11. (15 points) .1 Sections 2.32.4 Suppose f (z) : 113(3)\

II. (15 points) Sections 2.32.4 Suppose f (z) u(x, y) + iv(x, y) is analytic in a domain in the complex plane. a. b. Show that the family of level curves u(x, y) Cl and v(x, y) = c2 are orthogonal. More precisely, at any point of intersection zo of the two curves, the tangent lines to the two curves are perpendicular, if f' (zo) # 0. (Hint: Compute the gradient vectors of u and v and use the Cauchy-Riemann equations.) Let f (z) 22. Show that at z 0 (f' (0) 0), in order for the two sets of curves have intersection that passing through z = 0, Cl = c2 0 is the only possibility. Explain why the above result fails. (Hint: Don't forget the curves must pass through z = 0 in order to make the condition f '(20) # 0 in the previous part invalid.)