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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
  • 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
  • Properties of infinities, infinitesimals - Mathematics Stack Exchange
    @Eliot : There is the Nelson variant of NSA where infinitesimals and infinities are just "not observable" members of the real number line, numbers that will never occur in numerical computation, regardless of what multi-precision mechanism you use Astonishingly, such a separation of numbers in standard and non-standard ones can be done in extension of the usual set theory Then indeed there
  • 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?
  • Whats an example of an infinitesimal? - Mathematics Stack Exchange
    If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimal
  • Do limits leave residual infinitesimals, or do they resolve exactly . . .
    Thus, "residual infinitesimals" are indeed involved in evaluating a limit, but they disappear once we apply the standard part For an elementary axiomatic approach to infinitesimal analysis (that does not involve either the axiom of choice or ultrafilters), see this introduction
  • 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
  • Infinitesimally small time intervals - Mathematics Stack Exchange
    In the hyperreal numbers an infinitesimal is a positive number that is smaller than any positive real number (That statement can be made precise, but I am going for the concept rather than for mathematical rigor)
  • 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





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