Surfaces

Abstract

A surface in R³ can be defined in several ways in terms of functions of several variables. For a surface S, we may have a real-valued function f : R³ → R, such that S = (x,y,z) | f (x,y,z) = 0. Alternatively, we may have a vector-valued function p : Q , R² → R³, such that S = p(u, v) | (u, v) ∈ Q , R². The component functions of p are identified separately as p₁(u, v), p₂(u, v), and p₃(u, v). We thus have the surface S defined parametrically in terms of three separate coordinate functions of two independent parameters, u and v, as S = (P₁(u, v), p₂(u, v), p₃(u, v) | (u, v) ∈ Q , R², where the coordinate functions p₁ p₂, and p₃ map from Q , ∈² into ∈. The variables u and u are the coordinates of a parameter point in Q , R,², and we can imagine the surface S is formed by distorting the region Q. The variables u and v can also be considered to be coordinate values of a point on the surface S located with respect to a curvilinear coordinate grid inscribed on the surface. This coordinate grid is composed of the space curves p(u,v) with u or u fixed and the remaining variable v or u changing in order to trace out a grid curve on S. When S is a so-called single-valued surface, then there is a real-valued function z, such that S = (x, y, z(x, y)) | (x, y) ∈ U , R², where z : U , R² → R. By single-valued, we mean that a line parallel to the z-axis intersects the surface at most once. This latter case is a trivial form of parametric representation with p₁(u,v) = u, p₂(u,v) = v, and p₃(u, v) = z(u, v). Generally, we shall suppose that all the partial derivatives of the functions we consider exist, and that all mixed partials are invariant with respect to the order of differentiation.