In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
the differential calculus over commutative algebras is a part of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
based on the observation that most concepts known from classical differential
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
can be formulated in purely algebraic terms. Instances of this are:
# The whole topological information of a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is encoded in the algebraic properties of its
-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
of smooth functions
as in the
Banach–Stone theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
In brief, the Banach–Stone theorem allows one to recove ...
.
#
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s over
correspond to projective finitely generated
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
s over
via the
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
which associates to a vector bundle its module of sections.
#
Vector fields on
are naturally identified with
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
s of the algebra
.
# More generally, a
linear differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
of order k, sending sections of a vector bundle
to sections of another bundle
is seen to be an
-linear map
between the associated modules, such that for any
elements
:
where the bracket
is defined as the commutator
Denoting the set of
th order linear differential operators from an
-module
to an
-module
with
we obtain a bi-functor with values in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
-modules. Other natural concepts of calculus such as
jet space
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
s,
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s are then obtained as
representing objects of the functors
and related functors.
Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.
Replacing the real numbers
with any
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
, and the algebra
with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
secondary calculus. Moreover, the theory generalizes naturally to the setting of
graded commutative algebra, allowing for a natural foundation of calculus on
supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
s,
graded manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commuta ...
s and associated concepts like the
Berezin integral
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is n ...
.
See also
*
*
*
References
* J. Nestruev, ''Smooth Manifolds and Observables'', Graduate Texts in Mathematics 220, Springer, 2002.
*
* I. S. Krasil'shchik, "Lectures on Linear Differential Operators over Commutative Algebras". Eprin
DIPS-01/99
* I. S. Krasil'shchik, A. M. Vinogradov (eds) "Algebraic Aspects of Differential Calculus", ''Acta Appl. Math.'' 49 (1997), Eprints
* I. S. Krasil'shchik, A. M. Verbovetsky, "Homological Methods in Equations of Mathematical Physics", ''Open Ed. and Sciences,'' Opava (Czech Rep.), 1998; Eprin
arXiv:math/9808130v2
* G. Sardanashvily, ''Lectures on Differential Geometry of Modules and Rings'', Lambert Academic Publishing, 2012; Eprin
arXiv:0910.1515 ath-ph137 pages.
* A. M. Vinogradov, "The Logic Algebra for the Theory of Linear Differential Operators", ''Dokl. Akad. Nauk SSSR'', 295(5) (1972) 1025-1028; English transl. in ''Soviet Math. Dokl.'' 13(4) (1972), 1058-1062.
* A. M. Vinogradov, "Cohomological Analysis of Partial Differential Equations and Secondary Calculus", AMS, series: Translations of Mathematical Monograph, 204, 2001.
* A. M. Vinogradov, "Some new homological systems associated with differential calculus over commutative algebras" (Russian), Uspechi Mat.Nauk, 1979, 34 (6), 145-150;English transl. in ''Russian Math. Surveys'', 34(6) (1979), 250-255.
{{Manifolds
Commutative algebra
Differential calculus