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332mt2soln

332mt2soln - B U Department of Mathematics Spring 2006 Math...

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B U Department of Mathematics Spring 2006 Math 332 - Real Analysis II, Second Midterm, 22/5/2006, 17:00-19:30 Whatever theorem, proposition, lemma you use, Full Name : you must be sure that it is applicable. Student ID : Justify all your claims in your proofs. over 40 1. [8] What does a mathematician mean exactly when he/she says: ”A function f : R m R n is not interesting away from its critical points.” Here we might have m n or n m . Solution: A critical point for an R -valued function is a point where the derivative map is zero, i.e. the rank of the matrix of the derivative map is not full rank. Make the same definition for a function f : R m R n . Let m = n . Then by Inverse Function Theorem, f is nothing but a local coordinate change; i.e. up to a local coordinate change, f is the identity map. Is it interesting? Not much. If m n , then by Domain Straightening Theorem f turns out to be just a projection up to a local coordinate change. If m n , then by Range Straightening Theorem f turns out to be just an insertion up to a local coordinate change. Hence, away from critical points, maps are either identity,

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332mt2soln - B U Department of Mathematics Spring 2006 Math...

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