Parametric representation of a point set M in three-dimensional Euclidean space:
where are real-valued variables defined in a simply-connected, bounded domain .
We make the following assumptions about the parametric representations:
(1) is of class in B. Each point of corresponds to just one ordered pair in .
(2) The Jacobian is of rank 2 in :
A representation satisfying these assumptions is called an allowable representation.
Partial derivatives will be denoted with the following notation:
If is of rank for a particular , the corresponding point is called a singular point with respect to the representation and it is called singular if it is singular with respect to every allowable representation.
By imposing a transformation (with ), we can obtain a new parametric representation of . The transformation has to satisfy the following assumptions:
(1) The domain of the transformation includes .
(2) are of class everywhere in and are one-to-one.
(3) The Jacobian is different from zero everywhere in :
A transformation satisfying these assumptions is called an allowable coordinate transformation. As in the theory of curves, allowable coordinate transformations define equivalence classes of allowable parametric representations.
Definition 24.1: A point set in which can be represented by the allowable representations of an equivalence class is called a portion of a surface. and are called the coordinates on . The curves and are called the coordinate curves of the coordinate system.
Definition 24.2: A union of portions of surfaces is called a surface if every two portions , of can be joined by finitely many portions such that the intersection of two subsequent portions , is a portion of a surface.