Surfaces

Regular surfaces

defintion of regular surfaces

A subset \(S\subset \mathbb{E}^3\) is a regular surface if for each \(p\in S\), there exsits a neighborhood \(V\) in \(\mathbb{E}^3\) and a map \(\pmb{x}: U\rightarrow V\cap S\) of an open set \(U\in \mathbb{E}^2\) onto \(V\cap S\subset \mathbb{E}^3\) such that

(i) \(\pmb{x}\) is differentiable, i.e.

\[ \pmb{x}(u,v)=(x(u,v), y(u,v), z(u,v)),\quad (u,v)\in U \]

whose component functions have continuous partial derivatives of all orders in \(U\).

(ii) \(\pmb{x}\) is a homeomorphism.

(iii) regularity condition. For each \(q\in U\), the differential \(d\pmb{x}_q: \mathbb{E}^2\rightarrow \mathbb{E}^3\) is one-to-one.