Differential Geometry of Curves and Surfaces, 2nd Edition

CHAPTER 5

Regular Surfaces

5.1 Parametrized Surfaces

There are many approaches that one can take to introduce surfaces. Some texts immediately build the formalism of differentiable manifolds, some texts first encounter surfaces in ℝ³ as the solution set to an algebraic equation F (x, y, z) = 0 with three variables, and other texts present surfaces as the images of vector functions of two variables. In this section, we introduce the last of these three options and occasionally refer to the connection with the surfaces as solution sets to algebraic equations. Only later will we show why one requires more technical definitions to arrive at a workable definition that matches what one typically means by a “surface.”

We imitate the definition ...