21-356 Lecture 3

N for all s r and x r write x i1 xi ei then by the

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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.

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