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Unformatted text preview: MAT 257Y Solutions to Practice Term Test 2 (1) Let f : R n R m be C 1 where n > m . Suppose [ df ( x )] has rank m . Show that there exists > 0 such that for any y B ( f ( x ) , ) there exists x R n such that f ( x ) = y . Solution By the Corollary to Implicit Function Theorem there exist an open set U R n , an open set V R n containing x and a diffeomorphism : U V such that f ( ( x, y )) = y for any ( x, y ) U . Here x R n- m , y R m . Let p = ( a, b U be the preimage of x . i.e. ( p ) = x . Since U is open there is an > such that B ( p. ) U . Then for any y B ( f ( x ) , ) there exists u U such that f ( ( u )) = y . (2) Let A be a rectangle in R n and let S A be a set of measure zero which is rectifiable. Show that S has content zero. Hint: Use that R A S exists and must be equal to zero. Solution Let f = S . Then R A f exists and since f = 0 ex- cept on a set of measure 0, we must have R A f = 0 by a theorem from class. Leta theorem from class....
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