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infinitesimal    音标拼音: [,ɪnfɪnɪt'ɛsɪməl]
a. 极小的,极微的,无限小的
n. 极小量,极微量,无限小

极小的,极微的,无限小的极小量,极微量,无限小

infinitesimal
adj 1: infinitely or immeasurably small; "two minute whiplike
threads of protoplasm"; "reduced to a microscopic scale"
[synonym: {infinitesimal}, {minute}]
n 1: (mathematics) a variable that has zero as its limit

Infinitesimal \In`fin*i*tes"i*mal\, a. [Cf. F. infinit['e]simal,
fr. infinit['e]sime infinitely small, fr. L. infinitus. See
{Infinite}, a.]
Infinitely or indefinitely small; less than any assignable
quantity or value; very small.
[1913 Webster]

{Infinitesimal calculus}, the different and the integral
calculus, when developed according to the method used by
Leibnitz, who regarded the increments given to variables
as infinitesimal.
[1913 Webster]


Infinitesimal \In`fin*i*tes"i*mal\, n. (Math.)
An infinitely small quantity; that which is less than any
assignable quantity.
[1913 Webster]


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  • What is the meaning of infinitesimal? - Mathematics Stack Exchange
    An infinitesimal is either a positive infinitesimal, a negative infinitesimal, or zero In $\mathbb {R}$ there is only one infinitesimal, zero - this is precisely the Archimedean property of $\mathbb {R}$
  • Definition of an Infinitesimal - Mathematics Stack Exchange
    Covering 1 4 of Keisler's Elementary Calculus, "Slope and Velocity; The Hyperreal Line" That chapter defines: A number $\\epsilon$ is said to be infinitely small, infinitesimal, if: $-a lt; \\epsil
  • How do you understand Infinitesimals? - Mathematics Stack Exchange
    There is an $\epsilon$ (infinitesimal) thrown in there as well How do you understand these extremely small values and what do I need to do to account for them when calculating very precise values with them? I know that they are too small to make a difference when dealing with smaller numbers, but when does it start to impact your results?
  • Precisely how is infinitesimal calculus meaningfully different from . . .
    How exactly is "infinitesimal" calculus different from "limit-based" calculus? I've heard people argue over which is the "best approach" to the subject, and I've read numerous books and articles that emphasize the distinction, yet I've never seen someone lay out precisely what makes the approaches unique
  • ordinary differential equations - What exactly is a infinitesimal . . .
    The term infinitesimal generator is often used in physics, where it refers to the Lie algebra elements of a Lie group The group reflects the flows, the Lie algebra the vector field Hence, when a physicist uses the term, then he means a Lie algebra element, a tangent vector of the left-invariant vector field the Lie algebra is defined as
  • Are infinitesimals equal to zero? - Mathematics Stack Exchange
    By far the most direct way to talk about "infinitely short line segments" is to use nonstandard analysis In standard mathematics, there are various ways to make sense of 'infinitesimal' geometry, which is basically what calculus is secretly doing, and what differential geometry does more explicitly
  • Integral Calculus, Infinitesimal - Mathematics Stack Exchange
    The biggest problem with the concept of an infinitesimal in my mind is that they suggest that there is a 'smallest possible number' Actually, when we are working with the standard real numbers, there is no such thing This should be intuitively obvious: however low you go, you can always go lower
  • Is it possible to extend a function in Smooth Infinitesimal Analysis . . .
    In "A primer of infinitesimal analysis" theorem 2 1 says "The only detachable parts of R are R itself and its empty part " This result is explained in the following way: " ( ) the smooth line is indecomposable in the sense that it cannot be split in any way whatsoever into two disjoint nonempty parts"
  • Why Cauchys definition of infinitesimal is not widely used?
    You ask "why Cauchy's definition of infinitesimal, along with his 'basic approach' was superseded?" The answer is that Cantor, Dedekind, Weierstrass and others developed a foundation for analysis to deal with certain difficulties related to Fourier series, uniform continuity, and uniform convergence
  • calculus - infinity times infinitesimal - what happens? - Mathematics . . .
    and define an infinitesimal number as the difference between a convergent geometric series and its sum: $ x+1 -\displaystyle\sum_ {i=0}^ {n\rightarrow\infty} \left (\frac {x} {x+1}\right)^i$ If the x is the same in both the infinity and the infinitesimal their product will converge to the finite number x (x+1) as n increases without bound





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