infinitesimal


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Related to infinitesimal: infinitesimal calculus

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 classic literature ?
And if history has for its object the study of the movement of the nations and of humanity and not the narration of episodes in the lives of individuals, it too, setting aside the conception of cause, should seek the laws common to all the inseparably interconnected infinitesimal elements of free will.
You will conceive a bunch of grapes," said he, "which are covered by some infinitesimal but noxious bacillus.
For that infinitesimal space of time he was to all purposes a frightened tiger filled with rage and terror at the apprehension of the trap.
Fifteen of the boys were to be landed at Su'u and with wild gesticulations and cries they began to recognize and point out the infinitesimal details of the landfall of the only spot they had known on earth prior to the day, three years before, when they had been sold into slavery by their fathers, uncles, and chiefs.
That great, infinitesimal question - I can't shake it out of my thoughts.
Thus, he could avoid the leap of another dog, or the drive of its fangs, and at the same moment could seize the infinitesimal fraction of time in which to deliver his own attack.
Were it but the pitifullest infinitesimal, fraction of a Product, produce it, in God's name
When two legs of my level were on the shore and the third on the ice, and the sights were directed over the latter, a rise or fall of the ice of an almost infinitesimal amount made a difference of several feet on a tree across the pond.
In point of fact the three actions of perceiving, determining, and responding were sequential; but so infinitesimal were the intervals of time between them that they appeared simultaneous.
I grant you she's that, but are you aware of the infinitesimal part looks play in married life?
The vividness of the thought checked him and for an infinitesimal fraction of a second his fingers closed again on the small object as though it had been the symbol of all these little habits that chain us to the weary round of life.
It was as though he had dreamed for an infinitesimal fraction of time of some dark print of the Inquisition.