Engineering Calculus Notes 442

Engineering Calculus Notes 442 - 430 CHAPTER 4. MAPPINGS...

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: 430 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS → we need to estimate this in terms of △− . Now we know from the claim x that there exists M2 > 0 such that → → → D F− 0 (E2 (△− )) ≤ M2 E2 (△− ) ; x x y Using the triangle inequality as well as our previous estimates, we see that for → △− ≤ max δ2 , x δ1 M1 + ε2 we have → → → E (△− ) ≤ M2 E2 (△− ) + E1 (△− ) x x y → − + M ε △− → ≤ M2 ε2 △ x y 21 → = [M2 ε2 + M2 ε1 (M1 + ε2 )] △− . x Thus, if we pick ε2 < ε 2M2 ε1 < ε 2M2 M1 and then → → E (△− ) ≤ [M2 ε2 + M2 ε1 (M1 + ε2 )] △− x x → − < [M ε + M M ε ] △ x 22 1 21 ε ε → → < △− + x △− x 2 2 → = ε △− , x as required. Let us consider a few special cases, to illustrate how this chain rule subsumes the earlier ones. First, a totally trivial example: if f and g are both real-valued functions of one real variable, and y0 = g(x0 ), then the Jacobian matrix of each is a 1 × 1 matrix Jf (y0 ) = [f ′ (y0 )] Jg(x0 ) = [g′ (x0 )] ...
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