Engineering Calculus Notes 441

Engineering Calculus Notes 441 - 4.2. DIFFERENTIABLE...

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: 4.2. DIFFERENTIABLE MAPPINGS 429 where → → E2 (△− ) ≤ ε2 △− . x x → → → Note that our expression for G(− 0 + △− ) lets us express △− in the form x x y → → → → △− = DG− 0 (△− ) + E2 (△− ) . y x x x → Applying the claim to DG− 0 we can say that for some M1 > 0 x → → → D G− 0 (△− ) ≤ M1 △− x x x so → → → → △− = D G− 0 (△− ) + E2 (△− ) y x x x → −. ≤ (M + ε ) △ x 1 2 Thus for → △− ≤ max δ2 , x δ1 M1 + ε2 we have → △− ≤ δ1 y so → → → → → F (− ) − F (− 0 ) = DF− 0 (△− ) + E1 (△− ) y y y y y → → with E1 (△− ) < ε1 . Substituting our expression for △− into this, and y y → using the linearity of DF− 0 , we have y → → → → → → → F (− ) − F (− 0 ) = DF− 0 D G− 0 (△− ) + E2 (△− ) + E1 (△− ) y y x x y y x → − ) + DF− (E (△− )) + E (△− ) → → → → → = (DF− 0 ◦ DG− 0 )(△ x x y 2 1 y x y0 so in Equation (4.2), we can write → → → → E (△− ) = DF− 0 (E2 (△− )) + E1 (△− ) ; x x y y ...
View Full Document

This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

Ask a homework question - tutors are online