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Unformatted text preview: (3/23/08) Section 14.5 Directional derivatives and gradient vectors Overview: The partial derivatives f x ( x ,y ) and f y ( x , y ) are the rates of change of z = f ( x,y ) at ( x ,y ) 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 ,y ) 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 , y ), 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 + su 1 , y = y + su 2 , and the value of z = f ( x,y ) at the point s on the s-axis is F ( s ) = f ( x + su 1 , y + su 2 ) . (1) We call z = F ( s ) the cross section through ( x , y ) of z = f ( x,y ) in the direction of u . braceleftbigg x = x + su 1 y = y + su 2 Tangent line of slope F prime (0) = D u f ( x , y ) FIGURE 1 FIGURE 2 327 p. 328 (3/23/08) Section 14.5, Directional derivatives and gradient vectors If ( x ,y ) negationslash = (0 , 0), we introduce a second vertical z-axis with its origin at the point ( x , y , 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 , y , f ( x , y )). The directional derivative is denoted D u f ( x , y ), as in the following definition. Definition 1 The directional derivative of z = f ( x, y ) at ( x , y ) 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 , y ) = bracketleftbigg d ds f ( x + su 1 , y + 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....
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