Regular surfaces

A regular surface is defined using the following pieces:

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.

Copyright © 2005 Adrian Secord. All rights reserved.