Curves and Regular Surfaces in ℝ 3
Stephen C. Newman · 2019
This chapter takes a different approach to the problem that can be loosely described as follows: a “smooth surface” is defined to be a topological subspace of R3 that can be covered in a piecewise fashion by a collection of parametrized surfaces in such a way that the pieces “fit together nicely”. By definition, a regular surface is a patchwork of images of parametrized surfaces. The chapter shows that the existence of charts on regular surfaces makes it possible to answer questions about extended smoothness of maps on regular surfaces using methods developed for Euclidean smoothness. Having defined a regular surface and established some of its basic properties, the chapter also presents a rigorous definition of “tangent plane”. It defines four types of regular surfaces: open sets in regular surfaces, graphs of functions, surfaces of revolution, and level sets of functions.