The north and south stereographic charts of the unit sphere overlap away from the poles

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The north and south stereographic charts of the unit sphere overlap away from the poles, with transition map

u' = u/(u^2+v^2)

v' = v/(u^2+v^2)

. At the point of the sphere whose north-chart coordinates are

(u,v) = (1,1)

, a tangent vector has north-chart components

v^u = 2

v^v = -1

. Find the components

v^{u'}

v^{v'}

of the same tangent vector in the south chart.

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