Advanced Calculus fi..

Chapter 2 Differential ~alculus of Functiuns d Several Variables 1352. Evaluate aF/an, where n is the unit outer normal to the surface given, at a general point(x, y, z) of the surface given:a) F = x2 - y2, surface x2 + y2 + z2 = 4;b) F = xyz, surface x2 + 2y2 + 4z2 = 8.3. Prove that if u = f (x, y) and v = g(x, y) are functions such that--au av au ava- - - --ax ay' ay ax'~ ,-.-\then . 1c-Vcx~ = Va+n/~v- 9*. .r,for every angle a.4. Show that if u = f (x, y), then the directional derivatives of u along the line 0 = cdwand circles r = const (polar coordinates) are given respectively by5. Show that under the hypotheses of Problem 3 one hasau 1 av- - -- -- ---I a~ a~ar r ae' r ae ar'[Hint: Use Problem 4.16. Let arc length s be measured from the point (2,O) of the circle x2 + y2 = 4, starting in thedirection of increasing y . If u = x2 - y2, evaluate dulds on this circle. Check the resultby using both the directional derivative and the explicit expression for u in terms of s. Atwhat point of the circle does u have its smallest value?7. Under the hypotheses of Problem 3, show that &/as = av/an along each curve C of thedomain in which u and v are given, for appropriate direction of the normal n.8. Determine the points (x, y) and directions for which the directional derivative of u =3x2 + y2 has its largest value, if (x, y) is restricted to lie on the circle x2 + y2 = 1.9. A function F(x, y, 2) is known to have the following values : F(1, 1, 1) = 1, F(2, 1, 1) = 4,F(2,2, 1) = 8, F(2, 2,2) = 16. Compute approximately the directional derivatives:at the point (I, 1, 1).ViF, V,+jF, Vi+j+kFLet a function z = F(x, y) be given; its two partial derivatives az/ax and az/ay arethemselves functions of x and y:Hence each can be differentiated with respect to x and y; one thus obtains the foursecond partial derivatives: