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Rog_Sec_14_5_P_1

Rog_Sec_14_5_P_1 - Math 20C Lecture Examples Section 14.5...

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(8/17/08) Math 20C. Lecture Examples. Section 14.5, Part 1. Directional derivatives and gradient vectors in the plane The x -derivative f x ( a, b ) is the derivative of f at ( a, b ) in the direction of the unit vector u , and the y -derivative f y ( a, b ) is the derivative in the direction of the unit vector j . To find the derivative of z = f ( x, y ) at ( a, b ) in the direction of an arbitrary unit vector u = ( u 1 , u 2 ) , we introduce an t -axis, as in Figure 1, with its origin at ( a, b ), with its positive direction in the direction of u , and with the scale used on the x - and y -axes. Then the point at t on the t -axis has xy -coordinates x = a + tu 1 , y = b + tu 2 , and the value of z = f ( x, y ) at the point t on the t -axis is F ( t ) = f ( a + tu 1 , b + tu 2 ) . (1) We call z = F ( t ) the cross section through ( a, b ) of z = f ( x, y ) in the direction of u . Its t -derivative at t = 0 is the directional derivative of f at ( a, b ). braceleftbigg x = a + tu 1 y = b + tu 2 Tangent line of slope F prime (0) = D u f ( a, b ) FIGURE 1 FIGURE 2 Definition 1 The directional derivative of z = f ( x, y ) at ( a, b ) in the direction of the unit vector u = ( u 1 , u 2 ) is the derivative of the cross section function (1) at t = 0 : D u f ( a, b ) = bracketleftbigg d dt f ( a + tu 1 , b + tu 2 ) bracketrightbigg t =0 . (2) The directional derivative (2) is the rate of change of f at ( a, b ) in the direction of u . Its geometric meaning is shown in Figure 2. We introduce a second vertical z -axis with its origin at ( a, b ) as in Figure 2. Then the graph of z = F ( t ) is the intersection of the surface z = f ( x, y ) with the tz -plane and the directional derivative of z = f ( x, y ) is the slope of the tangent line to this curve in the positive t -direction at t = 0. Lecture notes to accompany Section 14.5, Part 1 of Calculus, Early Transcendentals by Rogawski. 1
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Math 20C. Lecture Examples. (8/17/08) Section 14.5, Part 1, p. 2 The next theorem is used to calculate directional derivatives from partial derivatives.
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