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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}}

Revision as of 10:01, 3 March 2014

Have you been wondering "how do I accelerate my computer" lately? Well possibilities are in the event you are reading this article; then you are experiencing one of numerous computer issues that thousands of individuals discover which they face on a regular basis.

Google Chrome crashes on Windows 7 by the corrupted cache contents plus problems with all the stored browsing information. Delete the browsing data plus well-defined the contents of the cache to solve this issue.

The error is basically a outcome of issue with Windows Installer package. The Windows Installer is a tool employed to install, uninstall plus repair the many programs on a computer. Let you discuss a limited aspects that helped a great deal of people whom facing the similar matter.

If that refuses to work you need to try plus repair the issue with a 'registry cleaner'. What occurs on countless computers is that their registry database becomes damaged plus unable to show your computer where the DLL files which it needs are. Every Windows PC has a central 'registry' database that stores information regarding all of the DLL files on the computer.

In a word, to accelerate windows XP, Vista business, it's quite significant to disable some business items and clean and optimize the registry. We can follow the steps above to disable unnecessary programs. To optimize the registry, I suggest you utilize a tuneup utilities 2014 software. Because it is actually especially dangerous for we to edit the registry by oneself.

Files with all the DOC extension are moreover susceptible to viruses, nevertheless this is solved by superior antivirus programs. Another problem is that .doc files may be corrupted, unreadable or damaged due to spyware, adware, and malware. These cases may prevent consumers from properly opening DOC files. This really is when powerful registry products become beneficial.

You want a choice to automatically delete unwanted registry keys. This usually save you hours of laborious checking through your registry keys. Automatic deletion is a key element whenever you compare registry cleaners.

Often the number one means is to read ratings on them and if many users remark about its efficiency, it's likely to be function. The right piece is that there are numerous top registry products that work; you only have to take your pick.

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