%\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}} \begin{document} \title{Analysis Seminar: Assignment 1} \maketitle \section*{Reading} Read sections five through seven in Munkres. \section*{Doing} I would like you to try to do all the problems in these sections. This may be an unreasonable expectation, so what I ask that you do is start with problems 5.2, 5.7, 6.4, 6.7, 6.9, 6.10, 7.3, and the one I have written below. After you have done these, spend the rest of the time you allocate to problem solving working on the other problems in what ever order makes sense to you. \begin{ex} Suppose that \( A \subset \R^{n} \) and that \( f : A @>>> \R^{m} \) is differentiable at every point of \( A \). If one thinks of \( Df({\bf a}) \) as an \( m \) by \( n \) matrix and one thinks of the collection of \( m \) by \( n \) matrices as \( \R^{nm} \) then \( Df \) may be thought of as a function \( A @>>> \R^{nm} \). Show that \( f \) is \( C^{1} \) if and only if \( Df \) is continuous. State and prove a generalization to \( C^{r} \) for \( r \geq 0 \). (I am asking you to supply a reasonable definition of \( C^{0} \) here.) \end{ex} \section*{Writing} Pick three or more of the high priority problems to write up carefully and hand in to me. \section*{Presentations} \begin{pres}[Jeremy] A careful solution to exercise 5.2 with illustrative examples using Mathematica and Spiderwoman. \end{pres} \begin{pres}[Sonya] Elucidation of Theorem 6.3. \end{pres} \begin{pres}[James] Elucidation of Theorem 7.1. \end{pres} \end{document}