Sphäre/Stereographische Projektion/Kartenwechsel/Diffeomorphismus/Aufgabe

Zeige, dass die Abbildung

\mathbb {R} ^{2}\setminus \{(0,0)\}\longrightarrow \mathbb {R} ^{2}\setminus \{(0,0)\},\,(u,v)\longmapsto \left({\frac {-u}{u^{2}+v^{2}}},\,{\frac {-v}{u^{2}+v^{2}}}\right),

ist.

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