Smoothness of Parametric Surfaces

Definition: Smoothness of Parametric Surfaces

Let $s:\mathcal{D}\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^3$ be a parametric surface.

We say that $s$ is smooth at $\mathbf{a}$ iff it is differentiable there and its normal vector at $\mathbf{a}$ is nonzero.

If $s$ is smooth at every $\mathbf{x}$ in some $S\subseteq {\mathbb{R}}^2$, then we say that $s$ is smooth on $S$. If $S=\mathcal{D}$, then we just say that $s$ is smooth.

Definition: Piecewise Smoothness

A piecewise smooth parametric surface is a function $\varphi :\mathcal{D}\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^3$ such that $\mathcal{D}$ can be represented as the union of finitely many disjoint sets ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ and $\varphi$ can be represented as

$$\varphi (\mathbf{x})=\{\begin{array}{l}{s}_1(\mathbf{x})\text{ if }\mathbf{x}\in {\mathcal{D}}_1\\ ⋮\\ s_k(\mathbf{x})\text{ if }\mathbf{x}\in {\mathcal{D}}_k\end{array}$$

where $s_1:{\mathcal{D}}_1\to {\mathbb{R}}^3,\dots ,s_k:{\mathcal{D}}_k\to {\mathbb{R}}^3$ are parametric surfaces which are smooth except at possibly finitely many places.