Differential Geometry Problems
Reading Assignment: 3.1 (through page 181), 3.2
Suggested Problems: 3.1: 1,3, 7, 15, 17
3.2: 1,5,11, 12
#1. Explain why the path parameterized by x (t )  (t 2 , t 3 ), 1  t  1 is not smooth.
#2. Suppose that x is a C 2 path with non-zero velocity and x  x  0 for all t. Show that x has
constant speed if and only if its velocity and acceleration vectors are always perpendicular to
one another.
#3. Prove that for a C2 path x :[a, b] 
3
,
d
 x (t )  v (t )   x (t )  a (t ) .
dt
#4. Find the curvature of each path at the given point.
a.
x (t )  (t 2 , t 3 ), 1  t  1
1 1
p , 
 4 8
b. x (t )  (et cos 2t , et sin 2t ),   t  


p  e 2 ,0
