UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

FALL-WINTER 2005-06

ASSIGNMENT #1. DUE ON OCTOBER 6

PROBLEMS

 

DO NOT SUBMIT YOUR SOLUTIONS WITHOUT THE COVER PAGE.

READ THE INSTRUCTIONS WRITTEN ON THAT PAGE.

 

1. Compute the area of the surface generated when the parametric curve

  is rotated about the line  x = 1 .

 

2. Determine the maximum and minimum values of the curvature at points of the polar curve  r = 3 + sin q  .

 

3. Let  p  denote a real number such that  0 < p < 1  and let  v 1 , v 2 , v 3 ,   be vectors in  R 3  such that    and    for all  k = 1 , 2 , 3 ,   Find the values of  p , if any, for which  .

 

4. Prove or disprove each of the following propositions:

a) If  C 1 , C 2 , C 3 ,   are all closed sets in  R n , then the set    is also a closed set in  R n .

b) If  A 1 , A 2 , A 3 ,   are all non empty compact sets in  R n  such that  A k + 1 A k  for all  k = 1 , 2 , 3 ,   , then the set    is also non empty.

 

 

 

 

 

 

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