## Regular surfaces

A regular surface is defined using the following pieces:

• a subset of (the surface),
• an arbitrary point ,
• a neighbourhood of ,
• a subset , and
• a mapping from onto .
The subset is called a regular surface if
1. is differentiable, that is, its component functions have continuous paritial derivatives of all orders,
2. is a homeomorphism, that is, it is one-to-one and has a continuous inverse , and
3. the Jacobian of is full rank.

The first condition simply guarantees that we can find derivative-related quantities on the surface such as tangent planes. The second condition forbids problematic features such as self-intersections, since it is again impossible to define tangent planes at these features. The existence and continuity of the inverse allows us to show that the various parameterizations available at a point are equivalent and not special. The third condition is called the regularity condition and ensures that coordinate lines in do not collapse and become colinear in .(TODO) The mapping is called the parameterization of the surface.