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 ) .
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