Rog_Sec_14_5_P_2

Rog_Sec_14_5_P_2 - (8/16/08) Math 20C. Lecture Examples....

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(8/16/08) Math 20C. Lecture Examples. Section 14.5, Part 2. Directional derivatives and gradient vectors in space The defnitions and results in Part 1 oF this section concerning gradient vectors and directional derivatives with two variables can be converted to the three-variable case by allowing For the extra variable in the Formulas. Defnition 1 (a) The gradient vector of w = f ( x, y, z ) at ( a, b, c ) is f ( a, b, c ) = a f x ( a, b, c ) , f y ( a, b, c ) , f z ( a, b, c ) A . (1) (b) The directional derivative D u f ( a, b, c ) of z = f ( x, y, z ) at ( a, b, c ) in the direction of the unit vector u = a u 1 , u 2 , u 3 A is the t -derivative of the cross section w = f ( a + tu 1 , b + tu 2 , b + tu 3 ) at t = 0 . The directional derivative D u f ( a, b, c ) is the rate oF change oF f with respect to distance in the direction oF the vector u . Theorem 1 (a) The directional derivative D u f ( a, b, c ) of w = f ( x, y, z ) is equal to the dot product of the gradient of f at
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Rog_Sec_14_5_P_2 - (8/16/08) Math 20C. Lecture Examples....

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