5. In this problem, you'll look at conservative vector fields and potential functions and compare them to source- free vector fields and stream functions. First, let's review properties of conservative vector fields. Let F = (f, g) = Vo = (4x, dy) (a) (3 points) Show that the curl of F, do _ d, is zero. (b) (3 points) Use Green's Theorem to show that the circulation OF . dr = 0 on all closed curves C. Also Recall that conservative vector fields are path independent: F . dr = $(B) - $(A) Now, let's consider source free vector fields and stream functions which we usually denote by : C (4 points) A vector field F = (f, g) is source free if its divergence, d + do, is zero. Compare this to what you did in 5a, noting that you can rearrange df dg = 0 df dg into dx dy dx dy For source free vector fields, you can find a stream function v so that appropriate partial derivatives of y give you f and g. In order to satisfy the condition that divergence of F is zero, which partial derivative of y gives f? Which gives g? Hint: compare this to o for conservative vector fields. d)(3 points) Use Green's theorem to show that the flux OF . nds = 0 on all closed curves C. Source free fields also have the nice property that their flux integrals are path independent! F . n ds = 1(B) - 4(A)