### Differential geometry notes 7

**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 :

$latex = \Large \begin{pmatrix}

\frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\

\frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \\

\frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2}

\end{pmatrix} $

A representation satisfying these assumptions is called an **allowable representation**.

Partial derivatives will be denoted with the following notation:

and .

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 :

$latex = \Large \frac{\partial (u^1,u^2)}{\partial (\bar{u}^1,\bar{u}^2)} = \begin{vmatrix}

\frac{\partial u^1}{\partial \bar{u}^1} & \frac{\partial u^1}{\partial \bar{u}^2} \\

\frac{\partial u^2}{\partial \bar{u}^1} & \frac{\partial u^2}{\partial \bar{u}^2}

\end{vmatrix} $

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.