Lab 5: Local Theory of Surfaces

Introduction

In this lab we introduce the local theory of surfaces in three-dimensional space. This subject is the natural extension of the study of the local theory of plane curves and space curves. As we did for plane curves and space curves, we will introduce coordinate representation and vector notation for surfaces. Next, we will define a particular family of plane curves on a surface, the parametric curves, and use these to define the metric coefficients, which help us to characterize classes of surfaces. The metric coefficients provide us with a tool for computing not only lengths of curves but also areas of regions and angles between curves on a surface.