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\begin{document}
\title{Analysis Seminar: Assignment 5}
\maketitle
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
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}