Regular Surfaces

Differential

Let \newcommand{\R}{\mathbb{R}}f\colon U\subset\R^{n}\to \R^{m} be a differentiable map. To each $p\in U$, we associate a linear map $\text{dd}{f}_p:{\mathbb{R}}^n\to {\mathbb{R}}^m$ called the differential of $f$ at $p$ and is defined as follows: For all $w\in {\mathbb{R}}^n$, and $\gamma :[-\epsilon ,\epsilon ]\to U$ be a differentiable curve such that $\gamma (0)=p$, ${\gamma }^{\prime }(0)=w$, then $\text{dd}{f}_p(w):=(f\circ \gamma {)}^{\prime }(0)$

Regular Surface

A subset $S\subset {\mathbb{R}}^3$ is a regular surface if

• for each $p\in S$, there exists a neighbourhood $V\subset {\mathbb{R}}^3$ of $p$ and a smooth homeomorphism $\mathbf{x}:U\to V\cap S$ such that $U\subset {\mathbb{R}}^2$ is an open set.
• For each $q\in U$, the differential ${\text{dd}}_q\mathbf{x}:{\mathbb{R}}^2\to {\mathbb{R}}^3$ is injective.

Such mapping $\mathbf{x}$ is called a local coordinate system or a chart of $p$.