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The University of British Columbia
Final Examination - April 19, 2007
Mathematics 317
Instructors: Jim Bryan and Hendryk Pfeiffer
Closed book examination Time: 3 hours
Name Signature
Student Number
Special Instructions:
- Be sure that this examination has 14 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 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 10 [10 points] Suppose the curve C is the intersection of the cylinder x^2 +y^2 = 1 with the plane x+y +z = 1.
(1.) [4 points] Find a parameterization of C. (2.) [3 points] Determine the curvature of C. (3.) [3 points] Find the points at which the curvature is maximum and determine the value of the curvature at these points.
Problem 3 of 10 [10 points] Consider the vector field
F(x, y, z) = − 2 y cos x sin x i + (cos^2 x + (1 + yz) eyz^ ) j + y^2 eyz^ k. (1.) [5 points] Find a real valued function f (x, y, z) such that F = ∇f. (2.) [5 points] Evaluate the line integral ∫
C
F · dr
where C is the arc of the curve r(t) = 〈t, et, t^2 − π^2 〉, 0 ≤ t ≤ π, traversed from (0, 1 , −π^2 ) to (π, eπ^ , 0).
Problem 4 of 10 [10 points] Evaluate the surface integral (^) ∫ ∫
S
F · dS
where F(x, y, z) = 〈cos z + xy^2 , x e−z^ , sin y + x^2 z〉^ and S is the boundary of the solid region enclosed by the paraboloid z = x^2 + y^2 and the plane z = 4.
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Problem 6 of 10 [10 points] Evaluate the line integral ∫
C
(x^2 + y ex) dx + (x cos y + ex) dy
where C is the arc of the curve x = cos y for −π/ 2 ≤ y ≤ π/2, traversed in the direction of increasing y.
Problem 8 of 10 [10 points] Suppose the curve C is the intersection of the cylinder x^2 + y^2 = 1 with the surface z = xy^2 , traversed clockwise if viewed from the positive z-axis, i.e. viewed “from above”. Evaluate the line integral (^) ∫
C
(z + sin z) dx + (x^3 − x^2 y) dy + (x cos z − y) dz.
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Problem 10 of 10 [10 points] Which of the following statements are true (T) and which are false (F)? You do not need to give reasons. This problem will be graded by answer only. [1 point each] (1.) If a smooth curve C is parameterized by r(s) where s is arc length, then the tangent vector r′(s) satisfies |r′(s)| = 1. (2.) If r(t) defines a smooth curve C in space that has constant curvature κ > 0, then C is part of a circle with radius 1/κ. (3.) Suppose F is a continuous vector field with open domain D. If ∫
C
F · dr = 0
for every piecewise smooth closed curve C in D, then F is conservative. (4.) Suppose F is a vector field with open domain D, and the components of F have contin- uous partial derivatives. If ∇ × F = 0 everywhere on D, then F is conservative. (5.) The curve defined by r 1 (t) = cos(t^2 ) i + sin(t^2 ) j + 2t^2 k, −∞ < t < ∞, is the same as the curve defined by r 2 (t) = cos t i + sin t j + 2t k, −∞ < t < ∞. (6.) The curve defined by r 1 (t) = cos(t^2 ) i + sin(t^2 ) j + 2t^2 k, 0 ≤ t ≤ 1 , is the same as the curve defined by r 2 (t) = cos t i + sin t j + 2t k, 0 ≤ t ≤ 1. (7.) Suppose F(x, y, z) is a vector field whose components have continuous second order partial derivatives. Then ∇ · (∇ × F) = 0. (8.) Suppose the real valued function f (x, y, z) has continuous second order partial deriva- tives. Then ∇ · (∇f ) = 0. (9.) The region D = { (x, y) | x^2 + y^2 > 1 } is simply connected.
(10.) The region D = { (x, y) | y − x^2 > 0 } is simply connected.
