The second picture is the reference to exercise 4.1.2.

1. Fix a, b, c > 0. Consider the ellipsoid E = {(x, y, 2) E R3 + + (= 1}. (a) Prove that E is a smooth surface. (Hint: Adapt the surface patches for the sphere which were given in exercise 4.1.2.) (b) Find one of the (non-trivial) transition maps between allowable patches in your atlas. (c) Find the tangent plane Tp E for p = (0, b V/2'1.1.2 Define surface patches of : U -> R3 for S? by solving the equation x2 + y? + 22 = 1 for x in terms of y and z: of (u, v) = (+V/1-u2 - v2, u, v), defined on the open set U = {(u, v) E R? | u2 + v2