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John P. D’Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801 [email protected] defnition Let f : R 3 R be a function. The partial deriva- tives of f are deFned as usual. ±or example ∂f ∂x ( x, y, z ) = lim h 0 f ( x + h, y, z ) f ( x, y, z ) h . and similar equations hold for partials with respect to y and z . We also have the notion of directional derivative: ∂f ∂v ( p ) = lim t 0 f ( p + tv ) f ( p ) t . Notice here that p = ( x, y, z ). Also note that the partial ∂f ∂x is the directional derivative in the i direc- tion; in other words, put v = (1 , 0 , 0). Then ∂f ∂x = ∂f ∂v . Similar statements hold for the other coordinate di- rections. The gradient f is extremely important in this course and in science in general. When the partial derivatives exist, we put them in a vector: f ( p ) = ( ∂f ∂x , ∂f ∂y , ∂f ∂z ) . (1) We have the following uses of the gradient. 1

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