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  2. Infinitesimal - Wikipedia

    en.wikipedia.org/wiki/Infinitesimal

    Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers.

  3. Leibniz's notation - Wikipedia

    en.wikipedia.org/wiki/Leibniz's_notation

    Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.. In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively ...

  4. Higher-order function - Wikipedia

    en.wikipedia.org/wiki/Higher-order_function

    In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).

  5. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as f ( − 1 ) ( x ) {\displaystyle f^{(-1)}(x)} for the first integral (this is easily confused with the inverse function f − 1 ( x ) {\displaystyle f ...

  6. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.

  7. Path-ordering - Wikipedia

    en.wikipedia.org/wiki/Path-ordering

    where + is the evolution operator over an infinitesimal time interval [, (+)]. The higher order terms can be neglected in the limit ε → 0 {\displaystyle \varepsilon \to 0} . The operator h j {\displaystyle h_{j}} is defined by

  8. Deformation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Deformation_(mathematics)

    The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties.This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. Spencer, after deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.

  9. Transcendental law of homogeneity - Wikipedia

    en.wikipedia.org/wiki/Transcendental_law_of...

    Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded. [2] Thus, if a {\displaystyle a} is finite and d x {\displaystyle dx} is infinitesimal, then one sets