Someone argues: "For stereographic projection from the north pole,

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

Someone argues: "For stereographic projection from the north pole, (u,v)=(x,y)/(1−z)(u,v) = (x,y)/(1-z), and on the unit sphere x2+y2=1−z2≤1x^2+y^2 = 1-z^2 \le 1. So u2+v2=x2+y2(1−z)2≤1(1−z)2≤1,u^2+v^2 = \frac{x^2+y^2}{(1-z)^2} \le \frac{1}{(1-z)^2} \le 1, and the whole image of the sphere lies inside the closed unit disc u2+v2≤1u^2+v^2 \le 1." Find the error in this argument, and state what u2+v2u^2+v^2 actually equals in terms of zz.

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