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In [[mathematics]], the term '''differential''' has several meanings.
 
==Basic notions==
* In [[calculus]], the [[differential of a function|differential]] represents a change in the [[linearization]] of a [[function (mathematics)|function]].
** The [[total differential]] is its generalization for functions of multiple variables.
* In traditional approaches to calculus, the [[differential (infinitesimal)|differentials]] (e.g. d''x'', d''y'', d''t'' etc...) are interpreted as [[infinitesimal]]s. There are several methods of defining infinitesimals rigorously, but it is sufficient to say that an infinitesimally small number is smaller than any real, positive number, and an infinitely large one is larger than any real number.
* The [[Total derivative|differential]] is another name for the [[Jacobian matrix]] of [[partial derivative]]s of a function from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> (especially when this [[Matrix (mathematics)|matrix]] is viewed as a [[linear map]]).
* More generally, the [[Pushforward (differential)|differential]] or ''[[Pushforward (differential)|pushforward]]'' refers to the derivative of a map between [[smooth manifold]]s and the pushforward operations it defines. The differential is also used to define the dual concept of [[pullback (differential geometry)|pullback]].
* [[Stochastic calculus]] provides a notion of [[stochastic differential]] and an associated calculus for [[stochastic process]]es.
* The [[integrator]] in a [[Stieltjes integral]] is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly as a differential: thus, the [[integration by substitution]] and [[integration by parts]] formulae for Stieltjes integral correspond, respectively, to the [[chain rule]] and [[product rule]] for the differential.
 
==Differential geometry==
The notion of a differential motivates several concepts in [[differential geometry]] (and [[differential topology]]).
*[[Differential form]]s provide a framework which accommodates multiplication and differentiation of differentials.
*The [[exterior derivative]] is a notion of differentiation of differential forms which generalizes the [[Total derivative|differential]] of a function (which is a [[differential 1-form]]).
*[[Pullback (differential geometry)|Pullback]] is, in particular, a geometric name for the [[chain rule]] for composing a map between manifolds with a differential form on the target manifold.
*[[Covariant derivative|Covariant derivatives or differentials]] provide a general notion for differentiating of [[vector field]]s and [[tensor field]]s on a manifold, or, more generally, sections of a [[vector bundle]]: see [[Connection (vector bundle)]]. This ultimately leads to the general concept of a [[Connection (mathematics)|connection]].
 
==Algebraic geometry==
Differentials are also important in [[algebraic geometry]], and there are several important notions.
* [[Abelian differential]]s usually refer to differential one-forms on an [[algebraic curve]] or [[Riemann surface]].
* [[Quadratic differential]]s (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
* [[Kähler differential]]s provide a general notion of differential in algebraic geometry
 
==Other meanings==
The term ''differential'' has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a [[cochain complex]] <math>(C_\bullet, d_\bullet)</math>, the maps (or ''coboundary operators'') ''d<sub>i</sub>'' are often called differentials. Dually, the boundary operators in a chain complex are sometimes called ''codifferentials''.
 
The properties of the differential also motivate the algebraic notions of a ''[[derivation (abstract algebra)|derivation]]'' and a ''[[differential algebra]]''.
 
==External links==
* {{MathWorld|title=Differentials|urlname=Differential}}
 
{{DEFAULTSORT:Differential (Calculus)}}
[[Category:Mathematical terminology]]
[[Category:Differential calculus]]
 
[[bg:Диференциал]]
[[de:Differential (Mathematik)]]
 
{{sia}}

Latest revision as of 13:16, 3 September 2014

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