McGill University April 16, 2014
Faculty of Science Final examination
Math 264
April 16, 2014
Time: 6PM-9PM
Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer
Student name (last, first) Student number (McGill ID)
INSTRUCTIONS
1. This is a closed book exam EXCEPT you are allowed ONE DOUBLED SIDED 8.5 x 11
SHEET of information. Calculators are NOT permitted.
2. Make sure you CAREFULLY READ the question BEFORE embarking on the solution.
3. Note the value of each question.
4. This exam consists of 15 pages (including the cover page). Pages 14 and 15 provide extra space
in case you need them. Please check that ALL pages are intact and provide all your answers
on this exam.
A paddle wheel illustration for Question 7:
Problem 1 2 3 4 5 6 7 8 9 10 Total
Mark
Out of 10 10 10 12 10 10 8 10 10 10 100
Math 264 Final Exam Page 2 April 16, 2014
Question 1a (5 pts):
Find the flux of the vector field F = 〈x, y, z〉 through the surface which is the part of the
paraboloid z = x2 + y2 which lies below the plane z = 4 (oriented with the upward normal).
Leave your answer as an iterated double integral either with respect to x and y or polar coordinates
r and θ.
Question 1b (5 pts): Find the work done by the force field F = 〈y, z, x〉 on moving a particle
along the curve
r(t) = 〈t, t2, t3〉, 0 ≤ t ≤ 1.
Math 264 Final Exam Page 3 April 16, 2014
Question 2a (5 pts). Write down a double iterated integral with respect to x and y which gives
the surface area of the top half of the ellipsoid
x2
a2
+
y2
b2
+
z2
c2
= 1.
Do not evaluate the integral.
Question 2b (5 pts). Write down an example of a conservative vector field for which the
previous ellipsoid is an equipotential surface.
Math 264 Final Exam Page 4 April 16, 2014
Question 3a (5 pts). Let S be the boundary surface, oriented with the outer normal, of the
region V which lies above the xy plane (i.e. z ≥ 0) and is bounded by the two planes and
cylinder:
y = 0, y = 2, x2 + z2 = 1.
Let
F = 〈x+ cos y, y + sin z, z + ex〉.
Evaluate the flux out of V , i.e. the flux integral∫∫
S
F · dS.
Question 3b (5 pts) Suppose a smooth vector field F has divergence equal to 3 at the origin, i.e.
div F(0, 0, 0) = 3. Use only this information to approximate the flux of F out of the sphere centred
at the origin of radius 2.
Math 264 Final Exam Page 5 April 16, 2014
Question 4a (8 pts). Use the change of variables
u = x+ y v = x− 2y
to write the integral ∫∫
R
(2x− y) dx dy,
where R is the trapezoidal region in the xy plane with boundary points (0,−1), (0,−2), (2, 0), (4, 0),
as an iterated integral with respect to u and v. Make sure you clearly state the limits of
integration.
Math 264 Final Exam Page 6 April 16, 2014
Question 4b (4 pts).Verify the Divergence Theorem for
F = 〈x, y, z〉,
with D the solid ball centred at the origin of radius 1 and S its boundary surface (the unit sphere).
Math 264 Final Exam Page 7 April 16, 2014
Question 5b (7 pts). Use Green’s Theorem to find the area of the region which lies inside
both circles
x2 + y2 = 4 and (x− 2)2 + y2 = 4.
Leave your answer as a single definite integral (or sum of two integrals) with respect to parameter
t.
Hint: first draw a picture. For the intersection points of the circles, consider the triangle formed
by one of these intersection points and the centres of the circles. What type of triangle is it?
Math 264 Final Exam Page 8 April 16, 2014
Question 6a (7 pts). Consider the unit cube with vertices (corner points)
(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 1).
Let S be the boundary of the cube minus (i.e. not including) the bottom square (the
side which lies in the xy plane). Orient S with the normal which points out of the cube. Let
F =
〈− y , x , y2ex〉.
Evaluate ∫∫
S
(curl F) · dS.
Make sure you carefully compute the curl. Hint: Use Stokes’ Theorem twice to write the flux
integral as an equal flux integral but over a different surface.
Math 264 Final Exam Page 9 April 16, 2014
Question 6b (3 pts). Let f(x, y, z) = x2 + y3 + z2 and let C be the helix with parametrization
r(t) = 〈2 cos t, 2 sin t, t2〉, t ∈ [0, pi].
Find ∫
C
(∇f) · dr.
Math 264 Final Exam Page 10 April 16, 2014
Question 7 (8 pts total). Consider the following experiment associated with a fluid moving in
space. Two paddle wheels of radius 1 lying in the xy plane are placed with centres at two different
points. In all cases, the centre is fixed but the wheel is free to turn (depending on what the fluid
is doing). For each spatial velocity vector field v of the fluid and choices of centres for the paddle
wheels, decide whether or not each paddle wheel will turn, and if so which way (clockwise or counter
clockwise in the xy plane).
a) v = 〈x, y, z〉 and centres at (x, y) = (0, 0) and at (x, y) = (4, 4).
b) v = 〈y,−x, 0〉 and centres at (x, y) = (0, 0) and at (x, y) = (4, 4).
c)
v =
〈 −y
x2 + y2
,
x
x2 + y2
, 0

and centres at (x, y) = (0, 0) and at (x, y) = (4, 4).
Math 264 Final Exam Page 11 April 16, 2014
Question 8 (10 pts total). Consider the function φ(x) = x2 + 1.
a) What are the Fourier sine series and the Fourier cosine series of φ(x) on the interval x ∈ (0, 1)?
Write down these series but leave your coefficients as integrals.
b) At the point x = 0, what number does the Fourier sine series converge to? What number does
the Fourier cosine series converge to? In other words, if we take more and more terms in the sum
and evaluate at x = 0, what number will we get close to?
c) For each series, plot the function that the Fourier series will get closer to (converge to) on the
interval x ∈ (−2, 2).
Math 264 Final Exam Page 12 April 16, 2014
Question 9a (5 pts). What are all the positive eigenvalues and corresponding eigenfunctions of
the BVP for y(x), x ∈ [0, 1]:
y′′ + λy = 0 y′(0) = 0 y(1) = 0.
Read the above line carefully, i.e. note the prime (derivative) in one of the boundary conditions.
Question 9b (5 pts). Consider the BVP for the heat equation:
ut = uxx for x ∈ (0, 1), t ≥ 0
ux(0, t) = 0 u(1, t) = 10 for t > 0,
u(x, 0) = x2 for x ∈ [0, 1].
What is a physical model for which this BVP applies, i.e. a physical interpretation of u, the
domain x ∈ [0, 1], the condition u(x, 0) = x2, and the boundary conditions.
Math 264 Final Exam Page 13 April 16, 2014
Question 10 (10 pts). Consider the BVP from the previous question:
ut = uxx for x ∈ (0, 1), t ≥ 0
ux(0, t) = 0 u(1, t) = 10 for t > 0,
u(x, 0) = x2 for x ∈ [0, 1].
Find the steady state solution and then use the method of separation of variables to solve the
BVP – leave all the coefficients as integrals.
Math 264 Final Exam Page 14 April 16, 2014
EXTRA SPACE – PLEASE REFERENCE THE QUESTION NUMBER
Math 264 Final Exam Page 15 April 16, 2014
EXTRA SPACE – PLEASE REFERENCE THE QUESTION NUMBER 