6. This problem addresses Laplace '3 equation V2u = v - Vii = 0 and the functions that satisfy it, called harmonic functions. This is a very important equation in heat flow, uid dynamics1 and electrostatics! (a) (2 points) Show that 7!} = % (U2 3:2) is a stream function for f = (91,95). (b) (2 points) Show that F above has zero divergence and zero curl. Vector elds that are both conser- vative and source free are called ideal ows, and can be used to model uids that are irrotational and incompressible. (c) (2 points) Find a potential function 4) for El. (d) (2 points) Show that the potential function and the stream function both satisfy two dimensional Laplace's equation um + uyy = 0. (Hence, they are both harmonic.) (e) (8 points) For a harmonic function u(a:, y, Z), Show that j/Vu-Hd8=0 s Hint: use the divergence theorem