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.     

Liouville在分数阶微积分概念方面的研究

分数阶微积分的概念是以整数阶微积分理论研究为基础,而分数阶微积分概念的建立经历了漫长的过程.探析此过程中数学家在研究分数阶微积分理论方面的贡献,进而整理Liouville在分数阶微积分概念方面的研究,进一步概括分数阶微积分第一定义的由来以及为后续相关研究奠定的坚实基础.     

林群院士和周向宇院士荣获全国创新争先奖状

<正>5月16日科技部发布了《关于全国创新争先奖拟推荐对象的公示》名单,其中我院的林群院士和周向宇院士入选。林群院士主要从事计算数学研究。二十多年来,林群一直致力于微积分的科普教育工作:画微积分连环画《画中漫游微积分》;出微积分读物《微分方程与三角测量》《微积分快餐》《微积分减肥快跑》;办微积分     

从历史的角度来讲微积分

要将微积分的历史发展与微积分的教学紧密结合起来,并运用数学史的观点与材料来重新组织微积分的教学.     

解读“微积分算术”

2011年在科学网博客上出现的微积分算术,把抽象而高深的微积分看作函数的算术,只用几步高中代数,就能避开极限而又不失严格讲解微积分.首先,用等式讲解多项式的微积分;然后用不等式讲解显式初等函数的微积分.但是,某些读者可能会存在疑问:真的能让微积分的门槛降低,而又不失严格?它到底具备什么样的特点?对学生群体的定位如何?需要进一步的解读,这就是本文的目的.     

基于本原问题的微积分概念教学浅析

利用匀速直线运动、自由落体运动、变速直线运动等问题,直观介绍微积分基本概念及其相互联系,阐明微积分本原问题在微积分概念教学中的作用.     

概率论中的微积分方法

目前，在我国高等学校所开设的概率论课程中，多效是以微积分知识做基础的，做为微积分课程的一门后续课程——概率论，如何正确、巧妙地运用微积分方法技巧是值得重视的问题，本试图归纳一些同题来说明微积分方法在概率论中有着广泛的用途，同时希望在学习微积分、概率论时，引起注意，从而产生更多、更好的微积分方法为概率论所应用。     

微积分MOOC数据的统计分析研究

MOOC(Massive Open Online Course)即"大规模在线开放课程"已成为国内外互联网在线学习的热点.北京理工大学微积分MOOC课程自2014年在中国大学MOOC网站上线,经过2年的开课,积累了丰富的课程数据.基于微积分MOOC数据,通过统计分析的方法,研究了微积分MOOC参与者的成绩分布,最终成绩与主观作业和客观测验成绩之间的关系,并给出了课程参与者数量变化模型.还结合两个样本班级学生的期中、期末卷面成绩,对比分析了参与微积分MOOC和未参与微积分MOOC学习的学生成绩,结果表明,参加微积分MOOC学习的同学期末较期中取得的进步程度要好于未参加微积分MOOC学习的同学,且这种促进效果对课堂学习微积分有困难的同学尤其明显.     

微积分基本定理——微积分历史发展的里程碑

微积分是经典数学的重要内容,曾引起马克思、恩格斯、列宁的关心和兴趣.他们从哲学家的角度,对微积分及其发展史进行深入地研究,并对微积分的本质进行了广泛的讨论.认为微分和积分是微积分的主要研究对象,它们之间的矛盾是微积分的主要矛盾.明确指出:微积分这门科学,是研究微分和积分这对矛盾的科学.为我们研究微积分及其历史提供了线索.本文以研究反映微分和积分内在联系的微积分基本定理发展为主线,简叙微积分发展历史.事物是普遍联系的,发现事物的一种联系,是一种创造.从哲学角度来说,事物相距越远,其发现难度就越大,就越能说明事物之间…     

探索微积分在中学数学中的必要性

美国的中学数学教育,为了顺应科学技术现代化发展的需要,早已将微积分的内容作为中学数学课程中的必修或选修内容．我国中学的微积分教学重视程度时起时落,虽然在现今的课程改革中将微积分纳入了高中的必修系列（上海课程除外）,但有关微积分教学的必要性改革仍是一个颇有争议的问题．     

Tensor calculus: unlearning vector calculus

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.     

On modeling with multiplicative differential equations

This work is aimed to show that various problems from different fields can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. Since multiplicative calculus is still in its infancy, some effort is put to explain its basic principles such as exponential arithmetic, multiplicative calculus, and multiplicative differential equations. Examples from finance, actuarial science, economics, and social sciences are presented with solutions using multiplicative calculus concepts. Based on the encouraging results obtained it is recommended that further research into this field be vested to exploit the applicability of multiplicative calculus in different fields as well as the development of multiplicative calculus concepts.     

Explicit algebras with the Leibniz–Mellin translation product

A sub‐calculus of the calculus (“algebra”) of all complete conormal symbols arising in the edge pseudodifferential calculus is constructed. This calculus of complete conormal symbols is suitable for constructing sub‐calculi of the general edge pseudodifferential calculus, for which the edge‐degenerate pseudodifferential operators involved map conormal asymptotics of distributions near the edges in a prescribed manner. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

What does it mean for a student to understand the first-year calculus? Perspectives of 24 experts

This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of the first-year calculus, (b) construction of connections and relationships between and among concepts and skills, (c) the ability to use the ideas of the first-year calculus, and (d) a deep sense of the context and purpose of the calculus. Each end goal for student understanding is explored in detail and the potential for using the framework as an organizational tool is discussed.     

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-     

Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts

In this paper we provide a first overview of the landscape with respect to calculus teaching in European classrooms, an area where research is very limited. In particular through a small expert-based survey and a literature review, we trace the development of calculus teaching at schools in a number of European countries and identify commonalities and differences. In the current curriculum developments, we notice a reduction in the content of calculus and a more informal approach. The use of digital tools has started to be integrated in calculus teaching in most countries. However in some nations, teaching of calculus in the classroom is rather traditional, focusing on procedural aspects of knowledge. Moreover, in cases where more informal and conceptual teaching approaches are used in the classroom, often contradictions seem to exist with other contextual matters such as examination requirements. Finally, we discuss the future of calculus teaching in Europe.     

F—fuzzy演绎系统

本文建立了一种演绎系统FFCS,在该演义系统中可以处理具有模糊性的推理过程,区别与其他对模糊推理进行形式化的逻辑系统,FFCS对模糊假言推理FMP做了完全形式化的处理。     

A coding method for a sequent calculus of propositional logic

The paper deals with a coding method for a sequent calculus of the propositional logic. The method is based on the sequent calculus. It allows us to determine if a formula is derivable in the calculus without constructing a derivation tree. The main advantage of the coding method is its compactness in comparison with derivation trees of the sequent calculus. The coding method can be used as a decision procedure for the propositional logic.     

Understanding Qualitative Calculus: A Structural Synthesis of Learning Research

More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative calculus. This revised version was published online in July 2006 with corrections to the Cover Date.