. In this problem, let f (x, y,z) be a scalar-valued function of class C1, and let D be

Chapter 7, Problem 21

(choose chapter or problem)

. In this problem, let f (x, y,z) be a scalar-valued function of class C1, and let D be a region in space to which Gausss theorem applies. Let n = (n1, n2, n3) be the outward unit normal vector to S = D. (a) If a is any constant vector and F = f a, show that F = f a. (b) Use part (a) with a = i to show that S f n1 d S = D f x dV. Also obtain similar results by letting a equal j and k. (c) Define a vector quantity S f dS = S f n d S by S f n d S = S f n1 d S, S f n2 d S, S f n3 d S . With notations and definitions as above, show that S f n d S = D f dV. (Note that the right side is a triple integral of a vector-valued expression, so it is also computed by integrating each scalar component function.)