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The University of British Columbia
Final Examination - April 17, 2009
Mathematics 317 Section 201
Instructors: Mahta Khosravi and Hendryk Pfeiffer
Closed book examination. Time: 2.5 hours
Name Signature
Student Number
Special Instructions:
- Be sure that this examination has 15 pages. Write your name on top of each page.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location.
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBC- card for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action.
(a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the ex- aminers. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candi- dates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any exam- ination material from the examination room without permission of the invigilator.
- Candidates must follow any additional examination rules or direc- tions communicated by the instructor or invigilator.
Total 100
Problem 1 of 9 [10 points]
Assume the paraboloid z = x^2 + y^2 and the plane 2x + z = 8 intersect in a curve C. C is
traversed counter-clockwise if viewed from the positive z-axis.
(1) [4 points] Parameterize the curve C.
(2) [6 points] Find the unit tangent vector T, the principal normal vector N, the binormal vector B and the curvature κ all at the point 〈 2 , 0 , 4 〉.
Problem 3 of 9 [14 points]
Consider the vector field F defined as
F(x, y, z) =
(1 + ax^2 )ye^3 x
2 − bxz cos(x^2 z), xe^3 x
2 , x^2 cos(x^2 z)
where a and b are real valued constants.
(1) [4 points] Compute curl F.
(2) [2 points] Determine for which values a and b the vector field F is conservative.
(3) [5 points] For the values of a and b obtained in part (2), find a potential function f such that ∇f = F.
(4) [3 points] Evaluate the line integral
∫
C
ye^3 x
2
dx + xe^3 x
2 dy + x^2 cos(x^2 z) dz,
where C is the arc of the curve
t, t, t^3
starting at the point 〈 0 , 0 , 0 〉 and ending at the
point 〈 1 , 1 , 1 〉. Hint: Notice the difference between this vector field and the conservative vector field.
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Problem 5 of 9 [10 points]
Suppose S is the part of the hyperboloid x^2 + y^2 − 2 z^2 = 1 that lies inside the cylinder
x^2 + y^2 = 9 and above the plane z = 1 (i.e. for which z ≥ 1).
Which of the following are parameterizations of S? Write your answer ‘yes’ (Y) or ‘no’ (N)
in the following box. No explanation required. [2 points for a correct answer, 1 point if you
do not answer, 0 if wrong]
Y/N
(1) The vector function
r(u, v) = u i + v j +
u^2 + v^2 − 1 √ 2
k
with domain D = { (u, v) | 2 ≤ u^2 + v^2 ≤ 9 }.
(2) The vector function
r(u, v) = u sin v i − u cos v j +
u^2
2
k
with domain D = { (u, v) |
3 ≤ u ≤ 3 , 0 ≤ v ≤ 2 π }.
(3) The vector function
r(u, v) =
1 + 2v^2 cos u i +
1 + 2v^2 sin u j + v k
with domain D = { (u, v) | 0 ≤ u ≤ 2 π, 1 ≤ v ≤ 2 }.
(4) The vector function
r(u, v) =
1 + u sin v i +
1 + u cos v j +
u
2
k
with domain D = { (u, v) | 2 ≤ u ≤ 8 , 0 ≤ v ≤ 2 π }.
(5) The vector function
r(u, v) =
u cos v i −
u sin v j +
u + 1 √ 2
k
with domain D = { (u, v) | 3 ≤ u ≤ 9 , 0 ≤ v ≤ 2 π }.
Problem 6 of 9 [14 points] Consider the ellipsoid S given by
x
2
y^2
4
z^2
9
with the unit normal pointing outward.
(1) [4 points] Parameterize S. Hint: Use polar coordinates in a suitable way. Do not forget
to specify the range of the parameters.
(2) [6 points] Compute the flux
S F^ ·^ dS^ of the vector field
F(x, y, z) = 〈x, y, z〉.
(3) [4 points] Verify your answer in (2) using the divergence theorem.
Problem 7 of 9 [12 points]
Evaluate the line integral
∫
C
z +
1 + z
dx + xz dy +
3 xy −
x
(z + 1)
2
dz,
where C is the curve parameterized by
r(t) =
cos t, sin t, 1 − cos
2 t sin t
, 0 ≤ t ≤ 2 π.
Hint: The curve C bounds the surface z = 1 − x^2 y, x^2 + y^2 ≤ 1.
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Problem 9 of 9 [8 points]
Which of the following statements are true (T) and which are false (F)? Write your answers
in the following box. You do not need to give reasons. All real valued functions f (x, y, z) and
all vector fields F(x, y, z) have domain R^3 unless specified otherwise. [1 point for a correct
answer, 0.5 points if you do not answer, 0 if wrong]
T/F
(1) If f is a continuous real valued function and S a smooth oriented surface, then
∫ ∫
S
f dS = −
−S
f dS,
where ‘−S’ denotes the surface S but with the opposite orientation.
(2) Suppose the components of the vector field F have continuous partial derivatives. If ∫∫ S curl^ F^ ·^ dS^ = 0 for every closed smooth surface, then^ F^ is conservative.
(3) Suppose S is a smooth surface bounded by a smooth simple closed curve C. The orientation of C is determined by that of S as in Stokes’ theorem. Suppose the real
valued function f has continuous partial derivatives. Then
∫
C
f dx =
S
∂f
∂z
j −
∂f
∂y
k
· dS.
(4) Suppose the real valued function f (x, y, z) has continuous second order partial deriva-
tives. Then (∇f ) × (∇f ) = ∇ × (∇f ).
(5) The curve parameterized by
r(t) =
2 + 4t^3 , −t^3 , 1 − 2 t^3
, −∞ < t < ∞,
has curvature κ(t) = 0 for all t.
(6) If a smooth curve is parameterized by r(s) where s is arc length, then its tangent vector
satisfies |r′(s)| = 1.
(7) If S is the sphere x^2 + y^2 + z^2 = 1 and F is a constant vector field, then
S F^ ·^ dS^ = 0.
(8) There exists a vector field F whose components have continuous second order partial
derivatives such that curl F = 〈x, y, z〉.
