Understanding Vector Fields on the Sphere S^2: A Student's Guide

[jem05]

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 don't 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

 

[quasar987]

Take any smooth nonvanishing vector field V on R² ($\partial / \partial x$ 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.

 

[jem05]

thank you that was very helpful!