differential calculus

Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of math.x^math.n is math.nmath.x^{math.n − 1}, (2) the derivative of sin math.x is cos math.x, and (3) the derivative of the exponential function math.e^math.x is itself. The rules are: (1) (math.amath.f + math.bmath.g)' = math.amath.f' + math.bmath.g', (2) (math.fmath.g)' = math.fmath.g' + math.gmath.f', (3) (math.f/math.g)' = (math.gmath.f' − math.fmath.g')/math.g², and (4) (math.f(math.g))' = math.f'(math.g)math.g', where math.a and math.b are constants, math.f and math.g are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. Seealso integration.

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

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.

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

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.