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Math Help - Differential Geometry HW HELP!!!!

  1. #1
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    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!!!
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  2. #2
    Junior Member
    Joined
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    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.
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