UNIVERSITY OF TORONTO

DEPARTMENT OF MATHEMATICS

MAT 237 Y - MULTIVARIABLE CALCULUS

FALL-WINTER 2005-06

ASSIGNMENT #2. DUE ON OCTOBER 20.

PROBLEMS

 

DO NOT SUBMIT YOUR SOLUTIONS WITHOUT THE COVER PAGE.

READ THE INSTRUCTIONS WRITTEN ON THAT PAGE.

 

1.  Question 1-21 from Spivak:

a) If  A  is closed and  x A , prove that there is a number  d > 0  such that    for all  y A .

b) If  A  is closed,  B  is compact, and  A B = , prove that there is  d > 0  such that   for all  y A  and  x B .

Hint: For each  b B  find an open set  U  containing  b  such that this relation holds for  x U B .

c) Give a counterexample in  R 2  if  A  and  B  are closed but neither is compact.

 

2. Define the  closure  of  A  as

 = { x For every open rectangle  U  containing  xU A } .

a) Show that   = A bd A .

b) Prove  bd A =    = bd ( R n A ) .

Note:  bd A   means  the boundary of the set  A .

 

3. Show that every bounded set in  R  has a least upper bound.

 

4. Prove the following for a vector function  f : R R n :

 is constant  if and only if  f ( x )  is perpendicular to  f ( x ) .

 

 

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