nLab
smooth map

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

A function on (some open subset of) a cartesian space n\mathbb{R}^n with values in the real line \mathbb{R} is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if A nA \subseteq \mathbb{R}^n is any subset, a function f:Af: A \to \mathbb{R} is defined to be smooth if it has a smooth extension to an open subset containing AA.

By coinduction: A function f:f : \mathbb{R} \to \mathbb{R} is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

For A nA \subseteq \mathbb{R}^n, a smooth map ϕ:A m\phi: A \to \mathbb{R}^m is a function such that πϕ\pi \circ \phi is a smooth function for every linear functional π: m\pi: \mathbb{R}^m \to \mathbb{R}. (In the case of finite-dimensional codomains as here, it suffices to take the π\pi to range over the mm coordinate projections.)

The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).

A manifold whose transition functions are smooth maps is a smooth manifold. The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.

Yet more generally, the morphisms between generalised smooth spaces are smooth maps.

For functions between manifolds that fall short of full smoothness, see differentiable map.

Properties

Basic facts about smooth functions are

Examples

Every analytic functions (for instance a holomorphic function) is also a smooth function.

A crucial property of smooth functions, however, is that they contain also bump functions.

Examples of sequences of infinitesimal and local structures

first order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on November 10, 2013 09:05:17 by Todd Trimble (67.80.199.186)