21-356 Lecture 2

F f simply the derivative of g at t 0 v x0 g 0

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: artial derivative of f with respect !f to xi and is denoted ! xi (x0 ) or fxi (x0 ) or Di f (x0 ). Remark 4 Let F := {t # R : x0 + tv # E } $ R. If we consider the function !f of one variable g (t) := f (x0 + tv), t # F , then ! v (x0 ), when it exists, is !f !f simply the derivative of g at t = 0, ! v (x0 ) = g " (0). If ! v (x0 ) exists, then g is di!erentiable in t = 0 and so it is continuous at t = 0. Thus, the function f restricted to the line L is continuous at x0 . In view of the previous remark, one would be tempted to say that if the directional derivatives at x0 exist and are finite in every direction, then f is continuous at x0 . This is false in general, as the following example shows. Example 5 Let f (x, y ) := ! 1 0 if y = x2 , x %= 0, otherwise. Given a direction v = (v1 , v2 ), the line L through 0 in the direction v intersects the parabola y = x2 only in 0 and in at most one point. Hence, if t is very small, f (0 + tv1 , 0 + tv2 ) = 0. It follows that !f f (0 + tv1 , 0 + tv2 ) " f (0, 0) 0"0 (0, 0) = lim = lim = 0. t!0 t!0 !v t t " # However, f is not continuous in 0, since f x, x2 = 1 & 1 as x & 0, while f (x, 0) = 0 & 0 as x & 0. Exercise 6 Let f (x, y ) := $ x2 y x4 +y 2 0 if (x, y ) %= (0, 0) , if (x, y ) = (0, 0) . Find all directional derivatives of f at 0 and prove that f is not continuous at 0....
View Full Document

Ask a homework question - tutors are online