---
tag: 05788J51WZ
kind: definition
name: Parametrised surface
msc: [53A05]
defines: 0ZG9QF40HE
uses: [05AXG3QRKR, 0TCBZMEYGN, 0JF1XXGWB4]
sources:
  - url: https://en.wikipedia.org/wiki/Parametric_surface
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Parametric_surface
    locator: Examples
  - url: https://en.wikipedia.org/wiki/Parametric_surface
    locator: Tangent plane and normal vector
---

## 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|0TCBZMEYGN]] 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|05AXG3QRKR]] $r_u = \partial r / \partial u$ 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|0JF1XXGWB4]] 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.