The University of British Columbia
Final Examination - December 7th, 2005
Mathematics 317
Instructor: Jim Bryan
Closed book examination
Time: 2.5 hours
Signature
Name
Student Number
Special Instructions:
- Be sure that this examination has 12 pages. Write your name on top of each page.
- No calculators or notes are permitted.
- 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 should be prepared to produce her/his
library/AMS card upon request.
• No candidate shall be permitted to enter the examination
room after the expiration of one half hour, or to leave during
the first half hour of examination.
• Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
• CAUTION - Candidates guilty of any of the following or
similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers, or memoranda, other
than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other
candidates.
1
10
2
20
3
10
4
10
5
10
6
10
7
10
8
10
9
10
Total
100
December 2005
Math 317
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Page 2 out of 12
Problem 1. (10 points.) Let r(t) be a vector valued function. Let r0 , r00 , and r000 denote
3
dr d2 r
,
, and ddt3r respectively. Express
dt dt2
d
[(r × r0 ) · r00 ]
dt
in terms of r, r0 , r00 , and r000 . Select the correct answer.
1. (r0 × r00 ) · r000
2. (r0 × r00 ) · r + (r × r0 ) · r000
3. (r × r0 ) · r000
4. 0
5. None of the above.
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Page 3 out of 12
Problem 2. (2 points each.) Say whether the following statements are true (T) or false
(F). You may assume that all functions and vector fields are defined everywhere and have
derivatives of all orders everywhere. You do not need to give reasons; this problem will be
graded by answer only.
1. The divergence of ∇ × F is zero, for every F.
R
2. In a simply connected region, C F · dr depends only on the endpoints of C.
3. If ∇f = 0, then f is a constant function.
4. If ∇ × F = 0, then F is a constant vector field.
RR
5. If div F = 0, then S F · dS = 0 for every closed surface S.
R
6. If C F · dr = 0 for every closed curve C, then ∇ × F = 0.
7. If r(t) is a path in three space with constant speed |v(t)|, then the acceleration is
perpendicular to the tangent vector, i.e. a · T = 0.
8. If r(t) is a path in three space with constant curvature κ, then r(t) parameterizes part
of a circle of radius 1/κ.
9. Let F be a vector field and suppose that S1 and S2 are oriented surfaces with the same
boundary curve C, and
RR C is given the
RR direction that is compatible with the orientations
of S1 and S2 . Then S1 F · dS1 = S2 F · dS2 .
10. Let A(t) be the area swept out by the trajectory of a planet from time t0 to time t.
is constant.
Then dA
dt
December 2005
Math 317
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Problem 3. (10 points.)
Find the speed of a particle with the given position function
√
r(t) = 5 2ti + e5t j − e−5t k
Select the correct answer:
1. |v(t)| = (e5t + e−5t )
√
2. |v(t)| = 10 + 5et + 5e−t
√
3. |v(t)| = 10 + e10t + e−10t
4. |v(t)| = 5(e5t + e−5t )
5. |v(t)| = 5(et + e−t )
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Page 5 out of 12
Problem 4. (10 points.) Find the correct identity, if f is a function and G and F are
vector fields. Select the true statement.
1. div(f F) = f curl(F) + (∇f ) × F
2. div(f F) = f div(F) + F · ∇f
3. curl(f F) = f div(F) + F · ∇f
4. None of the above are true.
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Page 6 out of 12
Problem 5. (10 points.) Let S be the part of the paraboloid z + x2 + y 2 = 4 lying between
the planes z = 0 and z = 1. For each of the following, indicate with a yes or a no whether
it correctly parameterizes the surface S. You do not need to give reasons; only the yes/no
answer will be graded.
r(u, v) = ui + vj + (4 − u2 − v 2 )k,
(u, v) ∈ {0 ≤ u2 + v 2 ≤ 1}
√
√
r(u, v) = ( 4 − u cos v)i + ( 4 − u sin v)j + uk,
r(u, v) = (u cos v)i + (u sin v)j + (4 − u2 )k,
(u, v) ∈ {0 ≤ u ≤ 1, 0 ≤ v ≤ 2π}
√
(u, v) ∈ { 3 ≤ u ≤ 2, 0 ≤ v ≤ 2π}
December 2005
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Problem 6. (10 points.) Let S be the part of the plane
x+y+z =2
that lies in the first octant oriented so that N has a positive k component. Let
F = xi + yj + zk.
Evaluate the flux integral
ZZ
F · dS.
S
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Problem 7. (10 points.) Consider the vector field F(x, y, z) = 2xi + 2yj + 2zk.
1. Compute curl F.
2. If
R C is any path from (0, 0, 0) to (a1 , a2 , a3 ) and a = a1 i + a2 j + a3 k, show that
F · dr = a · a.
C
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Problem 8. (10 points.)
Let
F = x sin yi − y sin xj + (x − y)z 2 k.
Use Stoke’s theorem to evaluate
Z
F · dr
C
along the path consisting of the straight line segments successively joining the points P0 =
(0, 0, 0) to P1 = (π/2, 0, 0) to P2 = (π/2, 0, 1) to P3 = (0, 0, 1) to P4 = (0, π/2, 1) to
P5 = (0, π/2, 0), and back to (0, 0, 0).
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Problem 9. (10 points.) Let S be the hemisphere {x2 + y 2 + z 2 = 1, z ≥ 0} oriented with
N pointing away from the origin. Evaluate the flux integral
ZZ
F · dS
S
where
F = (x + cos(z 2 ))i + (y + ln(x2 + z 5 ))j +
p
x2 + y 2 k.
December 2005
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