Search results
Results from the WOW.Com Content Network
Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω) In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-eth" item in a sequence.
Faà di Bruno's formula; Reynolds; Integral. ... Originally called infinitesimal calculus or "the calculus of infinitesimals", ... [1] It is the "mathematical ...
If the functional [] attains a local minimum at , and () is an arbitrary function that has at least one derivative and vanishes at the endpoints and , then for any number close to 0, [] [+]. The term ε η {\displaystyle \varepsilon \eta } is called the variation of the function f {\displaystyle f} and is denoted by δ f . {\displaystyle \delta ...
One-to-one correspondence between an infinite set and its proper subset. A different form of "infinity" is the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ 0), the cardinality of the set of natural numbers.
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach.
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε , where Δ y = f ( x + Δ x ) − f ( x ...
1.1 Infinitesimal generator. 1.2 Johansson formula. 2 Harish-Chandra-Itzykson-Zuber integral formula. 3 Ginibre formula. 4 References. ... Johansson formula ...
Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.