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{\Large\bf Project 1}\\[8pt]
{(due 02/14/02)}
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\bigskip\bigskip
{\em\footnotesize Discuss the questions below in a concise and precise essay,
at most 2 typed pages long. If you use a reference, quote it and
do not copy. Use your own words. You might be interested how to typeset
mathematics well (also physics, chemistry, or anything). There is a powerful
computer language for that called TeX. Everyone having to write math papers needs
to learn it sooner or later. Maybe you can pick it up along the way this semester.
I am attaching a source file for this page, so you can take a first glimpse
at TeX.}
\MS\MS\MS
Consider a continuous curve $\, c:[\alpha ,\beta]\longrightarrow{\bf R}^2$.
For any (finite) subdivision \, $\tau = (t_0,t_1,...,t_n)$ of the
parameter interval, $\, \alpha = t_0 < t_1 < ... < t_n = \beta$, set
$$ L_{\sigma} = \sum_{i=1}^{n} |c(t_i)-c(t_{i-1})| \;.$$
Interpret $L_\sigma$ as the naive length of an inscribed polygon. Why should
these numbers not exceed the intuitive arc length of $c$?
Let $S=\{L_\sigma\}$ be the set of all lengths of inscribed finite polygons.
The curve $c$ is called {\em rectifiable} if $S$ is bounded. In that case,
the least upper bound of $S$ is called the {\em length} $L$ of $c$, so
$$L = \sup S \;.$$
\MS\MS
{\bf 1.} \,\, Explicitly describe an example of a continuous curve $\, c:[0,1]
\longrightarrow\bf R^2$ which is not rectifiable. \\
{\bf 2.} \,\, Carefully argue that if $c$ is rectifiable and $\sigma$ is any
subdivision as above, then all the restrictions $c_i = c|[t_{i-1},t_i]$ for $i=1,...n$
are rectifiable, and if $L_i$ denotes the length of $c_i$, then $L=L_1+...+L_n$. \\
{\bf 3.} \,\, Prove in detail that if $c$ is a $C^1$ curve, then $c$ is rectifiable
and we have the familiar formula
$$L = \int_{\alpha}^{\beta}|c'(t)|\, dt \;\;.$$
{\bf 4.} \,\, The curve $c$ is {\em piecewise $C^1$} if there is a subdivision $\sigma$
so that the restrictions $c_i$ as above are all $C^1$ curves (but $c$ may have `corners'
at the time $t_i$, so the velocity field $c'$ is possibly discontinuous at $t_i$).
Show that in this case $c$ is still rectifiable and we have the same integral formula
as before, if we interpret the integral in the usual way allowing finitely many `jump
discontinuities'. \\
{\bf 5.} \,\, Define what it means for $c$ to be {\em piecewise linear}. Such curves we
might want to call {\em polygons}. Notice that they can look very complicated, self-intersect
etc. Usually one likes to think of {\em polygons} in a much more restricted way as being
{\em convex} perhaps. Argue that if you parametrize the above inscribed polygons for a given
subdivision $\sigma$ in the natural way piecewise linearly on the interval $[\alpha,\beta]$ (how?),
then their length is $L_\sigma$. Make sure you realize and fully appreciate why this is a
non-trinial question!
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