Differential Geometry: Lecture 13 part 4: diffeomorphism of surfaces

here we sketch the proof of the inverse function theorem for surfaces as it lifts from the usual theorem on Rn. Then several examples of surfaces which are diffeomorphic are given following O'neill as usual.

[federicopagano]

I dont understand example 3. Isn't a necessary condition to be a diffeomorphism F:M----N that dim (M) =dim (N) how does it suppose to have an inverse If it's jacobian matrix is not squared