Some Remarks on Variations and Differentials
Some Remarks on Variations and Differentials
复制标题
关于变化和差异的一些评论
DOI：
10.1080/00029890.1966.11970919
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发表时间：
1966
影响因子：
0.5
通讯作者：
M. Nashed
中科院分区：
文献类型：
--
作者：
M. Nashed
1. 介绍。多元函数的微分学处理在本科课程中经常被微分、梯度、方向导数等计算公式所主导。微积分的内在本质和这些概念的概念意义很少被揭示出来。多变量函数的演算在许多方面比单变量函数的演算更微妙。例如，可以定义不同的可微性概念；中值定理和泰勒公式有多种推广方法，其中极值理论涉及较多。人们普遍认为，在现代微分几何的背景下，而不是在传统的实变量理论的背景下，可以更好地研究多变量的微分和积分。有关此方法优点的有趣讨论，请参阅[12]。另一方面，可以用向量空间和线性算子给出微分学某些方面的抽象表述。今天，数学专业的本科生在学习线性代数和分析的过程中就会接触到这些概念，并且可以上升到这样的公式水平。这种方法阐明了多元微积分和赋范线性空间上的映射微积分的思想和论点，并统一了积分方程、变分学和数值分析中的许多概念和方法。例如，[1,9,11,14,16]。本文的目的是讨论实数域上定义域和值域在赋范线性空间中的映射的这种方法。
1. Introduction. The treatment of differential calculus for functions of several variables is often dominated in undergraduate courses by computational formulas for differentials, gradients, directional derivatives, etc. The intrinsic nature of calculus and the conceptual meaning of these notions are seldom brought to light.The calculus of functions of several variables is in many respects more subtel than the calculus of one real variable. For example, different notions of differentiability can be defined; the mean value theorem and Taylor's formula can be generalized in several ways and the theory of extrema is more involved. It is generally recognized that the differential and integral calculus of several variables can be best studied in the setting of modern differential geometry, rather than in the traditional setting of real variable theory. For an interesting discussion of the merits of this approach, see [12]. On the other hand, an abstract formulation of some aspects of differential calculus can be given using vector spaces and linear operators. The undergraduate student in mathematics today is exposed to these notions in his study of linear algebra and analysis, and can climb up to such levels of formulations. This approach sheds light on the ideas and arguments of multivariate calculus and the calculus of mapping on normed linear spaces, and unifies many notions and methods in analysis related to integral equations, the calculus of variations and numerical analysis. See, for instance,[1, 9, 11, 14, 16]. The purpose of this exposition is to discuss this approach for mappings whose domain and range are in normed linear spaces over the field of real numbers.