Let S2 = {(x, y, z) 2 R3 | x2 + y2 + z2 = 1}, and R2 = {(x, y, z) 2 R3 | z = 0}.

(a) If (s, t, 0) 2 R2, the line through (s, t, 0) and (0, 0, 1) intersects S2 at a point other

than (0, 0, 1). Denote this point by ~x(s, t). Compute ~x(s, t) and show that it is a coordinate

patch.

(b) Replace the point (0, 0, 1) in part (a) with the point (0, 0,−1), and define ~y(u, v) in

the same way. In other words, ~y(u, v) is the point other than (0, 0,−1) at which the line

between (u, v, 0) and (0, 0,−1) intersects S2. Show that ~y(u, v) is a coordinate patch.

(c) Prove that ~x and ~y are coordinate charts for S2, by defining an appropriate coordinate

transformation.