In mathematics, differential refers to several related notions^[1] derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.^[2]

The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.

Introduction

The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise.

Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula

where ${\displaystyle {\frac {dy}{dx}}\,}$denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.

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.

History and usage

Infinitesimal quantities played a significant role in the development of calculus. Archimedes used them, even though he did not believe that arguments involving infinitesimals were rigorous.^[3] Isaac Newton referred to them as fluxions. However, it was Gottfried Leibniz who coined the term differentials for infinitesimal quantities and introduced the notation for them which is still used today.

In Leibniz's notation, if x is a variable quantity, then dx denotes an infinitesimal change in the variable x. Thus, if y is a function of x, then the derivative of y with respect to x is often denoted dy/dx, which would otherwise be denoted (in the notation of Newton or Lagrange) ẏ or y′. The use of differentials in this form attracted much criticism, for instance in the famous pamphlet The Analyst by Bishop Berkeley. Nevertheless, the notation has remained popular because it suggests strongly the idea that the derivative of y at x is its instantaneous rate of change (the slope of the graph's tangent line), which may be obtained by taking the limit of the ratio Δy/Δx as Δx becomes arbitrarily small. Differentials are also compatible with dimensional analysis, where a differential such as dx has the same dimensions as the variable x.

Calculus evolved into a distinct branch of mathematics during the 17th century CE, although there were antecedents going back to antiquity. The presentations of, e.g., Newton, Leibniz, were marked by non-rigorous definitions of terms like differential, fluent and "infinitely small". While many of the arguments in Bishop Berkeley's 1734 The Analyst are theological in nature, modern mathematicians acknowledge the validity of his argument against "the Ghosts of departed Quantities"; however, the modern approaches do not have the same technical issues. Despite the lack of rigor, immense progress was made in the 17th and 18th centuries. In the 19th century, Cauchy and others gradually developed the Epsilon, delta approach to continuity, limits and derivatives, giving a solid conceptual foundation for calculus.

In the 20th century, several new concepts in, e.g., multivariable calculus, differential geometry, seemed to encapsulate the intent of the old terms, especially differential; both differential and infinitesimal are used with new, more rigorous, meanings.

Differentials are also used in the notation for integrals because an integral can be regarded as an infinite sum of infinitesimal quantities: the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as

the integral sign (which is a modified long s) denotes the infinite sum, f(x) denotes the "height" of a thin strip, and the differential dx denotes its infinitely thin width.

Approaches

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There are several approaches for making the notion of differentials mathematically precise.

1. Differentials as linear maps. This approach underlies the definition of the derivative and the exterior derivative in differential geometry.^[4]
2. Differentials as nilpotent elements of commutative rings. This approach is popular in algebraic geometry.^[5]
3. Differentials in smooth models of set theory. This approach is known as synthetic differential geometry or smooth infinitesimal analysis and is closely related to the algebraic geometric approach, except that ideas from topos theory are used to hide the mechanisms by which nilpotent infinitesimals are introduced.^[6]
4. Differentials as infinitesimals in hyperreal number systems, which are extensions of the real numbers that contain invertible infinitesimals and infinitely large numbers. This is the approach of nonstandard analysis pioneered by Abraham Robinson.^[7]

These approaches are very different from each other, but they have in common the idea of being quantitative, i.e., saying not just that a differential is infinitely small, but how small it is.

Differentials as linear maps

There is a simple way to make precise sense of differentials, first used on the Real line by regarding them as linear maps. It can be used on ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {R} ^{n}}$, a Hilbert space, a Banach space, or more generally, a topological vector space. The case of the Real line is the easiest to explain. This type of differential is also known as a covariant vector or cotangent vector, depending on context.

Differentials as linear maps on R

Suppose ${\displaystyle f(x)}$ is a real-valued function on ${\displaystyle \mathbb {R} }$. We can reinterpret the variable ${\displaystyle x}$ in ${\displaystyle f(x)}$ as being a function rather than a number, namely the identity map on the real line, which takes a real number ${\displaystyle p}$ to itself: ${\displaystyle x(p)=p}$. Then ${\displaystyle f(x)}$ is the composite of ${\displaystyle f}$ with ${\displaystyle x}$, whose value at ${\displaystyle p}$ is ${\displaystyle f(x(p))=f(p)}$. The differential ${\displaystyle \operatorname {d} f}$ (which of course depends on ${\displaystyle f}$) is then a function whose value at ${\displaystyle p}$ (usually denoted ${\displaystyle df_{p}}$) is not a number, but a linear map from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$. Since a linear map from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$ is given by a ${\displaystyle 1\times 1}$ matrix, it is essentially the same thing as a number, but the change in the point of view allows us to think of ${\displaystyle df_{p}}$ as an infinitesimal and compare it with the standard infinitesimal ${\displaystyle dx_{p}}$, which is again just the identity map from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$ (a ${\displaystyle 1\times 1}$ matrix with entry ${\displaystyle 1}$). The identity map has the property that if ${\displaystyle \varepsilon }$ is very small, then ${\displaystyle dx_{p}(\varepsilon )}$ is very small, which enables us to regard it as infinitesimal. The differential ${\displaystyle df_{p}}$ has the same property, because it is just a multiple of ${\displaystyle dx_{p}}$, and this multiple is the derivative ${\displaystyle f'(p)}$ by definition. We therefore obtain that ${\displaystyle d{f}_p=f^{\prime }(p)d{x}_p}$Zdroj:https://en.wikipedia.org?pojem=Differential_(infinitesimal)
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