“正本清源”在力学之数学及专业基础知识体系建立中的作用.doc

  “正本清源”在力学之数学及专业基础知识 体系建立中的作用# 谢锡麟* 5 10 15 20 25 30 35 40 （复旦大学力学与工程科学系，上海 200433） 摘要：本文将力学之数学及专业基础知识体系分别归结为微积分和现代张量分析以及基于其 上的连续介质力学；借鉴具有一流水平的国内外教程或专著，给出了上述基础知识体系的基 本构成。提出以知识点以及知识要素组织知识体系，并分析了微积分知识体系的辐射性发展 特征；提出隶属不同知识体系的知识点其所属知识要素可能是同一数学结构或形式，称之为 数学通识。我们把数学作为认识自然及非自然世界的系统的思想及方法；叙述了数学知识体 系同力学知识体系间的关系。本文引述微积分、张量分析、微分几何、连续介质力学等知识 体系中的有关知识以阐述上述观点；所涉及的微积分中 Stokes 公式的统一性证明，张量分 析中张量梯度的可微性观点以及微分几何中 Lie 导数的场观点定义及结论等均为我们自己 认识。 关键词：知识体系；知识点；知识要素；数学通识；微积分；张量分析；微分几何；连续介 质力学 中图分类号：G642.0 The Roles of “To Radically Reform To Thoroughly Overhaul” in the Set Up of the Fundamental Mathematical and Mechanical Knowledge Systems of the Mechanics XIE Xilin (Department of Mechnaics and Engineering Science, Fudan University, ShangHai 200433) Abstract: In this paper, the fundamental mathematical and mechanical knowledge systems of the mechanics are concluded as calculus and modern tensor analysis with continuum mechanics based on it. As referred to the related national and international textbooks and monographs with the first levels, the fundamental constitutions of the above mentioned knowledge systems are presented. It is put forward that the knowledge points with the corresponding knowledge elements are suitable to recognize one knowledge system, and the radical development property of the knowledge system of calculus is expressed. The concept termed as “Mathematical Generality” is put forward that are just some mathematical structures or forms as the so-called knowledge elements of some knowledge points with respect even to different knowledge systems. Mathematics is taken as the systematic ideas and methods to recognize the natural and unnatural worlds in the present paper, and the relationships between mathematical knowledge system and mechanical knowledge system are represented to some extents. Some cases originated from the knowledge systems of calculus, tensor analysis, differential geometry and continuum mechanics are adopted to expound the arguments that are raised in the