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Final Examination for Math 317 at University of British Columbia - December 2006, Exams of Mathematics

Acharya Nagarjuna UniversityMathematics

The final examination for math 317 at the university of british columbia, held on december 7th, 2006. The examination covers various topics in vector calculus, including integrals, line integrals, surface integrals, and vector fields. Students were required to evaluate integrals, determine unit tangent and normal vectors, compute curvature, and identify conservative vector fields.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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The University of British Columbia
Final Examination - December 7th, 2006
Mathematics 317, joint final
Instructors: Jim Bryan and Alexander Roitershtein
Closed book examination Time: 2.5 hours
Name Signature
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 in-
vigilators, 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 12
2 13
3 12
4 13
5 12
6 13
7 12
8 13
Total 100
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Download Final Examination for Math 317 at University of British Columbia - December 2006 and more Exams Mathematics in PDF only on Docsity!

The University of British Columbia Final Examination - December 7th, 2006 Mathematics 317, joint final Instructors: Jim Bryan and Alexander Roitershtein

Closed book examination Time: 2.5 hours

Name Signature

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 in- vigilators, 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.

Total 100

Problem 1. (12 points.) Evaluate the integral (^) ∫

C

xy dx + yz dy + zx dz

around the triangle with vertices (1, 0 , 0), (0, 1 , 0), and (0, 0 , 1), oriented clockwise as seen from the point (1, 1 , 1).

Problem 3. (12 points.) Let S be the surface given by the equation

x^2 + z^2 = sin^2 (y)

lying between the planes y = 0 and y = π. Evaluate the integral

∫ ∫

S

1 + cos^2 (y) dS.

Problem 4. (13 points.) Let S be the part of the sphere x^2 + y^2 + z^2 = 4 between the planes z = 1 and z = 0 oriented away from the origin. Let

F = (ey^ + xz) i + (zy + tan(x))j + (z^2 − 1)k.

Compute the flux integral (^) ∫ ∫

S

F · dS.

Problem 5. (12 points.) Let

r(t) = cos^3 t i + sin^3 t j +

sin t cos t k.

Reparameterize r(t) with respect to arclength measured from the point t = 0 in the direction of increasing t.

Problem 6. (13 points.) Let r(t) = t^2 i + 2t j + ln t k.

Compute the unit tangent and unit normal vectors T(t) and N(t). Compute the curvature κ(t). Simplify whenever possible!

Problem 7. (12 points.) Show that the following line integral is independent of path and evaluate the integral.

∫

C

(yex^ + sin y)dx + (ex^ + sin y + x cos y)dy,

where C is any path from (1, 0) to (0, π/2).

Problem 8. (13 points.) Let

F =

−z x^2 + z^2

i + y j +

x x^2 + z^2

k.

  1. Determine the domain of F.
  2. Determine the curl of F. Simplify if possible.
  3. Determine the divergence of F. Simplify if possible.
  4. Is F conservative? Give a reason for your answer.