Problem 5 (polar coordinates; Haberman 1.5.3-9) This problem is about differential operators (and modeling heat diffusion in particular) in two dimensions n = 2. I involves the polar coordinates map v : [0, co) x R -> R2 given by (r, 0) = (rcos 0, r sin (). 2 (a) Discuss the invertibility and inverse of the polar coordinates map. (b) Given a temperature function u = u(r, y, t) in spatial rectangular coordinates satisfying 24 = KAu, assume the value of u is constant along each circle {(x, y) : 1 + y? =r?}. Use the chain rule to find the partial differential equation satisfied by the correspond- ing function w = w(r, t) = u(r, y, t) in polar coordinates given by w(r, t) = u((r, 0), t) = u(r cos 0, r sin 0, t). (1) Hint(s): Applly the chain rule to (1) to compute w,, we, w,, and was. Then consider wry + Woo/T2. (From the expressions for w, and we, you can express the first order derivatives of u in terms of first order derivatives of 0. (c) Use the equation you found in the previous part to determine the equilibrium temperature distribution in the annular/ring region R = {(I,y) : "