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In mathematics, the term differential has several meanings.
In calculus, the differential represents a change in the linearization of a function. In traditional approaches to calculus, the differentials (e.g. dx, dy, dt etc...) are interpreted as infinitesimals.
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 differential or pushforward refers to the derivative of a map between smooth manifolds and the pushforward operations it defines.
- The differential is also used to define the dual concept of pullback.
Formally, the differential appearing under the integral behaves exactly as a differential, thus,
for Stieltjes integral
correspond, respectively, to:
- the chain rule and
- the product rule
for the differential.
- Differential forms provide a framework which accommodates multiplication and differentiation of differentials.
- The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form).
- 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 derivatives or differentials provide a general notion for differentiating of vector fields and tensor fields on a manifold, or, more generally, sections of a vector bundle: see Connection . This ultimately leads to the general concept of a connection.
Differentials are also important in algebraic geometry, and there are several important notions.
- Abelian differentials usually refer to differential one-forms on an algebraic curve or Riemann surface.
- Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann surfaces.
- Kähler differentials provide a general notion of differential in algebraic geometry
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 , the maps (or coboundary operators) di 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 and a differential algebra.
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- What is the intuition behind the definition of the differential of a function
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