[Related concepts: Inverse mapping theorem, transformations and coordinate systems] 1) For each of the following transformations (u,v) = f(x,y), (i) compute det Df, (ii) find formulas for the local inverses of f when they exist. a) u=x^2, v=y/x b) u=(e^x) cos y, v=(e^x) sin y I got part (i) easily, but I don't understand part (iii) at all. Here are the answers from the solutions manual: For 1a)(iii), the answer is f^-1 (u,v) = (sqrt u, v (sqrt u) ) For 1b)(iii), the answer is x= ln(u^2 + v^2) / 2 y is given up to multiples of 2pi by arctan(v/u) when u>0, pi/2 - arctan(u/v) when v>0, pi + arctan(v/u) when u<0, 3pi/2 - arctan(u/v) when v<0 But they didn't show any of the steps, nor do they show me how they arrive at the answers. Can someone please explain this part? How exactly can I find the formula for the local inverse? I am terribly confused... Thanks a million!