Surfaces in Space

Abstract

In Sections 19.1 and 19.2 we consider some basic notions of a parametrized surface and a regular surface (analogous definitions for curves were studied in Section 5.1). In Section 19.3 we use a number of MAPLE commands to produce surfaces by various methods. In Section 19.4 we calculate and plot tangent planes and normal vectors of a surface. As an application we solve the conditional extremum problems in space (see the two-dimensional case in Section 5.6). In Section 19.5 we use changes in coordinates and linear transformations in space to calculate and plot an osculating paraboloid at the point of a surface. This elementary approach is given only for methodical reasons. In Section 19.6 we consider parametrized and implicitly defined surfaces with singularities.