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\begin{center}
\large{\textbf{Math 4250/6250 Final Exam}}
\end{center}
\bigskip
This take-home exam covers material from the entire class. Please
don't be afraid to read over the notes and the sections in the book as
you work on the problems-- there is more in the notes than we were
able to cover in the lectures, and some of those extra facts might be
helpful to you as you work on the exam problems.
As before, please pick 4 of the following problems. You must pick one
from each of the four categories: Curve theory, The First Fundamental
Form, The Second Fundamental Form, and Geodesics and Gauss-Bonnet. In
each category there is an ``easier'' problem and a ``harder''
problem. If you are a graduate student, you must attempt \textbf{two}
''harder'' problems. If you are an undergraduate, you will get some
bonus credit for doing ``harder'' problems.
\begin{tabular}{p{3in}p{3in}}
You \textbf{are} permitted to use & You \textbf{are not} permitted to
use \\ \hline Maple (or Mathematica or MATLAB) & The internet \\ A
calculator (or graphing calculator) & \\ DoCarmo & Other books \\ Your
notes & Other people's notes \\ Your brain & Other people's brains
\\ Class notes posted on the website & \\
\end{tabular}
\noindent\textbf{Hints on exam-writing}
At this stage in your mathematical life, you are ready to learn to
communicate your ideas to other people in a clear and organized
way. When you work in industry or academia after graduation, a
solution to a problem or a good idea that you can't convince others of
will be worse than useless-- even if you get the right answer! Nobody
trusts a derivation that they can't understand.
So remember the following things when you write your exams:
\begin{enumerate}
\item Mathematics is written in complete english sentences which
explain not just what you are doing but why you are doing it. Every
time you write down an equation, you should write a corresponding
sentence which explains how you got there.
\item There are few things more dangerous than a mathematical fact
which you (think) you know, but can't find any reference for. Often
these things are just wrong. And even when you are exactly right,
when you are working with other people, nobody will believe you if
you can't cite a reference. So it's very important to cite a
specific page (either in your book or in the course notes) when you
use a fact, formula, or definition.
\item You are being graded not just on your solution but on how well
you explain it. If I can't understand your solution after an honest
attempt to follow your logic, you will fail the ``explanation''
portion of the exam and you will not get credit (even if the math is
right!). This is exactly what happens in industry and research-- you
will get no credit with your boss or with your company for solving a
problem if your coworkers can't easily understand and check your
solution.
\end{enumerate}
Here are two examples of a solution, one done the wrong way and one
done the right way. We'll start with the incorrect solution:
\begin{aligned}
\alpha(t) &= (3 \sin t, t^2, 2 \cos t) \\
\alpha'(t) &= (3 \cos t, 2t, -2 \sin t) \\
\alpha''(t) &= (-3 \sin t, 2, -2 \cos t) \\
\alpha'(t) \times \alpha''(t) &= 2(-2 t \cos t + 2 \sin t, 3, 3 \cos t + 3t \sin t). \\
|\alpha'(t)| = \left[ 4 (1 + t^2) + 5 \cos^2 t \right]^{1/2}.\\
\kappa(t) = \frac{2 \sqrt{
The problems are organized by subject area (easier and harder):
\begin{problems}
\item \textbf{(Curve theory, easier)} Suppose we have a strictly convex curve $\alpha$ in the plane (strictly convex means that curvature is everywhere greater than zero). We can parametrize the curve by $\theta$ so that the tangent lines to $\alpha$ are all in the form
\begin{equation}
\label{eq:tv}
(\cos \theta) x + (\sin \theta) y = p(\theta).
\end{equation}
We call $p(\theta)$ the support function of $\alpha(\theta)$. Prove that
\begin{enumerate}
\item
The line in equation~\eqref{eq:tv} is tangent to $\alpha$ at the point
\begin{equation*}
\alpha(\theta) = (p(\theta) \cos\theta - p'(\theta) \sin\theta, p(\theta) \sin\theta + p'(\theta) \cos\theta).
\end{equation*}
\item The curvature at $\alpha(\theta)$ is given by $1/(p(\theta) + p''(\theta))$.
\item The length of $\alpha$ is given by $\int_0^{2\pi} p(\theta) d\theta$.
