Differential (calculus)

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

[edit] Basic notions

• 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 Rⁿ 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.

[edit] Differential geometry

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.

[edit] Algebraic geometry

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

[edit] 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 , 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.