In this problem, you will use the result of Exercise 31 to determine an expression for

Chapter 7, Problem 32

(choose chapter or problem)

In this problem, you will use the result of Exercise 31 to determine an expression for curl F in cylindrical coordinates. Begin by writing F = Fr er + F e + Fz ez. (a) Find the ez-component of curl F by considering the planar path shown in Figure 7.48. The pairs of opposite edges of the approximately rectangular x y z ez C r r Figure 7.48 The path C of Exercise 32(a). path C correspond to the values r r/2 and r + r/2, and /2 and + /2 (all with constant z-coordinate). Note that the area enclosed by C is approximately r r. Approximate the line integral " C F ds by using the fact that, for small and r, each edge of C is roughly straight. Show that ez curl F = 1 r Fr + 1 r r (r F ). (b) Use the path in Figure 7.49 to show that er curl F = 1 r Fz F z . z y x er z r Figure 7.49 The path C of Exercise 32(b). (c) Use the path in Figure 7.50 to show that e curl F = Fr z Fz r . Combine this with the results of parts (a) and (b) to obtain curl F = 1 r er re ez /r / /z Fr r F Fz . (See Theorem 4.5 of Chapter 3.)