
\chapter{Differentiable surface}




\begin{definition}[Regular surface]
A \emph{regular surface} is a set $\Sigma \subset \mathbb{R}^3$ such that any $p \in \Sigma$ has a neighborhood $V$ open in $\mathbb{R}^3$ and a differentiable homeomorphism  $\mathbf{x} : U \to V \cap \Sigma$ (where $U$ is an open set of $\mathbb{R}^2$) such that for any $q\in U$, its differential $\mathrm{d}\mathbf{x}_q : \mathbb{R}^2 \to \mathbb{R}^3$ is injective.
\end{definition}


\begin{proposition}
 $\mathrm{d}\mathbf{x}_q : \mathbb{R}^2 \to \mathbb{R}^3$ is injective iff are linearly independent.
\end{proposition}


By the implicit function theorem, level sets are either regular surfaces or empty sets

\begin{proposition}
Given a smooth $\mathbb{R}^2$ function defined on an open set, its graph is a regular surface.
	\[ \{ \begin{bmatrix}u \\ v \\ f(u,v) \end{bmatrix}  : \begin{bmatrix}u \\ v\end{bmatrix} \in \mathbb{R}^2 \}   \]
\end{proposition}


\begin{proposition}
Let $f : U \subset mathbb{R}^3 \to \mathbb{R}$ be a dfferentiable function and $p \in f(U)$ be a regular (non-critical) point, then $f^{-1}(p)$ is a regular surface.
\end{proposition}

\begin{proposition}
Let $S \subset \mathbb{R}^3$ be a regular surface with $p\in S$, then there exists some neighborhood $V$ of $p$ where $V$ is the graph of either of the differentiable functions $f(x,y),f(x,z),f(y,z)$.
\end{proposition}



Imagine considering the derivative on surfaces; we would want this to be irrespective of parametrization and the open sets used.

Let $\Sigma$ be a regular surface containing $p$. Let $x$ and $y$ parametrize different neighborhoods of $p$ on $\Sigma$, and define $x(U) \cap y(V) = W$. Then $\mathbf{x}^{-1} \circ \mathbf{y} : y^{-1}(W) \to x^{-1}(W)$ is a diffeomorphism between the coordinate systems.

\begin{definition}[Differentiable function on a regular surface]
	Let $\Sigma$ be a regular surface. A \emph{differentiable function on $\Sigma$ at $p$} is a function $f : V \subset S \to \mathbb{R}$, where some parametrization $\mathbf{x}$, $\mathbf{f} \circ \mathbf{x}$ is differentiable.
\end{definition}



\begin{definition}[Parametrized surface]
	Let $\Sigma$ be a regular surface. A \emph{differentiable function on $\Sigma$ at $p$} is a function $f : V \subset S \to \mathbb{R}$, where some parametrization $\mathbf{x}$, $\mathbf{f} \circ \mathbf{x}$ is differentiable.
A parametrized surface is the image of a differentiable function $f : U \subseteq \mathbb{R}^2 \to \mathbb{R}^3$ whose differential of any point is injective.
\end{definition}


Any point of a regular parametrized surfaces has a neighborhood that is a regular surface.

