Answer to Question #139063 in Differential Geometry | Topology for anjali

Question #139063
Show that the Mercator projection
σ(u ,v) = (sech u cos v, sech u sin v ,tanh u)
is a regular surface patch of the unit sphere
1
Expert's answer
2020-10-19T18:16:59-0400

Jacobian=  =  . Now if we look at the minor deleting the first row we get  Now  , hence minor vanishes if  Now taking the minor after deleting 2nd row gives,  This shows  But both  cannot be zero hence one of the two minors never vanish showing the Jacobian has rank two. Also, all points lie on the sphere since 

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