Consider the Ellipsoid E:= {(x, y, z) R³ |++= 1). 1. Show that E is a regular surface. 2. Find, using Sterographical projections, two parametrizations X, and X2 such that their images cover E. (You do not need to show that X, and X2 are parametrizations.) (Use the north pole N to find X, and the south pole S to find X2). 3. Show that X, is a parametrization. 4. Compute the change of parameters map h = Xo X2. (Precise also its domain and its range). 5. Check if h is a diffeomorphism. 6. Consider the Elliptic Cylinder C:= {(x, y, z) R³ | += 1). Find a diffeomorphism F from E\ (N, S) onto C. 7. Show that Fo X, is a parametrization for C. 8. Does the images of FoX; and Fo X2 cover C? 9. Interpret, geometrically, the effect of Fo X2.