21-356 Lecture 4

# 21-356 Lecture 4 - Monday If all the partial derivatives of...

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If all the partial derivatives of f at x 0 exist, the vector ! ! f ! x 1 ( x 0 ) ,..., ! f ! x N ( x 0 ) " ! R N is called the gradient of f at x 0 and is denoted by " f ( x 0 ) or grad f ( x 0 ) or Df ( x 0 ) . Note that part (ii) of the previous theorem shows that df x 0 ( v )= T ( v " f ( x 0 ) · v = N # i =1 ! f ! x i ( x 0 ) v i . (2) for all directions v .Hence ,on lya t interior points of E ,tocheckd i ! erentiability it is enough to prove that lim x ! x 0 f ( x ) # f ( x 0 ) #" f ( x 0 ) · ( x # x 0 ) \$ x # x 0 \$ =0 . (3) Exercise 12 Let f ( x,y ):= \$ x 2 y x 2 + y 2 if ( ) % =(0 , 0) , 0 if ( )=(0 , 0) . Prove that f is continuous at 0 , that all directional derivatives of f at 0 exist but that the formula ! f ! v (0 , 0) = ! f ! x (0 , 0) v 1 + ! f ! y (0 , 0) v 2 fails. Exercise 13 f ( \$ x 2 y x 2 + y 4 if ( ) % , 0) , 0 if ( , 0) . Find all directional derivatives of f at 0 . Study the continuity and the di ! eren- tiability of f at 0 . Theorems 9 and 10 give necessary conditions for the di ! erentiability of f at x 0 .L e t sp rov etha tthe second i t ion sa reno t ,how ev e r ,su " cient. Example 14 f ( % x if y = x 2 ,x % , 0 otherwise. Given a direction v =( v 1 ,v 2 ) , the line L through 0 in the direction v intersects the parabola y = x 2 only in 0 and in at most one point. Hence, if

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21-356 Lecture 4 - Monday If all the partial derivatives of...

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