%\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 6} \maketitle \section*{Reading} Read sections 19, 20, 21 (Omit the proof of Theorem 21.4 if you are pressed for time.) and 22. \section*{Doing} \begin{itemize} \item Do a selection from among the problems at the end of section 19. These are all computational. Problem 6 appeared in a different guise last week. I believe problem 7 has an interesting relationship to a problem 7 from last week. \item All the problems at the end of section 20 should be straightforward. Do at least problem 4--you will need to have this straight when we start talking about differential forms. Here is another problem which should be straightforward but isn't. \begin{defn} Let \( {\bf a_{0}},\dots, {\bf a_{k}} \) be vectors in \( \R^{n} \). We define the \( k \)-dimensional {\bf simplex} \( {\cal S} = {\cal S}({\bf a_{0}},\dots, {\bf a_{k}}) \) to be the set of all \( {\bf x} \) in \( \R^{n} \) such that \[ {\bf x} = c_{0} {\bf a_{0}} + \dots c_{k}{\bf a_{k}} \] for scalars \( c_{i} \) with \( 0 \leq c_{i} \) and with \[ \sum_{i=0}^{k}c_{i} = 1. \] The vectors \( {\bf a_{0}},\dots, {\bf a_{k}} \) are called the {\bf vertices} of \( {\cal S} \). \end{defn} \begin{ex} Find a formula for the \( n \) dimensional volume of a \( k \) dimensional simplex in \( \R^{n} \). \end{ex} \item Read the problems at the end of section 21. Do at least the first one. \begin{ex} Find a formula for the \( k \) dimensional volume of a \( k \) dimensional simplex in \( \R^{n} \). \end{ex} \item The problems at the end of section 22 are all good. Do at least numbers one and numbers two. Number four is involved but quite interesting. Those of you who want an extra challenge may want to try to do an \( n \)-dimensional version of this using simplices instead of triangles. \end{itemize} \section*{Writing} Expertly write up at least one page of work you are proud of from this week. Hand this in no later than than noon on Tuesday following Seminar. \section*{Presentations} \begin{pres}[Jeremy and James] The proof of the change of variables theorem. \end{pres} \begin{pres}[Sonya and Aaron] The proof and meaning of Theorem 22.1. This will involve some discussion of the material in the previous two sections. Include some examples including number 3 from the text. \end{pres} \end{document}