\item \textbf{Bonus question.} The area inside $\alpha$ is given by $\frac{1}{2} \int_0^{2\pi} (p(\theta)^2 - p'(\theta)^2) d\theta$.
\end{enumerate}
\item \textbf{(Curve theory, harder)} Suppose that we have a plane curve $\alpha(t)$ which is smooth but might not be parametrized by arclength. Prove that if $\| \alpha(s) - \alpha(t) \|$ depends only on $\| s - t \|$ then $\alpha$ is part of a line or a circle.
\item \textbf{(The First Fundamental Form, easier)} Gerardus Mercator developed his map projection in 1569. We can think of the Mercator projection as a system of local coordinates on the spherical Earth given by
\begin{equation*}
x(u,v) = (\sech u \cos v, \sech u \sin v, \tanh u)
\end{equation*}
Prove that these are conformal coordinates: angles measured in the u-v plane of the map agree with angles measured on the sphere.
\textbf{Hint.} The following are standard calculus facts about hyperbolic trig functions:
\begin{enumerate}
\item $\sech x = 1/\cosh x$, $\tanh x = \sinh x/\cosh x$.
\item $\frac{d}{dx} \cosh x = \sinh x$, $\frac{d}{dx} \sinh x = \cosh x$.
\item $\cosh^2 x - \sinh^2 x = 1$.
\end{enumerate}
\item \textbf{(The First Fundamental Form, harder)} The torus of revolution with major radius $R$ and minor radius $r$ is parametrized by
\begin{equation*}
x(u,v) = ( (R + r \cos v) \cos u, (R + r \cos v) \sin u, r \sin v )
\end{equation*}
There are two obvious families of (round) circles in this surface. Find a third family of circles and draw a picture showing how they fit into the surface. You will almost certainly have to use Maple or Mathematica to complete this problem. (Hint: Look for a plane which is tangent to the torus at exactly \textbf{two} points.)
\item \textbf{(The Second Fundamental Form, easier)} Prove or give a counterexample:
\begin{enumerate}
\item If a curve is both an asymptotic curve \textbf{and} a line of curvature, then it must be a plane curve.
\item If a curve is both an asymptotic curve \textbf{and} a plane curve, then it must be a line.
\end{enumerate}
\item \textbf{(The Second Fundamental Form, harder)} Suppose that $S$ is a surface with no umbilic points and one principal curvature $k_1 \neq 0$ constant. Prove that $M$ lies on a tube of radius $1/|k_1|$ around a curve. (That is, $M$ is the union of circles of radius $1/|k_1|$ in planes normal to the tangents of a curve $\alpha$ centered at corresponding points on $\alpha$.)
\textbf{Hints:} Choose a special parametrization where the coordinate curves $u = \text{constant}$ and $v = \text{constant}$ are lines of curvature with principal curvatures $k_1$ and $k_2$. Use the Mainardi-Codazzi equations to show that the $u = \text{constant}$ curves are planar curves of curvature $|k_1|$ (that is, circles). Define $\alpha$ to be the curve formed by the centers of these circles, and check that $\alpha$ is a regular curve.
\item \textbf{(Geodesics and Gauss-Bonnet, easier)} Let $S$ be the paraboloid given by $x(u,v) = (u \cos v, u \sin v, u^2)$. Let $S_r$ be the potion of the paraboloid with $0 \leq u \leq r$. Then
\begin{enumerate}
\item Calculate the geodesic curvature of the boundary of $S_r$ and compute $\int_{\partial S_r} \kappa_g(s) ds$.
\item Calculate the Euler characteristic $\chi(S_r)$.
\item Use Gauss-Bonnet to compute $\iint_{S} K d\text{Area}$.
\item Compute $\iint_{S} K d\text{Area}$ explicitly by calculating the curvature of $S_r$. (Be careful to integrate $d\text{Area}$ and not $du\, dv$.)
\end{enumerate}
\item \textbf{(Geodesics and Gauss-Bonnet, harder)}
The usual tiling of the plane by squares parallel to $x$ and $y$ axes is a tiling by congruent geodesic quadrilaterals so that four such quadrilaterals meet at each vertex of the tiling. Is it possible to tile the sphere with congruent geodesic quadrilaterals in the same way (so that four such quadrangles meet at each vertex)?
\end{problems}
\end{document}