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Vector Valued Function - Mathematics - Exam, Exams of Mathematics

Acharya Nagarjuna UniversityMathematics

This is the Exam of Mathematics which includes Vector Valued Function, Denote, Statements, Vector Equation, Equation, Plane Containing, Derivatives of Inverse, Notation, Derivatives etc. Key important points are: Vector Valued Function, Denote, Statements, Defined, Give Reasons, Problem, Derivatives, Simply Connected Region, Divergence, Endpoints

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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The University of British Columbia
Final Examination - December 7th, 2005
Mathematics 317
Instructor: Jim Bryan
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 10
2 20
3 10
4 10
5 10
6 10
7 10
8 10
9 10
Total 100
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The University of British Columbia Final Examination - December 7th, 2005 Mathematics 317 Instructor: Jim Bryan 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. (10 points.) dr Let r(t) be a vector valued function. Let r′, r′′, and r′′′^ denote dt ,^ d (^2) r dt^2 , and^ d (^3) r dt^3 respectively. Express d dt [(r^ ×^ r

′) · r′′]

in terms of r, r′, r′′, and r′′′. Select the correct answer.

  1. (r′^ × r′′) · r′′′
  2. (r′^ × r′′) · r + (r × r′) · r′′′
  3. (r × r′) · r′′′
  4. 0
  5. None of the above.

Problem 3. (10 points.) Find the speed of a particle with the given position function

r(t) = 5√ 2 ti + e^5 tj − e−^5 tk

Select the correct answer:

  1. |v(t)| = (e^5 t^ + e−^5 t)
  2. |v(t)| = √10 + 5et^ + 5e−t
  3. |v(t)| = √10 + e^10 t^ + e−^10 t
  4. |v(t)| = 5(e^5 t^ + e−^5 t)
  5. |v(t)| = 5(et^ + e−t)

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.

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 (^) ∫ ∫

S^ F^ ·^ dS.

Problem 7. (10 points.) Consider the vector field F(x, y, z) = 2xi + 2yj + 2zk.

  1. Compute curl F.
  2. If∫ C is any path from (0, 0 , 0) to (a 1 , a 2 , a 3 ) and a = a 1 i + a 2 j + a 3 k, show that C F^ ·^ dr^ =^ a^ ·^ a.

[blank page]

Problem 9. (10 points.) Let S be the hemisphere {x^2 + y^2 + z^2 = 1, z ≥ 0 } oriented with N pointing away from the origin. Evaluate the flux integral ∫ ∫ S^ F^ ·^ dS where F = (x + cos(z^2 ))i + (y + ln(x^2 + z^5 ))j + √x^2 + y^2 k.