共查询到20条相似文献,搜索用时 78 毫秒
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普遍认为利用等价无穷小代换求代数和及复合函数的极限时常常隐藏着两个误区,通过对其进行探讨可以发现.只要加以适当的条件,代数和各部分为无穷小量以及复合函数的中间变量为无穷小量时.对这些部分是可以进行等价无穷小代换的. 相似文献
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在求函数极限的过程中,利用等价无穷小代换常常会使计算简单,但对形如的极限,若无穷小量有时权限并不相等.这里的关键是与的选取不当.本文着重讨论这种情况下利用Taylor公式选取适当的等价无穷小量作代换保持权限不变.为叙述方便,我们只讨论的情况.同时假定下文中所涉及的都是当时的连续且具有任意阶导数的函数无穷小量,以后不再交代.引理1若引理2若则,这里不全为0引理3若a在处的Taylor展开式(带皮亚诺余项)为(按前面假设,常数项为0):证明显然,从而实际上,通常我们所用的等价无穷小都是取Tayfor展式的第一个非零项.如… 相似文献
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将二重变上限积分看作是一类特殊的一元诱导函数,本文给出了两种二重变上限积分的定义方式,分别对积分限和被积函数做相应的等价无穷小量替换.在一定的条件下,替换后的二重变上限积分与替换前的二重变上限积分是等价无穷小,从而得到一类求极限的方法,并给出了应用实例. 相似文献
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利用微分中值定理和L’Hospital法则证明含有变上限积分的等价无穷小代换问题,并提供两个实例作为应用. 相似文献
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The ring of Fermat reals 总被引:1,自引:0,他引:1
Paolo Giordano 《Advances in Mathematics》2010,225(4):2050-2075
We give the definition of the ring of Fermat reals, a simple extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from smooth infinitesimal analysis, but provides a powerful theory of actual infinitesimals without any need of a background in mathematical logic. In particular it is consistent with classical logic. We face the problem to decide if the product of powers of nilpotent infinitesimals is zero or not, the identity principle for polynomials, the characterization of invertible elements and some applications to Taylor's formulas. 相似文献
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针对微积分教材关于"无穷小的积是否是无穷小"的命题的模糊叙述,证明"一致有界的一族无穷小的积仍是无穷小". 相似文献
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对文[6]提出的质疑给出回答,表明由于不同的无穷小量趋近于0的速度有快有慢,因此无穷多个无穷小量的乘积∏∞k=1{x_n~(k)}∞n=1,有可能不是无穷小量(其中对每个正整数k,{x_n~(k)}_(n=1)~∞表示极限为0的数列),而验证∏∞k=1{x_n~(k)}∞n=1是否是无穷多个无穷小量的乘积,只需验证对每个正整数k,当n→+∞时,{x_n~(k))_(n=1)~∞是否趋近于0,而无需考虑函数列{{x_n~(k)}_(n=1)~∞}_(k=1)~∞的极限limk→∞x_n~(k)是不是无穷小量.进而,对无穷多个无穷小量的乘积是无穷小量或不是无穷小量给出了一些充分条件, 相似文献
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The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented. 相似文献
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Alpha-theory: An elementary axiomatics for nonstandard analysis 总被引:1,自引:0,他引:1
The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals. 相似文献
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A. Madanshekaf 《Southeast Asian Bulletin of Mathematics》2002,25(3):435-442
An isomorphism between the object of units of the real line and the automorphism group object of first order infinitesimals is established.AMS Mathematics subject classification
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