21-356 Lecture 15

Moreover 1 jf 1 y jf f 1 y proof we apply the implicit

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: . Proof. We apply the implicit function theorem to the function h : U RN RN defined by h (x, y) := f (x) y. Let b = f (a). Then h (a, b) = 0 and det h (a, b) = det Jf (a) = 0. x Hence, by the implicit function theorem there exists B (a, r0 ) RN and B (b, r1 ) RN such that B (a, r0 ) B (b, r1 ) U RN and a function g : B (b, r1 ) B (a, r0 ) of class C m such that h (g (y) , y) = 0 for all y B (b, r1 ), that is, f (g (y)) = y for all y B (b, r1 ). This implies that g = f 1 . Moreover, 1 g h h (y) = (g (y) , y) (g (y) ,...
View Full Document

This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online