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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
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.
Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series.Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India.
Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing that they lead from everything from the usual "epsilontic" definitions or to the notions of non-standard analysis. The consensus is ...
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 ...
Leibniz used the principle to extend concepts such as arithmetic operations from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. The transfer principle provides a mathematical implementation of the law of continuity in the context of the hyperreal numbers .
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic .
In non-standard calculus the limit of a function is defined by: = if and only if for all , is infinitesimal whenever x − a is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers.