Thread: Differential Geometry HW HELP!!!!

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.

Any help with this problem would be much appreciated!!!

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.

a) ~x(s,t)=(0,0,1)+p(s,t,-1) when p=0 we have ~x(s,t)=(0,0,1) when p=1
~x(s,t)=(s,t,0).
b) The same as in a), i.e, ~y(u,v)=(0,0,-1)+q(u,v,1).
c) As far as I can tell you need to find a trsnaformation from ~x to ~y, i.e
~y=(qu,qv,(q-1)) ~x=(qu,qv,(1-q))
T(qu,qv,q-1)=(qu,qv,1-q) this is the appropiate transformation.