Unformatted text preview: From the textbook, solve 1 . 9 . 1 and 1 . 9 . 2 . 6. ( Friday, February 5. ) Pathological functions and second order derivatives. Put f ( x,y ) = ( xy · x 2y 2 x 2 + y 2 if ( x,y ) 6 = (0 , 0) if ( x,y ) = (0 , 0) . (i) Calculate the ﬁrst derivative f x at (0 , 0) directly from the deﬁnition. (ii) Calculate the ﬁrst derivative f x at any other point ( x,y ) 6 = (0 , 0) . (iii) Differentiate once more i.e. calculate f xy (0 , 0) using the deﬁnition. Conﬁrm that f xy (0 , 0) = 1 . (iv) Repeat for the derivative f y (0 , 0) then f yx (0 , 0) . Conﬁrm that f yx (0 , 0) =1 . Observe that f xy (0 , 0) 6 = f yx (0 , 0) . 7. ( Friday, February 5. ) Spaces of matrices. From the textbook, solve 1 . 8 . 13 and 1 . 10 . 30 . 1...
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 Winter '11
 DragosOprea
 Calculus, Chain Rule, Derivative, Continuous function, Holomorphic function, ﬁrst derivative fx

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