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
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
In other words, we have just shown that the Euclidean distance has been scaled by . Therefore, the standard metric on induced from fulfills
Hence the standard volume element on satisfies
Thus the Jacobian of the stereographic projection at the point is the coefficient .