smooth map (Rev #22, changes) in nLab

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Contents

Idea

A function which is differentiable function to arbitrary order is called a smooth function.

Between Cartesian spaces

A function on (some open subset of) a cartesian space ℝ n\mathbb{R}^n with values in the real line ℝ\mathbb{R} is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if A⊆ℝ nA \subseteq \mathbb{R}^n is any subset, a function f:A→ℝf: 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:ℝ→ℝf : \mathbb{R} \to \mathbb{R} is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

For A⊆ℝ nA \subseteq \mathbb{R}^n, a smooth map ϕ:A→ℝ m\phi: A \to \mathbb{R}^m is a function such that π∘ϕ\pi \circ \phi is a smooth function for every linear functional π:ℝ m→ℝ\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).

Between smooth manifolds

A topological manifold whose transition functions are smooth maps is a smooth manifold. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of atlases, hence which is a kk-fold differentiable function (see there for more details), for all kk The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.

Between generalized smooth spaces

There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.

For details see for example at smooth set.

Properties

Basic facts about smooth functions are

• the Hadamard lemma

• Borel's theorem

• the Tietze extension theorem

• the Steenrod-Wockel approximation theorem

• embedding of smooth manifolds into formal duals of R-algebras

• derivations of smooth functions are vector fields

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.

• differentiable function

• smooth function with rapidly decreasing derivatives

• smooth homotopy, smooth isotopy

• nonlinear functional, linear functional

Examples of sequences of local structures

References

Early An early account, in the context ofCohomotopy, cobordism theory and the Pontryagin-Thom construction:

• Lev Pontrjagin, Chapter I of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)