Definition:Regular Surface

Definition
A subset $S \subseteq \R^3$ is a regular surface 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