I want to propose an alternative way to calculate the Jacobian of the stereographic projection . In Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas

and

.

It is well-known that the Jacobian of the stereographic projection is

The way to calculate its Jacobian is to compare the ratio of volumes. First pick two arbitrary points and denote and .

The Euclidean distance between and is