So far, we've been able, given a specific definition of a surface S Ã R3, we can determine whether or not it is a regular surface. But given a different parametrization of the same surface, we'd like to be able to say that it's a regular surface simply from the fact that it is the same surface. In this section, it will be shown that given any parametrization, if just one is regular, they all are regular.

Change of Parameters.   Let p be a point of a regular surface S, and let x : U Ã R2 Æ S, y : V Ã R2 Æ S be two parametrizations of if S such that p Ã x(U)«y(V) = W. Then the "change of coordinates" h = x-1° y : y-1(W) Æ x-1(W) is a diffeomorphism; that is, h is differentiable and has a differentiable inverse h-1.

Basically, all that this proof is telling us is that, given two parameterizations, there exists a differential map between the two open sets. In other words, we can easily transform from one basis to another, without affecting the surface what so ever. This states that given one parameterization that we know creates a regular surface, any other parametrization can simply be mapped to the one we know works.

Proof. Even though h is composed of diffeomorphisms, that is not sufficient to conclude that h is a diffeomorphism. Yet, to do so is quite complicated. Basically, we need to create a suface which extends above U (as shown in Fig. 1) and show that it has an inverse when mapped back from S.
However, the proof is beyond the scope of this section, the text can provide further explination.

Example   The unit shpere can expressed by the parametriztions (u,v) Æ (sin(u)sin(v),sin(u)cos(v),cos(u)), and (u,v) Æ (sin(u)cos(v),cos(u),sin(u)sin(v)). The sphere has already been proven as a regular surface, so we know that these two parameterizations both map to regular surfaces. In fact, these two parametrizations are called "coverings" of the sphere, because each edge that would be undefined in one is defined in the other. This again verifies the regularity of the sphere.