infinitesimal

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infinitesimal

1. Maths of, relating to, or involving a small change in the value of a variable that approaches zero as a limit
2. Maths an infinitesimal quantity

Infinitesimal

 

in mathematics, a variable quantity that approaches a limit equal to zero. In order that the concept of an infinitesimal may have an exact meaning, it is necessary to indicate the process of variation in which the given quantity becomes an infinitesimal. For example, the quantity y = 1/x is an infinitesimal for an argument x that approaches infinity, but for an x that approaches zero it proves to be infinitely large. If the limit of the variable y is finite and equal to a, then lim (y – a) = 0, and conversely. Thus, the concept of an infinitesimal can serve as the basis of a general definition of the limit of a variable. The theory of the infinitesimal is one of the methods of constructing a theory of limits.

In considering several variables involved in the same process of variation, the variables y and z are called equivalent if lim (z/y) = 1; if in this case y is infinitesimal, the y and z are called equivalent infinitesimals. The variable z is called infinitesimal with respect to y if z/y is infinitesimal. The latter fact is often written in the form z = o(y), which reads “z is o small with respect to y”. If y is infinitesimal here, then it is said that z is an infinitesimal of a higher order than y. Often, among several infinitesimals that take part in the same process of variation, one of them, let us say y, is assumed to be the principal, and the rest are compared with it. Then it is said that z is an infinitesimal of order k > 0 if lim (z/yk) exists and is not zero. If this limit is equal to zero, then z is called an infinitesimal of order higher than k. The study of the orders of different kinds of infinitesimals is one of the important problems of mathematical analysis. For the case in which the variable quantity is a function of the argument x, the following explicit definition of an infinitesimal results from the general definition of a limit: the function f(x), defined in the neighborhood of the point x0, is called an infinitesimal for x approaching x0 if, for any positive number ∊, there is a positive number δ such that for all xx0 which satisfy the condition ǀx - x0ǀ < δ, the inequality ǀf(x)ǀ < ∊ is satisfied. This fact is written in the form Infinitesimal. In the study of the function f(x) near the point x0, one takes the increment of the independent variable Δx = x - x0 as the principal infinitesimal. The formula Δy = f(xox + ox), for example, expresses the fact that the increment Δy of a differentiable function coincides with its differential dy = f’(x0x with an accuracy within an infinitesimal of an order higher than the first.

The method of the infinitesimal, or (what is the same) the method of limits, is currently the primary method of substantiating mathematical analysis, which is why it is also called infinitesimal analysis. It replaced the exhaustive method of antiquity and the method of indivisibles. The method of the infinitesimal was outlined by I. Newton in 1666 and was universally accepted after the works of A. Cauchy. Using the infinitesimal, one can define such basic concepts of analysis as convergent series, the integral, the derivative, and the differential. In addition, the method of infinitesimals serves as one of the primary methods of applying mathematics to the problems of natural science. This is connected with the fact that most of the principles of mechanics and classical physics are expressed in the form of formulas relating infinitesimal increments of the quantities under investigation, and the infinitesimal is usually used to construct the differential equations of the problem.

S. B. STECHKIN

infinitesimal

(mathematics)
A function whose value approaches 0 as its argument approaches some specified limit.
References in periodicals archive ?
Bernoulli believes that Leibniz is committed to the existence of infinitesimals because Leibniz asserts that the continuum is actually divided into infinitely many parts.
if all the terms actually exist there will surely be infinitesimals and all
Bernoulli goes on to point out that in bodies all such divisions are (according to Leibniz) actual, so that, Bernoulli argues, infinitesimals, that is, infinitely small magnitudes, would be necessary.
In the same letter Bernoulli asserts that either all the terms in the series are not actually given, and then only finitely many are given and more could be given, or else all the terms are actually given, and there is an infinite number of them, hence infinitesimals.
Bernoulli agrees with Leibniz that such a number of the multitude of all numbers is impossible, and yet he nonetheless believes that if Leibniz is committed to an actual division of magnitude into an infinity of parts then Leibniz must be committed to the existence of infinitesimals.
opposites, the infinitesimals, have no place except in geometrical
In McLaughlin's recent article(35) we read that the strength of the infinitesimals consists in that being infinitesimal intervals they
can never be captured through measurement; infinitesimals remain forever
time interval are in fact densely packed with infinitesimal regions.
All we can say, again, is that if one argues that the arrow is moving in these infinitesimal segments, which are presumably different from 0, the absolutely indivisible, we are still faced with an abstract plurality that has not even slightly addressed the problem of the conceptualization of change.
refer to a process taking place during the infinitesimal open intervals .