21-356 Lecture 3

# Since x0 is an interior point there exists b x0 r e

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: directional derivatives of f at x0 exist and !f (x0 ) = dfx0 (v) , !v (ii) for every direction v, N # !f !f (x0 ) = (x0 ) vi . !v ! xi i=1 (1) Proof. Since x0 is an interior point, there exists B (x0 , r) \$ E . Let v # RN be a direction and x = x0 + tv. Note that for |t| &lt; r, we have that &amp;x % x0 &amp; = &amp;x0 + tv % x0 &amp; = &amp;tv&amp; = |t| &amp;v&amp; = |t| 1 &lt; r and so x0 + tv # B (x0 , r) \$ E . Moreover, x &quot; x0 as t &quot; 0 and so, since f is di!erentiable at x0 , f (x0 + tv) % f (x0 ) % T (x0 + tv % x0 ) f (x) % f (x0 ) % T (x % x0 ) = lim t!0 &amp;x % x0 &amp; &amp;x0 + tv % x0 &amp; f (x0 + tv) % f (x0 ) % tT (v) = lim . t!0 |t| 0 = lim x!x0 Since lim t!0 |t| t is bounded by one, it follows that f (x0 + tv) % f (x0 ) % tT (v) |t| f (x0 + tv) % f (x0 ) % tT (v) = lim = 0. t!0 t t |t| But then 0 = lim t!0 f (x0 + tv) % f (x0 ) % tT (v) f (x0 + tv) % f (x0 ) = lim % T (v) , t!0 t t 7 !f which shows that there exists ! v (x0 ) = T (v). !N Part (ii) follows from the linearity of T . Indeed, writing v = i=1 vi ei , by the linearity of T , &quot;N \$ N N # # # !f !f (x0 ) = T (v) = T vi ei = vi T (ei ) = (x0 ) vi . !v ! xi i=1 i=1 i=1 Remark 11 If in the previous theorem x0 is not an interior point but for some direction v # RN , the point x0 is an accumulation point of the set E ) L, where L is the line through x0 in the direction v, then as in the ﬁrst part of the proof !f we can show that there exists the directional derivative ! v (x0 ) and !f (x0 ) = dfx0 (v) . !v 8...
View Full Document

## This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online