第三代的微积分

自然杂志 ›› 2010, Vol. 32 ›› Issue (2): 67-.

第三代的微积分

张景中①冯勇②

①中国科学院院士，②教授，电子科技大学计算机推理与可信计算实验室，成都 610054

收稿日期:2009-12-07 修回日期:2010-01-07 出版日期:2010-04-25 发布日期:2010-04-25

The Third Generation of Calculus

ZHANG Jingzhong①, FENG Yong②

①CAS member, ②Professor, Laboratory of Computer Reasoning and Trustworthy Computation, University of Electronic Science and Technology of China, Chengdu 610054, China

Received:2009-12-07 Revised:2010-01-07 Online:2010-04-25 Published:2010-04-25

摘要/Abstract

摘要： 微积分已经有三百多年的历史了，由牛顿和莱布尼兹创建的微积分，是第一代微积分。第一代微积分，是说不清楚的微积分。以严格的极限理论为基础的微积分为第二代微积分。第二代微积分由于概念和推理繁琐迂回，成为了当今高等数学教学的难点。通过对导数和积分定义的研究，提出了甲函数和乙函数的概念，以此为基础建立起来的初等化微积分系统通俗易懂而又具有严格的理论基础，我们称之为第三代微积分。由于第三代微积分是以初等数学为基础，从而可大大降低教学的难度，另一方面又可利用目前在数学机械化取得的成果，实现微积分的机械化。

关键词: 初等微积分, 极限数学, 机械化

Abstract:

Calculus has been developing for almost three hundreds years. It was first introduced by Newton and Leibniz. Without the basis of rigorous theory, it was ambiguous, which is called the first generation of calculus. Later, it was established on the concept of limit in a rigorous and abstract way, called the second generation of calculus. However, it is difficult to understand for most beginners. In this paper, we introduce two concepts of function A and function B, on which a new calculus is built in a rigorous and simple way. The new calculus is called the third generation of calculus. Due to its basis being elementary mathematics, the difficulty of its teaching and learning is reduced greatly. In addition, it is expected that this new calculus will have a profound influence on mechanization of calculus.

Key words: elementarization of calculus, limit, mechanization of mathematics

张景中冯勇. 第三代的微积分[J]. 自然杂志, 2010, 32(2): 67-.

ZHANG Jing-Zhong;FENG Yong. The Third Generation of Calculus[J]. Chinese Journal of Nature, 2010, 32(2): 67-.