Ma 109b Geometry and Topology
Winter 2010
FINAL
Instructions
Open book, open notes. You may appeal to any result stated in Do Carmo. References to theorems from other sources will not be accepted.
There are ve questions. The time limit is four hours. No c
Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 1 SOLUTIONS
1. Write (t) in coordinates as
(t) = x(t) + y (t).
A point at which the image of is closest to the origin is a minimum of
|(t)|2 = x2 (t) + y 2 (t).
Taking the derivative with respect to time
Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 2 SOLUTIONS
1. a) A curve is parametrized by arc length if its velocity is 1. We calculate that
(s) = (a cos(s/c), a sin(s/c), bs/c)
| (s)| = 1, s.
b) We calculate that
t(s) =
(s)
= (a cos(s/c), a sin(s
Ma 109b Geometry and Topology
Winter 2009
MIDTERM
Instructions
Open book, open notes. You may appeal to any result stated in Do Carmo. References to theorems from other sources will not be accepted.
There are ve questions. The time limit is four hours. No