Differential Calculus

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

BASIC NOTIONS

In traditional approaches to calculus, the Differential 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 Derivative s of a function from R^m to Rⁿ (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 Manifold s 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 Process es.

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 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.

. This ultimately leads to the general concept of a 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.

Kahler 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

(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 ''.

	CATEGORIES ABOUT DIFFERENTIAL CALCULUS

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	functions and mappingsmathematical analysis

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Information About

Differential Calculus