Editing Differential (mathematics) (section)

=== Basic notions ===
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* 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, differentials (e.g. ''dx'', ''dy'', ''dt'', etc.) are interpreted as [[infinitesimal]]s.  There are several methods of defining infinitesimals rigorously, but it is sufficient to say that an infinitesimal number is smaller in absolute value than any positive real number, just as an infinitely large number 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 [[Stieltjes integral#Definition|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.