Math 102: Calculus II Quiz 11/12
Instructor: Dr. Masood Shah
December 5, 2024
Name:
Roll Number:
Question 1: [10 Marks]
Consider the following vector field:
F (x, y, z) = xy 2 i + x2 yj,
where C : r(t) = ⟨t + sin π2 t, t + cos π2 t⟩ for 0 < t < 1
i) Show that F is a conservative vector field.
[3 Marks]
ii) Determine a function f such that F = ∇f .
[4 Marks]
iii) Using the above or otherwise, find the line integral
1
R
C
F · dr along this line segment.
[3 Marks]
Question 2: [10 Marks]
Consider the following vector field:
F = ⟨y 3 , −x3 ⟩.
i) Find the divergence of this vector field.
[3 Marks]
ii) Find the curl of this vector field.
[3 Marks]
R
iii) Using Green’s theorem determine the line integral C F · dr along the curve C, where C is the circle
x2 + y 2 = 4. Using any method other Green’s theorem will not be given any marks.
[4 Marks]
2