In this problem, you will use the result of Exercise 27 to find an expression for F in

Chapter 7, Problem 29

(choose chapter or problem)

In this problem, you will use the result of Exercise 27 to find an expression for F in cylindrical coordinates. (See Theorem 4.5 of Chapter 3.) Begin by writing F = Fr er + F e + Fz ez, where Fr(r,,z), F (r,,z), and Fz(r,,z) denote the components of F in the er-, e -, and ez-directions (respectively). Let P have cylindrical coordinates (r,,z). Consider the small cylindrical coordinate cuboid S shown in Figure 7.45. The pairs of opposite faces correspond to values r r/2 and r + r/2; /2 and + /2; z z/2 and z + z/2. Note that the volume of the cuboid is approximately r r z. (a) Approximate S F dS (where S is oriented by outward unit normal) by noting that each face of S is roughly flat with an obvious unit normal vector and that F is approximately constant on each face. (b) Use your answer in part (a) to calculate the divergence in cylindrical coordinates as div F = Fz z + 1 r r (r Fr) + 1 r F . (This agrees with formula (4) of 3.4.) ez er e2 r z r Figure 7.45 The cylindrical coordinate cuboid of Exercise 29.