Prove that the rotation of a surface of revolution S about its axis are diffeomorphisms of S.
A rotation in space is given by $\displaystyle y=Ax$, where A is orthogonal with determinant 1.
A rotation of the surface $\displaystyle X(u,v),(u,v)\in D$ is given by $\displaystyle Y(u,v)=AX(u,v),(u,v)\in D$.
Prove that $\displaystyle S=\{X(u,v),(u,v)\in D\}$ and $\displaystyle R=\{Y(u,v),(u,v)\in D\}$ are diffeomorphic,
by proving that the map $\displaystyle \phi:S\rightarrow R, p=X(u,v)\rightarrow q=Y(u,v)$ is differentiable
with full rank.