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 Ï A , prove that there is a number  d > 0  such that    for all  y Î A .

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

Hint: For each  b Î B  find an open set  U  containing  b  such that this relation holds for  x Î U Ç 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 ½ For every open rectangle  U  containing  x ,  U Ç A ¹ Æ } .

a) Show that   = A È bd A .

b) Prove  bd A =  Ç  = 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 ) .