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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 realvalued 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 )] ...
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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.
 Spring '08
 ALL
 Calculus, Transformations

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