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 \centerline{\bf {Mathematics 438}} 
\centerline{\bf Fall Semester 2000} 
\centerline{\bf Midterm Exam 1 (take-home)} 
\centerline{\bf {Due October 30, 2000}} 
 
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\noindent 
 
 
 
 
{\bf Sign the pledge:}  \lq \lq {On my honor, 
I have neither given nor received unauthorized aid on this Exam}": 
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\centerline{\bf GOOD LUCK} 
 

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\noindent 
1. (20 points)  A rat runs around the circle of radius 10 feet centered at $(0, 0)$
counter-clockwise with unit speed beginning at $(10, 0)$; its position 
at time $s$ is therefore
$$
   R(s) = ( 10 \cos \frac{s}{10}, 10 \sin \frac{s}{10}).
$$
 A cat starts at the same time from $(0, 0)$ and runs with unit speed
with unit speed toward the rat at the times.
\roster
\item Find a formula for the position $c(s)$ of the cat at all times
(Hint: Is it true $\frac{d c(s)}{ds} = ( \cos \frac{s}{10},  \sin \frac{s}{10}) $?
Find $\frac{d c(s)}{ds}$ and then integrate).

\item At what time does the cat bite the rat ?

\endroster
 
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\noindent 
2. (10 points) (1) Check that the curve
$$
  \alpha(t) = (\tan t,  \sec t)
$$
for $- \frac{\pi}{2} = \le t \le \frac{\pi}{2}$ is a regular curve.

(2) Compute $k(t)$;


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\noindent 
3. (15 points) Let $\alpha: [0, L] \to \Bbb R^2$ be a smooth curve parametrized
by arc-length. 
\roster
\item If $k(s) \equiv  \frac{1}{r_0}$ for some positive constant $r_0$
and if $\alpha'(s) = (\cos \theta (s), \sin \theta (s))$, then 
use the Frenet formula to compute $\theta'(s)$;

\item Integrate $\theta'(s)$ and $\alpha'(s)$ to find $\alpha (s)$, 
where $\theta'(s)$ is given by (1);

\item Check that if $k(s) \equiv  \frac{1}{r_0}$ for some positive constant $r_0$
then $\alpha([0, L])$ must be a subset of some circle of radius $r_0$.

\endroster


 

 
 
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4. (15 points) Suppose that we have a differentiable curve in $\Bbb R^3$. Consider 
its orthogonal projection on the $xy$-plane. Then prove the second curve
has the length less than or equal that of the first. What can you say
if their lengths  are equal ?

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\noindent 
5.   (20 points) (1) Compute the first fundamental form of the surface
 $$
   X(u, v) = (e^v \cos u, e^v \sin u, \int \sqrt{1-e^{2v}} dv)
$$


(2) Use the formula (9) of page 162 to compute the curvature of 
the  surface:
$$
   X(u, v) = (e^v \cos u, e^v \sin u, \int \sqrt{1-e^{2v}} dv).
$$

 
 
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\noindent 
6. (20 points) Let $X(u, v) = (\cos u, \sin u, u) + v(-\sin u, \cos u, 1)$.

  (1) Compute the first fundamental form;

  (2) Compute the unit normal vector $N(u, v)$ of the surface 
 

(3) Compute the Gauss curvature of the surface.
 
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