This is a problem from real analysis and is attached as a picture. It involves the inverse function theorem or the implicit function theorem. This problem involves proofs.

2. Let F: R2 > R2 be a mapping dened by F(:1:, y) : (u, v) Where u : u($ y) : :13cos(y) (ray) = ycos(a:). Note that F(7r/3,7r/3) : (7r/6,7r/6). (i) Show that there exist neighborhoods U of (7T/3, 77/3), V of (7r/6, 7r/6), and a differentiable function G: V > U such that F restricted to U is one-toone, F(U) : V and G(F(:13,y)): (33,:g) for every (any) 6 U. (ii) Let U, V and G be as in part (i), and write G(u,v) : (mg), with :1: : $(u,v),y : y(u,v). Find 8 8 8:(7r/6,7T/6) and y(7T/6a7T/6)