Let n(x, y,z) be a unit normal to a surface S. The directional derivative of a

Chapter 7, Problem 25

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Let n(x, y,z) be a unit normal to a surface S. The directional derivative of a differentiable function f (x, y,z) in the direction of n is called a normal derivative of f , denoted f/n. From Theorem 6.2 of Chapter 2, we have f n = f n. (a) Let S denote the portion of the sphere x 2 + y2 + z2 = a2 in the first octant (i.e., where x 0, y 0, z 0), oriented by the unit normal that points away from the origin. Let f (x, y,z) = ln (x 2 + y2 + z2). Evaluate S f n d S. (b) Let D denote the piece of the solid ball x 2 + y2 + z2 a2 in the first octant; that is, D = {(x, y,z) | x 2 + y2 + z2 a2 , x 0, y 0, z 0}. Compute D ( f ) dV, where f is as in part (a). (c) Apply Gausss theorem to the integral in part (b), and reconcile your result with your answer in part (a).