygons such that the intersection of any two polygons is either empty or a common edge or vertex of each polygon.  Suppose that there are $F$ polygons, $E$ edges and $V$ vertices (a common edge or vertex of more than one polygon is counted only once.  Show that the sum of the angles of all the polygons is $2\pi V$.  Deduce that $V-E+F = 2$. (Euler characteristic of a sphere).
\item (6.4.5) Show that:
\begin{enumerate}
\item Every local isometry is an equiareal map
\item A map that is both conformal and equiareal is a local isometry.
\end{enumerate}
Give an example of an equiareal map that is not a local isometry.
\item Let $\alpha:(a,b) \to \mathbb{R}^3$ be a regular parametrized curve, and $\beta:(a,b)\to mathbb{R}^3$ a smooth nonvanishing function.  Given a surface $S$ parametrized by $\sigma(u,v) = \alpha(u)+v\beta(u)$ with $\alpha' \neq 0$ and $\|\beta\| =1$ (such a surface is called a \textbf{ruled surface},
\begin{enumerate}
\item Check that we may assume that $\alpha'(u)\cdot \beta(u) = 0$ for all $u$. (Hint: replace $\alpha(u)$ with $\alpha(u) + t(u)\beta(u)$ for a suitable function $t$. (see exercise 5.3.4))
\item Suppose, moreover, that $\alpha'(u)$,$\beta(u)$, and $\beta'(u)$ are linearly dependent for every $u$. Conclude that $\beta'(u) = \lambda(u)\alpha'(u)$ for some function $\lambda$.  Prove that
\begin{enumerate}
\item If $\lambda(u) = 0$ for all $u$, then $S$ is a (generalized) cylinder.
\item If $\lambda$ is a nonzero constant, then $S$ is a (generalized) cone.
\item If $\lambda$ and $\lambda'$ are both nowhere zero, then $S$ is a tangent developable (at least locally).
\end{enumerate}
\end{enumerate}
\end{enumerate}

\end{document}