### Differential geometry notes 7

Parametric representation of a point set M in three-dimensional Euclidean space:

$\mathbf{x}(u^1,u^2) = \big( x_1(u^1,u^2), x_2(u^1,u^2),x_3(u^1,u^2) \big)$

where $u^1, u^2$ are real-valued variables defined in a simply-connected, bounded domain $B$.

We make the following assumptions about the parametric representations:

(1) $\mathbf{x}(u^1,u^2)$ is of class $r \geq 1$ in B. Each point of $M$ corresponds to just one ordered pair $(u^1,u^2)$ in $B$.

(2) The Jacobian is of rank 2 in $B$:

$J$ $= \Large \begin{pmatrix} \frac{\partial x_1}{\partial u_1} & \frac{\partial x_1}{\partial u_2} \\ \frac{\partial x_2}{\partial u_1} & \frac{\partial x_2}{\partial u_2} \\ \frac{\partial x_3}{\partial u_1} & \frac{\partial x_3}{\partial u_2} \end{pmatrix}$

A representation satisfying these assumptions is called an allowable representation.

Partial derivatives will be denoted with the following notation:

$\Large \mathbf{x}_{\alpha} = \frac{\partial \mathbf{x}}{\partial u^{\alpha}}$ and $\Large \mathbf{x}_{\alpha \beta} = \frac{\partial^2 \mathbf{x}}{\partial u^{\alpha} \partial u^{\beta}}$.

If $J$ is of rank $R<2$ for a particular $(u^1,u^2)$, 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 $u^{\alpha} = u^{\alpha}(\bar{u}^1,\bar{u}^2)$ (with $\alpha = 1, 2$), we can obtain a new parametric representation $\mathbf{x}(\bar{u}^1,\bar{u}^2)$ of $M$. The transformation has to satisfy the following assumptions:

(1) The domain $\bar{B}$ of the transformation includes $B$.

(2) $u^{\alpha}$ are of class $r\geq 1$ everywhere in $\bar{B}$ and are one-to-one.

(3) The Jacobian $D$ is different from zero everywhere in $\bar{B}$:

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

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 $S$ in $\mathbb{R}^3$ which can be represented by the allowable representations of an equivalence class is called a portion of a surface. $u^1$ and $u^2$ are called the coordinates on $S$. The curves $u^1 = const$ and $u^2 = const$ are called the coordinate curves of the $u^1 u^2$ coordinate system.

Definition 24.2: A union $U$ of portions of surfaces is called a surface if every two portions $S$, $S^{\prime}$ of $U$ can be joined by finitely many portions $S = S_1, \ldots, S_n = S^{\prime}$ such that the intersection of two subsequent portions $S_i$, $S_{i+1}$ is a portion of a surface.