Can we define a 'rectangular coordinate' on a curved surface? We use rectangular coordinate on a flat plane, so can we use it in a curved surface, like the axis is somehow bent?
If yes, is there any application? Also, can we generalize this to higher dimension?

Preliminary
\begin{align*}
  \mathbf{x}(u,v)
  &= \begin{pmatrix} x(u,v) \\ y(u,v) \\ z(u,v) \end{pmatrix} \\
  \mathbf{x}_u &= \frac{\partial \mathbf{x}}{\partial u} \\
  \mathbf{x}_v &= \frac{\partial \mathbf{x}}{\partial v} \\
  \mathbf{N} &=
  \frac{\mathbf{x}_u \times \mathbf{x}_v}{|\mathbf{x}_u \times \mathbf{x}_v|}
  \tag{unit normal vector} \\
\end{align*}
First Fundamental Form
$$\mathbb{I}=
\begin{pmatrix} E & F \\ F & G \end{pmatrix}= 
\begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix}
\begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} $$
Second Fundamental Form
$$\mathbb{II}=
 \begin{pmatrix} e & f \\ f & g \end{pmatrix}=
-\begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix}
 \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix}$$

For surface $\mathbf{x}(u,v)$ such that $F=0$ (or simply $\mathbf{x}_u \cdot \mathbf{x}_v=0$), we call it orthogonal patch which means the family of curves for constant $u$ is perpendicular/orthogonal/normal to the family of curves for constant $v$.
In particular, $F=0$ and $f=0$, then $\mathbf{x}(u,v)$ is known as principal patch where $\mathbf{x}_u$ and $\mathbf{x}_v$ are respectively along the directions of the two principal curvatures.

