Differentials:

Leibniz introduced the language of differentials to describe the calculus of infinitesimals, which were later ridiculed by Berkeley as ghosts of departed quantities . Modern calculus texts mention differentials only in passing, if at all. Nonetheless, it is worth remembering that calculus was used successfully during those 150 years. In practice, many scientists and engineers continue to this day to apply calculus by manipulating differentials, and for good reason. It works.

Differentials are commonly used informally when making substitutions in integrals. For example, the power law in the form

d(u²) = 2u du

Differentials are equally effective for differentiation.

Differential is an infinitesimal increment in a variable. We can also say that it is the product of the derivative of a function containing one variable and the increment of the independent variable.

In calculus, a differential is an infinitesimal change in the value of a function. Differentials help to constitute derivatives and integrals. Outside of this context, they occur as elements of cotangent spaces in the guise of differential forms, a formal extension of the naive notion of differential first taught to students.

• 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. Although infinitesimals are difficult to give a precise definition, there are several ways to make sense of them 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.
• 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.

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.

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