Rog_Sec_14_5_P_1

Rog_Sec_14_5_P_1 - (8/17/08) Math 20C. Lecture Examples....

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Unformatted text preview: (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....
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This note was uploaded on 08/31/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.

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Rog_Sec_14_5_P_1 - (8/17/08) Math 20C. Lecture Examples....

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