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- Differential is a term used in calculus to refer to an infinitesimal change in some varying quantity12. It was first introduced via an intuitive definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential as an infinitely small change in the value of the function, corresponding to an infinitely small change in the function's argument2. In the system of hyperreal numbers, differential is defined as infinitesimals, which are extensions of real numbers that contain inverted infinitesimals and infinitely large numbers3.Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.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 Δ x (pronounced delta x). The differential dx represents an infinitely small change in the variable x.en.wikipedia.org/wiki/Differential_(mathematics)The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential as an infinitely small (or infinitesimal) change in the value of the function, corresponding to an infinitely small change in the function's argument.en.wikipedia.org/wiki/Differential_of_a_functionDiferensial sebagai infinitesimals dalam sistem bilangan hiper-real, yakni perluasan bilangan real yang mengandung infinitesimal terbalikkan dan bilangan yang tak hingga besarnya. Cara ini adalah pendekatan analisis non-standar yang dikembangkan oleh Abraham Robinson.id.wikipedia.org/wiki/Diferensial_(matematika)
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Differential (mathematics) | Wikipedia
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. See more
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.
The term is used in … See moreThe term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. … See more
There are several approaches for making the notion of differentials mathematically precise.
1. Differentials as linear maps. This approach underlies … See moreThe 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 $${\displaystyle (C_{\bullet },d_{\bullet }),}$$ See more
17th century CECalculus evolved into a distinct branch of mathematics.19th centuryCauchy and others gradually developed the Epsilon, delta approach to continuity, limits and derivatives, giving a solid conceptual foundation for calculus.20th centurySeveral 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.ArchimedesInfinitesimal 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.Isaac NewtonIsaac Newton referred to them as fluxions.Gottfried LeibnizIt was Gottfried Leibniz who coined the term differentials for infinitesimal quantities and introduced the notation for them which is still used today.17th and 18th centuriesDespite the lack of rigor, immense progress was made in the 17th and 18th centuries.Bishop Berkeley's 1734 The AnalystThe use of differentials in this form attracted much criticism, for instance in the famous pamphlet The Analyst by Bishop Berkeley.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. Isaac Newton referred to them as fluxions. However, it was See more
The notion of a differential motivates several concepts in differential geometry (and differential topology).
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WEBJan 1, 2021 · The modern concept of a differential as the principal part of the increment must be credited to J.L. Lagrange (1736–1813), and was finally fixed by Cauchy; the latter also gave a rigorous definition of an …
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WEBDec 11, 2022 · Rather than relying on the rigor of axioms and limit postulates to develop a fully consistent system from the ground up, Newton and Leibniz— the pioneers of modern Calculus’s main ideas— built their …
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WEBAug 28, 2019 · For f f a smooth, analytic function defined on the real line, the linear part of its Taylor Series approximation is given by the differential. Scheme-theoretically, one can view infinitesimal neighborhoods of a …
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Differential | Encyclopedia of Mathematics
WEBMar 26, 2023 · Differential. The main linear part of increment of a function. 1) A real-valued function $ f $ of a real variable $ x $ is said to be differentiable at a point $ x $ if it is defined in some neighbourhood of …
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