Definition:Regular Surface

Definition

A subset $S \subseteq \R^3$ is a regular surface if and only if for each $p \in S$ there exist:

a neighborhood $V \subseteq \R^3$ of $p$

an open set $U \subseteq \R^2$

a surjective mapping $\mathbf x : U \to V \cap S$, written as:

$\map {\mathbf x} {u, v} := \struct {\map x {u, v}, \map y {u, v}, \map z {u, v} }$

such that:

$(1): \quad \map x {u, v}, \map y {u, v}, \map z {u, v}$ are smooth

$(2): \quad \mathbf x: U \to V \cap S$ is a homeomorphism

$(3): \quad$ For each $q \in U$, the differential $\d_q \mathbf x: \R^2 \to \R^3$ of $\mathbf x$ at $q$ is one-to-one

A regular surface is also known as a surface.

It can also be referred to as a $2$-manifold.