多元微积分中的线性代数

在处理多元微积分的很多问题时,我们可以将该函数局部地看作一次式,也就是局部看作它的微分式,而用线性代数的观点来处理,会给理解和记忆带来很大便利.这样给出的证明简单明了,不失严谨,它本质上是微积分"以直代曲"的表现.本文给出几个常见的困扰教学问题的简单证明.     

第三次数学危机的前因后果

17世纪牛顿、莱布尼兹创立了微积分以后,在近一二百年的时间里,微积分一直缺乏一个严格的逻辑基础,它的一些基本概念的的表述,还有某些混乱和自相矛盾之处.从19世纪开始,柯西、维尔斯特拉斯等人进行了微积分理论的严格化工作,他们首先建立了极限理论,并把极限理论的基础归结为实数理论.后来又有代德金、康托尔、德涅等人的共同努力,完善了实数理论,那么实数理论的基础又该是什么呢?于是,康托尔提出了集合理论,他试图用集合理论来作为实数理论,乃至整个微积分理论体系的基础.用集合论中的观点来诠释各个数学概念的逻辑关系,真可谓是“天衣无…     

算术观点下的林氏微积分与传统教科书中微积分比较分析

微积分是大学里普及程度非常高的一门学科,数学系学生、理工科学生、文科学生都需要学习,传统教科书中的微积分复杂度较高,使很多学生望而生畏.而算术观点下的林氏微积分复杂度保持在乘法表的水平,大大降低了微积分的门槛,且直击微积分的核心:牛顿—莱布尼茨公式.以《数学分析》中的微积分部分为例,与林群的微积分做对比,以期为大学微积分的教学改革提供思考.     

微积分教学:从冰冷的美丽到火热的思考(续)

三、微积分教育形态的表现形式在微积分教学中,人们面对的是教科书中书写的学术形态,比较形式化的表达.那么如何用各种手段使它呈现为人们易于接受的教育形态呢?以下是一些具体的建议.1·平易近人重视人的原始观念切线,瞬时速度,都是人们具有的原始观念.我们应该把它作为微积分的出发点,而不是导数的几何解释和力学解释.切线,人人都懂.于是,我们可以启发学生用切线的斜率变化来研究函数y=x的性质,这和中学里采用的方法完全不同,立即能使得学生关注微积分的奥妙.瞬时速度,其实也是人们的原始概念.当后面的快车赶上慢车的那一刹那,快车的速度…     

数学名星皮亚诺

朱塞佩·皮亚诺以发现充满空间的曲线及提出算术公理闻名于世,但他又是微积分严格化方面的专家,晚年则致力于国际语的研究。本文简要而全面地介绍了他的生平和主要科学成就。     

一类分形插值函数的分数阶微积分

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘维尔分数阶积分、微分的概念和相关定理.由于分形插值函数满足应用分数阶微积分处理问题的条件,所以利用这些概念及分步积分的方法讨论了折线段分形插值函数的分数阶积分的连续性,可微性及哪些点是不可微的,进一步说明了该插值函数分数阶微分的连续性并指出其不连续点,用黎曼-刘维尔分数阶微积分与分形插值函数结合起来研究,目的是想设法跟经典微积分一样,能找出函数上在该点的微积分的具体的实际应用意义.这些理论为研究分形插值函数的分数阶微积分的实际应用意义提供了一些理论基础.     

变上限积分的特性及应用

变上限积分是一个很重要的函数,在我们的微积分教材中,用它来证明了微积分基本公式.许多其它问题用它处理也是既方便又简单.这些主要用到它的两条特性:①变上限积分在积分区间上是可导的,且其导函数就是被积函数;②变上限积分和被积函数比起来,其可导的阶数大1.下面举几例说明其特性及应用.     

