Differential (mathematics) - Basic Notions

Basic Notions

• 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 is another name for the Jacobian matrix of partial derivatives of a function from Rn to Rm (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 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.