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Section14_5 - Section 14.5 Directional derivatives and...

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(3/23/08) Section 14.5 Directional derivatives and gradient vectors Overview: The partial derivatives f x ( x 0 , y 0 ) and f y ( x 0 , y 0 ) are the rates of change of z = f ( x, y ) at ( x 0 , y 0 ) in the positive x - and y -directions. Rates of change in other directions are given by directional derivatives . We open this section by defining directional derivatives and then use the Chain Rule from the last section to derive a formula for their values in terms of x - and y -derivatives. Then we study gradient vectors and show how they are used to determine how directional derivatives at a point change as the direction changes, and, in particular, how they can be used to find the maximum and minimum directional derivatives at a point. Topics: Directional derivatives Using angles of inclination Estimating directional derivatives from level curves The gradient vector Gradient vectors and level curves Estimating gradient vectors from level curves Directional derivatives To find the derivative of z = f ( x, y ) at ( x 0 , y 0 ) in the direction of the unit vector u = ( u 1 , u 2 ) in the xy -plane, we introduce an s -axis, as in Figure 1, with its origin at ( x 0 , y 0 ), with its positive direction in the direction of u , and with the scale used on the x - and y -axes. Then the point at s on the s -axis has xy -coordinates x = x 0 + su 1 , y = y 0 + su 2 , and the value of z = f ( x, y ) at the point s on the s -axis is F ( s ) = f ( x 0 + su 1 , y 0 + su 2 ) . (1) We call z = F ( s ) the cross section through ( x 0 , y 0 ) of z = f ( x, y ) in the direction of u . braceleftbigg x = x 0 + su 1 y = y 0 + su 2 Tangent line of slope F prime (0) = D u f ( x 0 , y 0 ) FIGURE 1 FIGURE 2 327
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p. 328 (3/23/08) Section 14.5, Directional derivatives and gradient vectors If ( x 0 , y 0 ) negationslash = (0 , 0), we introduce a second vertical z -axis with its origin at the point ( x 0 , y 0 , 0) (the origin on the s -axis) as in Figure 2. Then the graph of z = F ( s ) the intersection of the surface z = f ( x, y ) with the sz -plane. The directional derivative of z = f ( x, y ) is the slope of the tangent line to this curve in the positive s -direction at s = 0, which is at the point ( x 0 , y 0 , f ( x 0 , y 0 )). The directional derivative is denoted D u f ( x 0 , y 0 ), as in the following definition. Definition 1 The directional derivative of z = f ( x, y ) at ( x 0 , y 0 ) in the direction of the unit vector u = ( u 1 , u 2 ) is the derivative of the cross section function (1) at s = 0 : D u f ( x 0 , y 0 ) = bracketleftbigg d ds f ( x 0 + su 1 , y 0 + su 2 ) bracketrightbigg s =0 . (2) The Chain Rule for functions of the form z = f ( x ( t ) , y ( t )) (Theorem 1 of Section 14.4) enables us to find directional derivatives from partial derivatives. Theorem 1 For any unit vector u = ( u 1 , u 2 ) , the (directional) derivative of z = f ( x, y ) at ( x 0 , y 0 ) in the direction of u is D u f ( x 0 , y 0 ) = f x ( x 0 , y 0 ) u 1 + f y ( x 0 , y 0 ) u 2 . (3) Remember formula (3) as the following statement: the directional derivative of z = f ( x, y ) in the direction of u equals the x -derivative of f multiplied by the x -component of u , plus the y -derivative of f multiplied by the y -component of u .
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