如何不用复数讲解常系数线性微分方程

在高等数学课程中,复指数函数及其导数知识的严格讲解,通常要比微分方程知识的讲解晚很多.这使得微分方程的教学在逻辑上有些不足.用复值函数解的复系数线性组合推导出实值函数解,在教学实践中,学生经常感到迷惑.不以复数的任何知识作为前提,给出了常系数微分方程的一种自然的讲解方法.     

Cauchy和严格微积分的起源

考察十七、十八世纪的数学状况,叙述早期微积分的技巧,讨论十八世纪微积分严格化的紧迫性,解释Euler、d’Alembert、Poisson和Lagrange所开创的数学技巧的发展过程,最后指出这些数学成果是如何被Cauchy用来建立新的严格微积分基础的.     

微分概念的历史发展及教学启示

微分的形式化定义是学生学习微分概念的主要困难.微分概念的历史发展表明,形式化的微分定义是微积分严格化的产物,朴素的微分定义更能体现微积分思想,而非标准分析给微分概念带来重生.在微积分学中应用非形式化的方法构建微分概念,以微分为主线(传统教材一般以导数为主线)进行微积分教学可以促进学生学习效果.     

The use of interactive visualizations to foster the understanding of concepts of calculus: design principles and empirical results

Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.     

Four calculus qualms

In the teaching of calculus, some misunderstandings can arise, which can be alleviated by slightly deeper investigation, to be shared with students as occasions demand. In this note, four such explorations are described: (1) the Fundamental Theorem of Calculus is not stated and then proven, but calculated. (2) The integral calculus could conceivably be developed before the diff erential calculus. (3) The Taylor series about any point for any ‘good’ function is not stated and then proven, but calculated from scratch. (4) The graphing of the functions b?x and x?b (for any given b > 0) on the same set of axes is done more accurately, and more interestingly.     

The Itô formula for anticipative processes with nonmonotonous time scale via the Malliavin Calculus

Summary This work is devoted to derive It?-type formulae for anticipative stochastic processes with nonmonotonous time using the Malliavin Calculus techniques and the fundamental theorem of the differential calculus. The same method is applied also to give an It?-Ventcell type formula in the anticipative case.     

Improving student learning in calculus through applications

Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.     

统计方法在研究微积分与后续课程相关性中的应用与实证分析

着眼于本科生后期专业课程的学习效果,探讨微积分课程学习效果的有效性.以北京理工大学某经管类专业全部学生大一大二两学年的学习成绩为依据,分析微积分课程学习与后续理科课程学习的相关性,提出评估微积分课程学习效果的量化指标θ值的概念,并基于回归分析和相关性分析给出其算法.在实例分析中,通过研究,发现学生微积分课程学习效果的好坏会影响他们对后续相关课程的学习,同时也发现学生将微积分知识运用到间接相关科目的能力比运用到直接相关的科目的能力薄弱.     

The First Fundamental Theorem of Calculus

<正>The Fundamental Theorem of Calculus establishes a connection between the two branches of calculus:differential calculus and integral calculus.Integral calculus arose from the area problem.Differential calculus arose from the tangent problem.Two parts seem unrelated.But the great Mathematician Newton realized that differentiation and integration are inverse processes.So the Fundamental Theorem of Calculus gives the pre-     

The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation

The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ? taking their values in a Banach space E, is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in Rⁿ and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.     

A Validated Real Function Calculus

We present a framework for validated numerical computations with real functions. The framework is based on a formalisation of abstract data types for basic floating-point arithmetic, interval arithmetic and function models based on Banach algebra. As a concrete instantiation, we develop an elementary smooth function calculus approximated by sparse polynomial models. We demonstrate formal verification applied to validated calculus by a formalisation of basic arithmetic operations in a theorem prover. The ultimate aim is to develop a formalism powerful enough for reachability analysis of nonlinear hybrid systems.     

Polynomial asymptotes

This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of real numbers and the elementary limit theorems from calculus. This note could be used as enrichment material in courses ranging from Calculus to Real Analysis to Abstract Algebra.