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differential calculus: see calculus.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of *math.x*^{math.n} is *math.n**math.x*^{math.n − 1}, (2) the derivative of sin *math.x* is cos *math.x*, and (3) the derivative of the exponential function *math.e*^{math.x} is itself. The rules are: (1) (*math.a**math.f* + *math.b**math.g*)' = *math.a**math.f*' + *math.b**math.g*', (2) (*math.f**math.g*)' = *math.f**math.g*' + *math.g**math.f*', (3) (*math.f*/*math.g*)' = (*math.g**math.f*' − *math.f**math.g*')/*math.g*^{2}, and (4) (*math.f*(*math.g*))' = *math.f*'(*math.g*)*math.g*', where *math.a* and *math.b* are constants, *math.f* and *math.g* are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. *Seealso* integration.

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Encyclopedia Britannica, 2008. Encyclopedia Britannica Online.

In mathematics, and more specifically, in differential calculus, the term differential has several interrelated meanings.

- In traditional approaches to calculus, the differential (e.g. dx, dy, dt, etc...) of a function represents an infinitesimal change in its value. Although this is not a precise notion, there are several ways to make sense of it rigorously.
- The differential is another name for the Jacobian matrix of partial derivatives of a function from R
^{n}to R^{m}(especially when this matrix is viewed as a linear map). - More generally, 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.
- Stochastic calculus provides a notion of stochastic differential and an associated calculus for stochastic processes.

The notion of a differential motivates several concepts in differential geometry (and differential topology).

- 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 (vector bundle). 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.
- Kahler 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 $(C\_bullet,\; d\_bullet)$, the maps (or coboundary operators) d_{i} 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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