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. (a) First, let's review properties of conservative vector fields. Let F = (f,g) = Vo = (62:$4) (a) (3 points) Show that the curl of F, 12 - , 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 . di' = 0(B) - $(A) (c) Now, let's consider source free vector fields and stream functions which we usually denote by : (a) (4 points) A vector field F = (f,g) is source free if its divergence, dr + dy; de, is zero. Compare this to what you did in la, noting that you can rearrange df dg - 0 into df dg dx dy dx dy For source free vector fields, you can find a stream function y so that appropriate partial derivatives of t give you f and g. In order to satisfy the condition that divergence of F is zero, which partial derivative of t gives f? Which gives g? Hint: compare this to o for conservative vector fields. (b) (3 points) Use Green's theorem to show that the flux F . nds = 0 on all closed curves C. Source free fields also have the nice property that their flux integrals are path independent! F . aids = (B) - 4(A) Page 2 (c) (1 point) Consider the vector field F = (y',x2). Show that F has zero divergence. (d) (1 point) Is F conservative? (e) (4 points) Integrate to find a stream function