Simple Surfaces, Coordinate Mapping
Definition 1 1
Let’s consider subsets $U \subset \mathbb{R}^{2}$ of a $2$-dimensional Euclidean space with coordinates $u_{1}$, $u_{2}$ to be open sets. If there exists a $C^{k}$ injective function $\mathbf{x} : U \to \mathbb{R}^{3}$ that satisfies the following for all $p \in U$, it is called a Simple Surface.
$$ {{ \partial \mathbf{x} } \over { \partial u_{1} }} (p) \times {{ \partial \mathbf{x} } \over { \partial u_{2} }} (p) \ne \mathbf{0} $$
Explanation
In the definition, the open set $U$ is drawn from the $2$-dimensional space, and it’s mapped into the $3$-dimensional space without overlapping parts (since it’s injective), whether it’s flat or curved. In that sense, a simple surface can be imagined as smoothly connecting flat pieces of $2$ dimensions within the $3$-dimensional space. It’s best to grasp the geometric definition of this surface as a function, but don’t worry if it doesn’t come to mind right away; just spend time getting familiar with it.
The reason for defining a surface as a mapping from 2-dimensional space to 3-dimensional space is to treat the surface as if it’s locally like a plane. For example, although the Earth is close in shape to a sphere, we experience its surface as if it’s a 2-dimensional plane from above. $U$ can be likened to a world map, and $\mathbf{x}(U)$ to a globe.
Meanwhile, the mathematical condition given in the definition is similar to the condition a regular curve must meet, as in $\displaystyle {{ d \mathbf{x} } \over { d u }} (p) \ne 0$. Intuitively, this means we’re immediately excluding any parts that are pointy or bizarrely twisted. Satisfying $\dfrac{ \partial \mathbf{x} }{ \partial u_{1} } (p) \times \dfrac{\partial \mathbf{x} }{ \partial u_{2} } (p) \ne \mathbf{0}$ means that any directional partial derivative is not singular (not $0$), implying in some sense that we’re considering the geometry using two linearly independent (curve) axes.
If the simple surface is explicitly presented with coordinates and a graph, it’s also called a Monge Patch. For instance, if the simple surface $f$ is $f(x,y) = x^{2} + y^{2}$, its graph is $$ \left\{ \left( x, y , x^{2} + y^{2} \right) : (x,y) \in \mathbb{R}^{2} \right\} $$ and can be referred to as a Monge Patch.
Definition 2 2
Let’s consider subsets $U \subset \mathbb{R}^{2}$ of a $2$-dimensional Euclidean space with coordinates $u_{1}$, $u_{2}$ to be open sets. If the mapping $\mathbf{x} : U \to \mathbb{R}^{3}$ is bijective and regular, then $\mathbf{x}$ is called a coordinate patch.
Explanation 3
For $\mathbf{x} : U \to \mathbb{R}^{3}$ to be regular means that the rank of the Jacobian matrix of $\mathbf{x}$ is the same as $2$. If we say $\mathbf{x}(u,v) = (x_{1}(u,v), x_{2}(u,v), x_{3}(u,v))$, the Jacobian matrix of $\mathbf{x}$ is as follows.
$$ J = \begin{bmatrix} \dfrac{\partial x_{1}}{\partial u} & \dfrac{\partial x_{1}}{\partial v} \\[1em] \dfrac{\partial x_{2}}{\partial u} & \dfrac{\partial x_{2}}{\partial v} \\[1em] \dfrac{\partial x_{3}}{\partial u} & \dfrac{\partial x_{3}}{\partial v} \end{bmatrix} $$
The rank of this matrix being $2$ means that the dimension of its column space is $2$, implying $\mathbf{x}_{u} = \left( \dfrac{\partial x_{1}}{\partial u}, \dfrac{\partial x_{2}}{\partial u}, \dfrac{\partial x_{3}}{\partial u} \right)$ and $\mathbf{x}_{v} = \left( \dfrac{\partial x_{1}}{\partial v}, \dfrac{\partial x_{2}}{\partial v}, \dfrac{\partial x_{3}}{\partial v} \right)$ are linearly independent. Therefore, their cross product is not $\mathbf{0}$.
$$ \mathbf{x}_{u} \times \mathbf{x}_{v} \ne \mathbf{0} $$
Thus, we can see that the two definitions are equivalent.