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:
 * $\paren 1 : \map x {u,v}, \map y {u,v}, \map z {u,v}$ are smooth
 * $\paren 2 : \mathbf x : U \to V \cap S$ is a homeomorphism
 * $\paren 3 :$ For each $q \in U$, the differential $h \mapsto \map {\d \mathbf x} {q; h}$ of $\mathbf x$ is a one-to-one mapping $\R^2 \to \R^3$.