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\begin{center}
\large{\textbf{Math 4220/6220 Exam \#1}}
\end{center}
\bigskip
This exam covers Chapter 1 of our book. You are allowed to use:
\begin{itemize}
\item A linear algebra book of your choice.
\item A multivariable calculus book of your choice.
\item The first four chapters of Munkres' Topology book.
\item MAPLE, Mathematica, Matlab, or any similar calculational aid
\item your book
\item your notes from class
\item the notes posted on the course webpage.
\end{itemize}
Please don't use any other resources. The exam will be due next
Saturday at 1pm. During the exam period, please refrain from
discussing course material with the other students. You are welcome to
ask me questions either by email or during class time (private
questions don't seem entirely fair).
\begin{problems}
\item Consider the space of $6$ sided polygons in $\R^2$, which we
will denote $C_6(\R^2)$. Prove that this space is a differential
manifold which is diffeomorphic to $\R^n$, and find $n$.
\item Let the space $\Pol_6(\R^2)$ denote the space of (closed)
6-sided polygons in $\R^2$. Prove that $\Pol_6(\R^2)$ is a smooth
submanifold of $C_6(\R^2)$ and find its dimension and
codimension. (Hint: Define a map from $C_5(\R^2) \rightarrow \R^2$
by taking the difference of the first and last vertices as a
vector. Can you express $\Pol_6$ as the inverse image of a regular
value?)
\item Consider the space $\Tri$ of triangles in $\R^2$. We observed in
class that this is a smooth manifold diffeomorphic to $\R^6$.
Consider the map $f: \Pol_6 \rightarrow \Tri$ given by taking the
midpoints of the line segments connecting the first, third, and
fifth vertices. Prove or disprove: This map is a submersion.
\item Consider the map from $\Tri \rightarrow \Pol_6$ given by adding
vertices at the midpoints of each side of the triangle. Prove or
disprove: this map is an immersion.
\item (Bonus question). Prove that the space of quadrilaterals whose
vertices lie on a common circle is a smooth submanifold of the space
of quadrilaterals in $\R^2$. Find its dimension and codimension.
\end{problems}
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