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Unformatted text preview: for all x, y # RN and T (sx) = sT (x)
!N
for all s # R and x # R . Write x = i=1 xi ei . Then by the linearity of T ,
N T (x) = T "N
# xi ei i=1 $ = N
# xi T (ei ) . i=1 Deﬁne b := (T (e1 ) , . . . , T (eN )) # RN . Then the previous calculation shows
that
T (x) = b · x
for all x # RN .
Deﬁnition 7 Let E $ RN , let f : E " R, and let x0 # E be an accumulation
point of E . The function f is di!erentiable at x0 if there exists a linear function
T : RN " R (depending on f and x0 ) such that
lim x!x0 f (x) % f (x0 ) % T (x % x0 )
= 0.
&x % x0 & provided the limit exists. The function T , if it exists, is called the di!erential of
f at x0 and is denoted df (x0 ) or dfx0 .
Exercise 8 Prove that if N = 1, then f is di!erentiable at x0 if and only there
exists the derivative f " (x0 ) # R.
The next theorem shows that di!erentiability in dimension N ! 2 plays the
same role of the derivative in dimension N = 1.
Theorem 9 Let E $ RN , let f : E " R, and let x0 # E be an accumulation
poi...
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This document was uploaded on 03/31/2014.
 Spring '14
 Derivative

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