%\section{\defs} \textwidth=7in \evensidemargin=-.25in \oddsidemargin=-.25in \newtheorem{thm}{Theorem} %[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{ex}[thm]{Exercise} \newtheorem{pres}{Presentation} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{notn}[thm]{Notation} \newtheorem{eg}[thm]{Example} \newcommand{\R}{{\Bbb R}} \newcommand{\GL}{\operatorname{GL}} \begin{document} \title{Analysis Seminar: Assignment 5} \maketitle \section*{Reading} Read sections sixteen, seventeen, and eighteen in Munkres. Where in this seminar did you see something like a ``primitive diffeomorphism'' before? \section*{Doing} \begin{itemize} \item In section 16, do problems 1 and 3. Try to figure out what I think is wrong with figure 16.1. In problem 3, the phrase ``well-defined'' means defined for all values of \( {\bf x} \in \R^{n} \). \item In section 17, do problems 1, 3, 4, 5, 6, and the following generalization of 7. (For the sake of counting up whether you have written enough problems, count this as problem 7.) \begin{ex} Let \( A \) be a bounded rectifiable set in the right half of the \( xz \) plane. Let \( V \) be the region in \( xyz \) space consisting of all points with cylindrical coordinates \( (r, \theta, z) \) where \( (r,0,z) \) is in \( A \). That is to say, \( V \) is the solid generated by rotating \( A \) about the \( z \) axis. Use the change of variables theorem to show that \[ \int_{A}x \] exists. Denote the value of this integral by \( r_{c} \). Show that \( V \) is rectifiable and has volume \( 2 \pi r_{c} \). Show how problem 7 is solved by this formula and give one more example of its application. \end{ex} You may consider the following problem as optional, but give it value 10 if you choose to write it up. \begin{ex} Define the solid \( n \)-dimensional ball and find and prove a formula for its volume. \end{ex} \item In section 18, do 1,3,4,5. \end{itemize} \section*{Writing} Expertly write up at least one problem from each section and enough of the problems that the problem numbers add up to at least 33 minus your age in years. Hand these in no later than than noon on Tuesday following Seminar. \section*{Presentations} \begin{pres}[Tom] Theorem 16.3 and Lemma 16.2. \end{pres} \begin{pres}[Mike] A discussion of section 17. \end{pres} \begin{pres}[Peter] Theorem 18.1. \end{pres} \begin{pres}[Aaron] Theorem 18.3 \end{pres} \end{document}