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 \centerline{\bf {Mathematics 438}} 
\centerline{\bf Fall Semester 2000} 
\centerline{\bf Final Exam (take-home)} 
\centerline{\bf {Due December 15, 2000}} 
 
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\noindent 
 {\bf This Exam is due December 15, 2000, 10:00 am}. 



Please hand in your
answers to Professor Cao  on
or before  December 15, 2000, 10:00 am.
 
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{\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. The equation $ r = e ^{- \theta}$  in polar coordinates describes a set of points in the plane as $\theta$ goes from $0$ to $\infty$.

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\noindent (a) Describe this set by a parametrized curve 
$\alpha : [0, \infty) \to \Bbb R^2$ with $\theta $ as the parameter.

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\noindent (b) Compute the arc length parameter $s(\theta) = \int^{\theta}_0
\| \alpha'(t) \| dt$.

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\noindent (c) Find $\lim_{\theta \to + \infty} s(\theta)$.

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\noindent 
2.   Let $\alpha: [0, L] \to \Bbb R^2$ be a smooth curve 
of the {\it unit } speed, $\alpha(t) = (x(t), y(t)) $
and $\alpha_c(t) = \alpha(t) + c N(t)$ where $N(t) = (- y'(t), x'(t))$.


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\noindent (a) Using the Frenet formulae (p19) to compute $ N'(t) $ and
$\alpha_c'(t)$.

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\noindent (b) Is it true that the vectors $\alpha_c'(t)$ and $\alpha'(t)$
are parallel for all $t$ and all $c$ ? Justify your answer.
 


 



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\noindent
3. Let $\alpha(t) = (t, \frac{1+t}{t}, \frac{1- t^2}{t})$.

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(a) Compute $ x - y + z = ?$

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\noindent
(b) Computer the torsion $\tau$. 

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(c) Why is $\tau = 0$ ?


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4. Show that the solution set of the equation 
$$
   xz + yz + xy = c
$$
is a regular surface for each value of the constant c, other than zero.


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\noindent
5. Take any Riemann metric of the form 
$$
   g = e^{2 \rho}(du^2 + dv^2)
$$
where $ \rho = \rho(u, v)$ is a smooth function of $u$ and $v$.

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\noindent
(a) Compute the connection $\nabla$  of $g$, (
$\nabla_{\frac{\partial}{\partial u}}{\frac{\partial}{\partial u}}$,
$\nabla_{\frac{\partial}{\partial u}}{\frac{\partial}{\partial v}}$,
etc.)


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\noindent
b) Compute the Gaussian curvature of the metric $g$,
(Hint, Use the formulae on p237, Exercise 1-2). 


 

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\noindent 
6. The crew of the Enterprise live in a world they believe to be Euclidean.
We know it is not and the physical metric of their world is 
$g = e^{2 \rho}(du^2 + dv^2)$
with  $e^{2 \rho} = \frac{1}{ (1- r^2)^2}$ where $r^2 = u^2 + v^2$. 
Use Problem 5 b) to compute the Gaussian curvature of their world.


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\noindent 
7. As a result of a celestial storm, the metric of the 
Enterprise world is changed utterly. Assume  that the new metric  
$\tilde{g}$ has Gaussian curvature $\tilde{K} < 0$ everywhere. Let 
two space ships leave the same point in different directions, each traveling along a geodesic of $\tilde{g}$. Show that the two ships will never meet 
again.


 (Hint: Apply Gauss-Bonnet formula (p. 268-269) to a geodesic 
``di-angle" in a surface with $K < 0$). 



 
 
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8. The spaceship Enterprise moves in the disk $D = \{ (x, y) | x^2 + y^2 < 1 \}$.
The time it takes to travel along the curve $ \alpha: [a, b] \to 
D$ is $T = \int_a^b \frac{\sqrt{(x'(t))^2 + (y'(t))^2}}{1- [(x'(t))^2 + (y'(t))^2]} dt$. If $\alpha(t) = ( r(t) \cos \theta (t) ,    r(t) \sin \theta (t))$ show that $T$  is greater than or equal to the time it takes to go from $(r(a), 0)$ to $(r(b), 0)$ along the $x$-axis. Deduce that the spaceship
will never reach $\partial D = \{ (x, y) | x^2 + y^2 = 1 \}$ in finite time
no matter what path it takes.



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\noindent 
9. Let $\Psi (x, y) = (u(x,y), v(x,y)) = (e^x \cos y , e^x \sin y)$,
$\Psi: \Bbb R^2 \to \Bbb R^2$.

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\noindent
a) Compute the matrix of the differential $\Psi_*|_{(x, y)} = 
\frac{\partial (u, v)}{\partial (x, y)}$ for any point $(x, y)$.

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\noindent
b) Is the map  $ \Psi$  conformal ? Justify your answer.
 
 
 
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