PLAN 343 Lecture : geog 343 jan 3.pdf

Let psi : R2 rightarrow R2 be defined by (u, v) = psi(x, y) = (x2 - y2, x2 + y2). Note that the inverse image under psi of the line u = a > 0 is a hyperbola, and the inverse image under psi of the line v = c > 0 is a circle. Show that psi is not injective on R2, but its restriction to Q = {(x, y) : x > 0, y > 0} is an injective map onto {(u, v) : v > |u|}. Let phi be the inverse of the restriction psi|Q and show that if 0