# Here from the definition of the differential we have

• 646

This preview shows page 58 - 63 out of 646 pages.

Here, from the definition of the differential, we have
Substituting in equation ( 2 ), we have du = df = f 1 dg + f 2 dh Observe now the close similarity of this result with the definition in equation ( 1 ). It is precisely this sort of similarity which could be exploited to effect the simplification referred to above. 8.2 MEANING OF THE DIFFERENTIAL The student is familiar with the fact that the equation of the tangent plane to the surface z = f ( x , y ) at the point ( x 0 , y 0 , z 0 ) of the surface is z z 0 = f 1 ( x 0 , y 0 )( x x 0 ) + f 2 ( x 0 , y 0 )( y y 0 ) By definition, dz at ( x 0 , y 0 ) is dz = f 1 ( x 0 , y 0 x + f 2 ( x 0 , y 0 y Fig. 3. so that the point Q , Figure 3 , with coordinates ( x 0 + Δ x , y 0 + Δ y , z 0 + dz ) lies on
that plane. If Δ z = f ( x 0 + Δ x , y 0 + Δ y ) − f ( x 0 , y 0 ) then the point N , ( x 0 + Δ x , y 0 + Δ y , z 0 + Δ z ), lies on the surface z = f ( x , y ). Hence MN = Δ z , MQ = dz . That is, | dz | is the length of the ordinate x = x 0 + Δ x , y = y 0 + Δ y cut off between the tangent plane and the plane z = z 0 . It is clear from the defining property of a tangent plane that dz will be nearly equal to Δ z for small values of Δ x and Δ y . Since dz is usually so much more easily computed than Δ z , the former is frequently used in place of the latter in approximate computations. We have hitherto assumed for simplicity that f ( x , y ) C 1 . But if we assume only that f ( x , y ) is differentiable at ( a , b ), the differential df is equally well defined at ( a , b ) by equation ( 1 ). Then equation ( 3 ) of §3.6 becomes Δ z = Δ f = dz + Δ x ρ(Δ x , Δ y ) + Δ x , Δ y ) and this shows, without any appeal to geometry, in what sense dz is nearly equal to Δ z when (Δ x , Δ y ) is near (0, 0). Thus E XAMPLE C. Find approximately how much x 2 + y 3 changes when ( x , y ) changes from (1, 1) to (1.1, .9). Approximate change in ( x 2 + y 3 ) is | 2(.1) + 3(−.1) | = .1 Actual change in ( x 2 + y 3 ) is .061. 8.3 DIRECTIONAL DERIVATIVES We now introduce a natural generalization of partial derivatives. In the definition of f 1 ( x 0 , y 0 ), the numerator of the difference quotient used invoh es the values of f ( x , y ) at two points ( x 0 + Δ x , y 0 ) and ( x 0 , y 0 ). As Δ x approaches zero,
the first point approaches the latter along the line y = y 0 . For f 2 ( x 0 , y 0 ) a point ( x 0 , y 0 + Δ y ) approaches ( x 0 , y 0 ) along the line x = x 0 . We now replace these two special lines by an arbitrary line through ( x 0 , y 0 ). A direction ξ α is defined as the direction of any directed line which makes the angle α with the positive x -axis (positive angles measured in the counterclockwise sense as usual). Thus the line segment directed from the point (0, 0) to the point (−1, −1) has the direction ξ 5π/4 or ξ −3π/4 . Definition 8. The directional derivative of f ( x , y ) in the direction ξ α at ( a , b ) is E XAMPLE D. f ( x , y ) = x 2 − 2 y , a = 1, b = 2, α = 3π/4. At each point ( x , y ) a function has infinitely many directional derivatives so that is a function of the three variables x , y , α . In computing a directional derivative of higher order, the variable α must, of course, be held constant. For example, if then Observe that
Theorem 9. 1. f ( x , y ) C 1 By Theorem 3 we have where 0 < θ 1 < 1, 0 < θ 2 < 1. Now, when Δ s approaches zero, we obtain the desired result.