21-356 Lecture 15

Moreover 1 g h h y g y y g y y yk x yk that

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Unformatted text preview: y) , yk x yk that is, 1 f 1 (y) = Jf f 1 (y) ek . yk The next exercise shows that dierentiability is not enough for the inverse function theorem. 42 Exercise 63 Consider the function f : R2 R2 defined by 0 if x = 0, f1 (x, y ) = 1 x + 2x2 sin x if x = 0, f2 (x, y ) = y. Prove that f = (f1 , f2 ) is dierentiable in (0, 0) and Jf (0, 0) = 1. Prove that f is not one-to-one in any neighborhood of (0, 0). The next exercise shows that the existence of a local inverse at every point does not imply the existence of a global inverse. Exercise 64 Consider the function f : R2 R2 defined by f (x, y ) = (ex cos y, ex sin y ) . Prove that det Jf (x, y ) = 0 for all (x, y ) R2 but that f is not injective. 43...
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This document was uploaded on 03/31/2014.

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