Global analysis. Exercises 1
Find the Jacoby matrices and the Jacobians (when exist) of the following maps and
say which of these maps are immersions, submersions and diffeomorphisms on their
images.
1) x = 3 cos t, y = 4 sin t, (, t)  (0, 1) × (0, 2).
2) x = cos u cos v, y = sin u cos v, z = sin v, (u, v)  (0, 2) × (-
2
, 
2
).
3) x =

 cos , y =

 sin , z = , (, )  (0, +) × (0, 2).
4) x = 2u
1+u2+v2 , y = 2v
1+u2+v2 , (u, v)  R × R.
1