Differential (mathematics)

In mathematics, differential refers to infinitesimal differences or to the derivatives of functions.^[1] The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.

Basic notions[]

In calculus, the differential represents a change in the linearization of a function.
- The total differential is its generalization for functions of multiple variables.
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 infinitesimal number is smaller in absolute value than any positive real number, just as an infinitely large number is larger than any real number.
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).

The differential (Pushforward) of a map between manifolds.
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 mean 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.
Kähler differentials 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.

References[]

^ "differential - Definition of differential in US English by Oxford Dictionaries". Oxford Dictionaries - English. Retrieved 13 April 2018.

External links[]

Weisstein, Eric W. "Differentials". MathWorld.

This article includes a list of related items that share the same name (or similar names).
If an incorrectly led you here, you may wish to change the link to point directly to the intended article.

[1] "differential - Definition of differential in US English by Oxford Dictionaries". Oxford Dictionaries - English. Retrieved 13 April 2018.

[1]

v t Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Graph of a function Linear function Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Order of approximation (ε, δ)-definition of limit
Differential calculus	Derivative Differential Differential equation Differential operator Mean value theorem Notation Leibniz's notation Newton's notation Rules of differentiation linearity Power Sum Chain L'Hôpital's Product General Leibniz's rule Quotient Other techniques Implicit differentiation Related rates Stationary points First derivative test Second derivative test Extreme value theorem Maxima and minima Further applications Newton's method Taylor's theorem
Integral calculus	Antiderivative Arc length Basic properties Constant of integration Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Weierstrass Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method
Vector calculus	Derivatives Curl Directional derivative Divergence Gradient Laplacian Basic theorems Line integrals Green's Stokes' Gauss'
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History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Differentiation rules List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions Secant Secant cubed List of limits Lists of integrals
Miscellaneous topics	Differential geometry curvature of curves of surfaces Euler–Maclaurin formula Gabriel's Horn Integration Bee Proof that 22/7 exceeds π Regiomontanus' angle maximization problem Steinmetz solid