“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

The Roles of “To Radically Reform & To Thoroughly Overhaul” in the Set Up of the Fundamental Mathematical and Mechanical Knowledge Systems of the Mechanics

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。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,并以自己的方式给出了所涉及的微积分中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。

The fundamental mathematical and mechanical knowledge systems of the mechanics were con- cluded 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 consti-tutions of the above mentioned knowledge systems are presented. It was 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＂ was 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 knowl-edge system are represented to some extents. Some cases originated from the knowledge systems of calcu-lus, tensor analysis, differential geometry and continuum mechanics are adopted to expound the argu-ments that are raised in the present paper. The related proof of the Stokes formula in calculus in the uni fied form, the interpretation of the tensor field＇s gradient in tensor analysis in the point of view of differ-ential and the definitions of the Lie-derivative in differential geometry viewed from field argument with the related results are our own cognitions.