synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A function which is differentiable function to arbitrary order is called a smooth function.
Let be the real numbers. A function is smooth if it comes with a sequence of functions and a sequence of functions in the positive rational numbers, such that
for every real number ,
for every natural number , for every positive rational number , for every real number such that , and for every real number ,
Unwrapping the recursive definition above, a function is smooth if it comes with a sequence of functions and a sequence of functions in the positive rational numbers, such that
for every real number ,
for every natural number , for every positive rational number , for every real number such that , and for every real number ,
Given a predicate on the real numbers , let denote the set of all elements in for which holds. A partial function is equivalently a function for any such predicate and set .
A function is smooth at a subset with injection if it has a function with for all , such that for all Archimedean ordered Artinian local -algebras with ring homomorphism and nilradical , natural numbers , and purely infinitesimal elements such that
Equivalently, let denote the ring of univariate formal power series on . is a Artinian local -algebra with homomorphism . A function is smooth at a subset with injection if it has a function with for all , such that for all natural numbers
A function is smooth at an element if it is smooth at the singleton subset , and a function is smooth if it is smooth at the improper subset of .
A function on (some open subset of) a cartesian space with values in the real line is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if is any subset, a function is defined to be smooth if it has a smooth extension to an open subset containing .
By coinduction: A function is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.
For , a smooth map is a function such that is a smooth function for every linear functional . (In the case of finite-dimensional codomains as here, it suffices to take the to range over the 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 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 -fold differentiable function (see there for more details), for all The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps between them.
There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.
For details see for example at smooth set.
Basic facts about smooth functions are
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 local structures
An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:
Last revised on October 5, 2023 at 04:38:52. See the history of this page for a list of all contributions to it.