4.20 Consider the change of variable to polar coordinates: x = r cos 0, y = r sin 0. Use the chain rule to obtain u, and up in terms of ux and uy and hence show that x = cos 00 1 r 1 sin dysin 08, += cos = r Hence, by considering 1 u = (cos 08- sin ) (cos , r - sin 08)u, r or otherwise, show that 1 1 Uxx + Uyy = Urr +-ur + 2 u00. r For what functions f and what values of n is u = r" f (0) a solution of Laplace's equation? Consider also u = = f(0) Inr.