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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
    Likewise, Infinitesimal is a concept; its value is smaller than any value you can imagine Check out this video and you will appreciate why Infinity and Infinitesimal cannot be "explained" to someone seeking to find "applications" "methodology"
  • Definition of an Infinitesimal - Mathematics Stack Exchange
    That chapter defines: A number $\epsilon$ is said to be infinitely small, infinitesimal, if: $-a < \epsilon < a$ And goes on to an introduction to the hyperreal line However, this definition seems to imply an infinitely small number ($\epsilon$) is one which is between $\pm a$, which seems to be a very large range if you choose, for example
  • How do you understand Infinitesimals? - Mathematics Stack Exchange
    While it is possible to create number systems with infinitesimals, that does not validate the intuition that you've stated, that somehow $0 \overline{9}$ and $1$ differ by an infinitesimal You lose a lot of nice properties of the real numbers when you add infinitesimals
  • ordinary differential equations - What is the Lie group infinitesimal . . .
    Our prof gave us the definition of the Lie group infinitesimal generator and it's kth-coordinate, he also explained why the kth-coordinate is called "the kth-coordinate", but nonetheless he didn't want to explain the Lie group infinitesimal generator because "we will understand it later as the course will continue"
  • Are infinitesimals equal to zero? - Mathematics Stack Exchange
    Hence, zero is also an infinitesimal But not necessarily exactly like other infinitesimals, because it seems you cannot add zero to itself any number of times and arrive to anything other than zero, while you can add other infinitesimals to themselves and arrive to real values
  • Whats an example of an infinitesimal? - Mathematics Stack Exchange
    Cauchy said that a sequence converging to zero becomes an infinitesimal rather than is an infinitesimal The idea that he meant a null sequence to generate an infinitesimal is confirmed by the fact that he was interested in rates of growth of sequences and even tried to classify those
  • infinity times infinitesimal - Mathematics Stack Exchange
    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
  • Precisely how is infinitesimal calculus meaningfully different from . . .
    Infinitesimal calculus is really how it was thought of when it was first created We know the area under a curve because we just add up all of these infinitesimally small slices However, and really think about this, how small is an infinitesimal? This is why I said that this is how calculus was thought of when it was first created We don't
  • Justification of algebraic manipulation of infinitesimals
    An infinitesimal is arbitrarily small, therefore in your above example $8x dx$ should also be neglected In SIA for example lone infinitesimals always cancel out so such situations never arise If someone doing actual physical measurements has a finite i e practical definition of 'infinitesimal' they should say so and not pretend it's genuine
  • calculus - Book recommendation on infinitesimals - Mathematics Stack . . .
    For example there's the book A Primer of Infinitesimal Analysis by John Bell It gives a rigorous theory that allows you to use infinitesimals a lot like ordinary numbers But this approach is called non-standard for a reason It might not be what you really want and there's some heavy duty mathematics in the background





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