Math 303 - Complex Variables
Homework due March 6
Question 1.
(a) Show that the ellipse
x2
a2+y2
b2= 1
is a smooth closed curve by providing a parameterization z(t).
(b) Sketch the ellipse with a= 2 and b= 3.
(c) Compute the derivative z0(t) of your parameterization.
(d) Compute the tangent vectors z0(t) for a few values of tand draw the corresponding tangent
vectors on your graph from (b).
(d) Verify that (i), (ii), and (iii’) of the definition from the book for smooth closed curve holds true
for your parameterization.
Question 2. Parameterize the contour consisting of the perimeter of the square with vertices −1−i,
1−i, 1 + i, and −1 + i. Does your parameterization give a smooth closed curve? Why or why not?
Question 3. Parameterize the below “ice cream cone”, which includes a semi-circle of radius 3 about
the origin as the “ice cream” and the “cone” portion that has its bottom point at −5i.
-7.5 -5 -2.5 0 2.5 5 7.5 10
-5
-2.5
2.5
5
Figure 1: The “ice cream cone”
1
