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In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real number. Infinitesimals ...
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.
A hyperreal r is infinitesimal if and only if it is infinitely close to 0. For example, if n is a hyperinteger, i.e. an element of *N − N, then 1/n is an infinitesimal. A hyperreal r is limited (or finite) if and only if its absolute value is dominated by (less than) a standard integer.
Then the same equation = ′ + holds with the same definition of Δy, but instead of ε being infinitesimal, we have = (treating x and f as given so that ε is a function of Δx alone). See also [ edit ]
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory , it views all functions as being continuous and incapable of being expressed in terms of discrete entities.
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. One expects, intuitively, that deformation theory of the first order should equate the Zariski tangent space with a moduli ...
The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler.It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". [1]
Related to this is the integral in which the infinitesimal increments are summed (e.g. to compute lengths, areas and volumes as sums of tiny pieces), for which Leibniz also supplied a closely related notation involving the same differentials, a notation whose efficiency proved decisive in the development of continental European mathematics.