Regular surfaces

A regular surface is defined using the following pieces:

• a subset $S$ of $\mathbb{R}^3$ (the surface),
• an arbitrary point $p \in S$ ,
• a neighbourhood $V \subset \ensuremath{\mathbb{R}^3}$ of $p$ ,
• a subset $U \subset \ensuremath{\mathbb{R}^2}$ , and
• a mapping $x$ from $U$ onto $V \cap S$ .

The subset $S$ is called a regular surface if

1. $x$ is differentiable, that is, its component functions have continuous paritial derivatives of all orders,
2. $x$ is a homeomorphism, that is, it is one-to-one and has a continuous inverse $x^{-1}$ , and
3. the Jacobian of $x$ 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 $U$ do not collapse and become colinear in $S$ .(TODO) The mapping $x$ is called the parameterization of the surface.