Regular surface

Builds on Parametrised surface Smooth function Homeomorphism Open set Jacobian matrix

Definition

Let $S \subseteq \R^3$ be a subset. The set $S$ is a regular surface if each $p \in S$ has the following property. There are an open set $V \subseteq \R^3$ containing $p$ , an open set $U \subseteq \R^2$ , and a map $x \colon U \to V \cap S$ such that:

The map $x$ is smooth as a map from $U$ to $\R^3$ .
The map $x$ is a homeomorphism from $U$ onto $V \cap S$ , with the subspace topology on $V \cap S$ .
For each $q \in U$ , the Jacobian matrix of $x$ at $q$ has rank $2$ , that is, $x_u(q)$ and $x_v(q)$ are linearly independent.

Such a map $x$ is a local parametrisation of $S$ at $p$ .

Remarks

Example: the unit sphere is a regular surface. At points with $z > 0$ , the map $x(u, v) = (u, v, \sqrt{1 - u^2 - v^2})$ on the open unit disc satisfies the three clauses, and five similar maps cover the rest. More generally, if $f \colon \R^3 \to \R$ is smooth and $\nabla f \ne 0$ on $f^{-1}(c)$ , then the level set $f^{-1}(c)$ , when nonempty, is a regular surface. Non-example: the double cone $x^2 + y^2 = z^2$ fails clause 2 at the origin. Indeed, removing the origin from a small neighbourhood of it in the cone leaves two pieces, while a punctured disc is connected. Each local parametrisation is a regular parametrised surface in the sense of a map, but a regular surface is a set, covered by such maps.

Reading path

To understand this page from the ground up, read these 62 pages in order. The foundations come first.

Element definition
Euclidean space object
Function concept
Limit point of a set definition
Proposition concept
Real numbers object
Set definition
Fréchet derivative definition
Function of several variables concept
Vector-valued function concept
Limit of a function definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Limit of a function of several variables concept
Intersection definition
Partial derivative concept
Union definition
Continuity at a point definition
Derivative definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Subset definition
Union of a family of sets definition
Continuous function definition
Differentiable function definition
Uniqueness quantifier definition
Binary relation definition
Differentiable on an open set definition
Higher partial derivatives definition
Topological space definition
Jacobian matrix definition
Derivative function definition
Iterated derivative definition
Function definition
Higher derivative definition
C^k function characterised by continuous derivatives statement
Codomain definition
Injective function definition
Metric space definition
Preimage of a set definition
Continuously differentiable function definition
Surjective function definition
Parametrised curve definition
Open ball definition
Smooth function definition
Bijective function definition
Parametrised surface definition
Metric topology definition
Inverse function definition
Open set definition
Continuous map of topological spaces definition
Homeomorphism definition
Regular surface definition

Tag	`DRFKZKE229`
URI	`m2://DRFKZKE229`
Kind	definition
Writing	W0 stub
MSC	53A05
Defines	Regular surface
Uses	Parametrised surface, Smooth function, Homeomorphism, Open set, Jacobian matrix
Changed	2 November 2026

Regular surface

Definition

Remarks

Reading path

Further reading

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