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\begin{document}
\title{Analysis Seminar: Assignment 6}
\maketitle
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
\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}