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UBC Math 263 Exam Dec 2009 - Temp Deriv, Lagrange Mult, Vector Calc, Line Integ, Stokes', Exams of Mathematics

Acharya Nagarjuna UniversityMathematics

The final examination for mathematics 263 at the university of british columbia, held in december 2009. The exam covers various topics in vector calculus, including directional derivatives, temperature level surfaces, lagrange multipliers, curl and divergence, line integrals, and stokes' theorem. Students are required to answer questions related to these topics, some of which involve calculating integrals and finding equations for tangent planes.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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The University of British Columbia
Final Examination - December 2009
Mathematics 263
Section 102
Closed book examination Time: 2.5 hours
Last Name: First: Signature
Student Number
Special Instructions:
- Be sure that this examination has 12 pages. Write your name at the top of each page.
- You are allowed to bring into the exam one 81
2×11 formula sheet filled on both sides. No
calculators or any other aids are allowed.
- 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
UBCcard 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, dishon-
est 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 mem-
ory aid devices, other than those authorized by the examiners;
(b) speaking or communicating with other candidates; and
(c) purposely exposing written papers to the view of other can-
didates or imaging devices. The plea of accident or forgetfulness
shall not be received.
•Candidates must not destroy or mutilate any examination mate-
rial; must hand in all examination papers; and must not take any
examination material from the examination room without permis-
sion of the invigilator.
•Candidates must follow any additional examination rules or di-
rections communicated by the instructor or invigilator.
1 20
2 15
3 20
4 20
5 10
6 15
Total 100
Page 1 of 12 pages
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Partial preview of the text

Download UBC Math 263 Exam Dec 2009 - Temp Deriv, Lagrange Mult, Vector Calc, Line Integ, Stokes' and more Exams Mathematics in PDF only on Docsity!

The University of British Columbia

Final Examination - December 2009

Mathematics 263

Section 102

Closed book examination Time: 2.5 hours

Last Name: First: Signature

Student Number

Special Instructions:

  • Be sure that this examination has 12 pages. Write your name at the top of each page.
  • You are allowed to bring into the exam one 8

1

2

× 11 formula sheet filled on both sides. No

calculators or any other aids are allowed.

  • 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

UBCcard 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, dishon-

est 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 mem-

ory aid devices, other than those authorized by the examiners;

(b) speaking or communicating with other candidates; and

(c) purposely exposing written papers to the view of other can-

didates or imaging devices. The plea of accident or forgetfulness

shall not be received.

  • Candidates must not destroy or mutilate any examination mate-

rial; must hand in all examination papers; and must not take any

examination material from the examination room without permis-

sion of the invigilator.

  • Candidates must follow any additional examination rules or di-

rections communicated by the instructor or invigilator.

Total 100

Page 1 of 12 pages

  1. Suppose the function T (x, y, z) describes the temperature at a point (x, y, z) in space,

with T (1, 1 , 1) = 10, and ∇T (1, 1 , 1) = 2ˆi −

j +

k. Suppose also that the position at

time t of a particle moving through space is (

1 + t, cos t, e

t

).

(a) Compute the directional derivative of T at (1, 1 , 1), in the direction of the vector

i + 2ˆj + 3

k.

(b) At (1, 1 , 1), in what direction does the temperature decrease most rapidly?

  1. Use Lagrange multipliers to find the points on the surface z = x

2

  • 2y

2 that are closest

to the point (0, 0 , 2). (Hint: Minimize the distance squared rather than the distance.)

Extra space (if needed)

Extra space (if needed)

  1. Let C be the closed curve oriented counterclockwise consisting of the line segment from

(0, 0) to (1, 0), the line segment from (1, 0) to (1, 1) and the part of the parabola y = x

2

from (1, 1) to (0, 0). Find

C

F · drˆ where

F (x, y) = xy

i + x

j by two methods:

(a) By calculating the line integral directly.

(b) By using Green’s Theorem.

  1. Use Stokes’ Theorem to evaluate

C

F · drˆ, where C is the curve in which the plane

y = 1 intersects the sphere x

2

  • y

2

  • z

2

= 5, oriented clockwise when viewed from the

positive y-axis, and

F (x, y, z) =

−y

2

  • e

x

2

i + ln(y

2

  • y) ˆj +

x +

z

2

  • 1

k.

  1. Let

F (x, y, z) = 〈 z tan

− 1 (y

2 ), z

3 ln(x

2

  • 1), z 〉. Find the flux of

F across the part of

the paraboloid x

2

  • y

2

  • z = 2 that lies above the plane z = 1 and is oriented upwards.