KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY
INSTITUT DES SCIENCES ET TECHNOLOGIE DE KIGALI
Avenue de l'Armée, B.P. 3900 Kigali, Rwanda
INSTITUTE EXAMINATIONS – ACADEMIC YEAR 2012/2013
FACULTY OF SCIENCE
DEPARTMENT OF APPLIED MATHEMATICS
SEMESTER II MAIN EXAM
MAT3224: Vector Analysis.
SECOND YEAR.
MAXIMUM MARKS: 60.
DATE:
/2013
TIME: 2 HOURS.
Instructions:
1. This paper contains two sections.
2. Section A is compulsory and carries 30 marks.
3. Section B contains three questions. You have to choose any two
Questions. It carries 30 marks.
4. Start every new question on a fresh page.
5. Do not write anything on this question paper.
SECTION A
Question 1
⃗⃗ .
a) Let 𝑟(𝑡⃗) = 𝑒 𝑡 𝑖⃗ + 𝑒 −𝑡 𝑗⃗ + √2𝑡𝑘
⃗⃗(𝑡).
Calculate the unit tangent vector 𝑇
What are the unit tangent vector at 𝑃 = 𝑟𝑙𝑛(2)?
(6 marks)
b) Let 𝐹⃗ (𝑥, 𝑦) = −𝑦𝑖⃗ + 𝑥𝑗⃗. Set 𝑃0 = (1,0) and 𝑃1 = (−1,0).
Consider two paths from 𝑃0 to 𝑃1 : C parameterized by
𝑟⃗(𝑡) = cos 𝑡 𝑖⃗ + sin 𝑡 𝑗⃗ ; 0 ≤ 𝑡 ≤ 𝜋
and 𝐶 ∗ parameterized by 𝑟⃗ ∗ (𝑡) = cos 𝑡 𝑖⃗ − sin 𝑡 𝑗⃗ ; 0 ≤ 𝑡 ≤ 𝜋
Is the line integral of 𝐹⃗ over C equal to the line integral of 𝐹⃗ over
𝐶 ∗?
(6 marks)
⃗⃗ .
c) Let 𝐹⃗ (𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑖⃗ + 𝑦𝑧𝑗⃗ − 2𝑥𝑧𝑘
Calculate 𝑑𝑖𝑣𝐹⃗ .
(2 marks)
d) Prove the following theorem: “ Let 𝑢 be a twice continuously
differentiable scalar valued function on a region 𝐺 in the plane or
space. Let 𝐹⃗ be a twice continuously differentiable vector field on
𝐺. Then 𝑑𝑖𝑣(𝑢𝐹⃗ ) = 𝑢𝑑𝑖𝑣𝐹⃗ + 𝑔𝑟𝑎𝑑 𝑢 ∙ 𝐹⃗
(6marks)
e) State the Stoke’s Theorem.
(2 marks)
f) Verify Green’s Theorem for 𝐹⃗ (𝑥, 𝑦) = −3𝑦𝑖⃗ + 6𝑥𝑗⃗ and
𝑅 = {(𝑥, 𝑦): 𝑥 2 + 𝑦 2 < 1}
(4marks)
g) Find the scalar and vector projection of 𝑏⃗⃗ onto 𝑎⃗ where
⃗⃗ , 𝑏⃗⃗ = 𝑖⃗ + 2𝑗⃗ + 2𝑘
⃗⃗.
𝑎⃗ = −2𝑖⃗ + 3𝑗⃗ + 𝑘
(4marks)
SECTION B
Question 1
a) Prove that the maximum value of 𝐷𝑢⃗⃗ 𝑓(𝑥⃗) is given by ‖∇𝑓(𝑥⃗)‖ and
will occur in the direction given by ∇𝑓(𝑥⃗).
(4 marks)
b) Find the tangent plane and normal line to 𝑥 2 + 𝑦 2 + 𝑧 2 = 30 at the
point (1, -3, 5).
(5 marks)
c) Determine if the following vector field is conservative or not
𝐹⃗ (𝑥, 𝑦) = (𝑥 2 − 𝑦𝑥)𝑖⃗ + (𝑦 2 − 𝑥𝑦)𝑗⃗.
(6marks)
Question 2
a) Let C be the circle 𝑥 2 + 𝑦 2 = 4, oriented counter-clockwise. Use
Green’s Theorem to evaluate the following integral
2
3
3
(8marks)
C (cosx  y )dx  x dy .
b) Suppose that 𝑅, 𝐶and 𝑟 satisfy the hypotheses of Green’s Theorem.
Then (Area of 𝑅) 
1
 ydx  xdy .
2 C
(7marks)
Question 3
a) State the divergence Theorem.
(4 marks)

b) Use the divergence Theorem to evaluate


1


 F  d S where
S

F  xy i  2 y j  z k and the surface consists of three surfaces,
2
z  4  3x 2  3 y 2 , 1  z  4 on the top and x 2  y 2  1 , 0  z  1
side and
z  0 on the bottom.
on the
(11marks)