332mt2soln

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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,to be just an insertion up to a local coordinate change....
View Full Document

This note was uploaded on 05/04/2011 for the course MATH 321 taught by Professor Talinbudak during the Spring '11 term at BU.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online