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Curl - Multivariable - Exam, Exams of Mathematics

Baddi University of Emerging Sciences and TechnologiesMathematics

This is the Exam of Multivariable which includes Interpret Mathematical, Integration, Region, Evaluate, Illustrate, Explanation Needed etc. Key important points are: Curl, Jacobian Matrix, Approximation, Defined, Differentiable Function, Suppose, Function, Directional Derivative, Direction, Point

Typology: Exams

2012/2013

Uploaded on 03/07/2013

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MATH206A MULTIVARIABLE CALCULUS - PROF. P.
WONG
EXAM II - NOVEMBER 3, 2006
NAME:
Instruction: Read each question carefully. Explain ALL your work and
give reasons to support your answers.
Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1. 20
2. 20
3. 20
4. 20
5. 20
Total 100
1
pf3
pf4
pf5

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MATH206A MULTIVARIABLE CALCULUS - PROF. P.

WONG

EXAM II - NOVEMBER 3, 2006

NAME:

Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.

Problems Maximum Score Your Score

  1. 20
  2. 20
  3. 20
  4. 20
  5. 20 Total 100

1

2 EXAM II - NOVEMBER 3, 2006

  1. Let F : R^3 → R^3 be given by F (x, y, z) = (yz, xz, xy). (5 pts) (i) Find div F.

(5 pts) (ii) Find curl F

(5 pts) (iii) What is the Jacobian matrix DF (1, 1 , 1) of F at (1, 1 , 1)?

(5 pts) (iv) Find an approximation of F (0. 9 , 1. 1 , 1 .1).

4 EXAM II - NOVEMBER 3, 2006

  1. Consider the function f (x, y, z) = x^3 y − yz^2 + z^5. (8 pts) (i) Find the directional derivative Duf (1, 1 , 0) of f at the point (1, 1 , 0) in the direction of u = i − j + 3k.

(12 pts) (ii) Find an equation for the plane tangent to the level surface f (x, y, z) = 9 at the point (3, − 1 , 2).

MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 5

  1. Consider the function f (x, y) = 4y − y^3 − x^2. (7 pts) (i) Find all the critical points of f.

(6 pts) (ii) For each of the critical point(s) a found in part (i), find the corresponding Hessian matrix Hf (a).

(7 pts) (iii) Use the second derivative test to classify each of the critical point(s) in part (i), i.e., determine whether the critical point is a local max, local min, or saddle point.