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Vector field

  1. Mar 23, 2010 #1
    i need a vector field on S^2 that vanishes at 1 point. there was a thread like this here, but the answer was ((v1/1+x^2),(v2/1+y^2)) and i really dont see how this vanishes at a point although i do get it intuitively.
    My professor hinted that i should take a non zero vector fiels in S^2 x R^2 pull it back by streographic projection via the north pole, then represent it by the south pole chart.
    ca someone help me understand this.
    thank you
     
  2. jcsd
  3. Mar 24, 2010 #2

    quasar987

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    Take any smooth nonvanishing vector field V on R² ([itex]\partial / \partial x[/itex] for instance). Because streographic projection via the north pole is a diffeomorphism, the pushfoward of V by the streographic projection via the north pole is a nonvanishing smooth vector field on S²\{south pole}. Now write that vector field in terms of the basis induced by stereographic projection via the south pole and notice/show that extending your vector field to all of S² by setting it equal to 0 at the south pole gives a smooth vector field on S² vanishing at only one point.
     
  4. Mar 25, 2010 #3
    thank you that was very helpful!
     
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