Parametrised surface

Definition

Let $D \subseteq \mathbb{R}^2$ be a set that is the closure of an open set. A parametrised surface is a map $r \colon D \to \mathbb{R}^3$, $r(u, v) = \big( x(u, v), y(u, v), z(u, v) \big)$, of class C^1 on an open set that contains $D$. Its image $S = r(D)$ is the surface that $r$ parametrises. The parametrisation is regular at $(u, v)$ if the partial derivatives $r_u = \partial r / \partial u$ and $r_v = \partial r / \partial v$ are linearly independent, that is, $r_u \times r_v \ne 0$.

Remarks

The graph of a function $f \colon D \to \mathbb{R}$ is parametrised by $r(x, y) = (x, y, f(x, y))$. Here $r_x \times r_y = (-f_x, -f_y, 1) \ne 0$, so this parametrisation is regular. The unit sphere is parametrised by $r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi)$ on $[0, \pi] \times [0, 2\pi]$. It is not regular at the poles $\varphi = 0$ and $\varphi = \pi$. The cylinder of radius $R > 0$ about the $z$-axis is parametrised by $r(\theta, z) = (R \cos \theta, R \sin \theta, z)$. One surface has many parametrisations. Areas and surface integrals do not depend on the choice.

Explanations

L1 human

A parametrised surface is to a surface what a parametrised curve is to a curve, with two parameters instead of one. Think of $(u, v)$ as coordinates on a flat sheet, and of $r$ as a map that bends the sheet into space.