【作者】 孙庆华；

【导师】 李文林；

【作者基本信息】 西北大学 ， 科学技术史， 2006， 博士

【摘要】 向量理论活跃在数学的各个分支,同时也是现代物理及其他科学技术领域中不可缺少的工具。本文在现有工作的基础上,从总体和概括的角度出发,对它的历史发展按照三条线索进行了分析和研究。主要研究成果如下: 一、较深入地考察了起源于力和速度的平行四边形法则的向量理论。我们发现由此发展起来的向量理论直到19世纪上半叶主要是和力学应用紧密结合在一起的,这一时期向量主要是以笛卡儿坐标的形式出现,并反过来为力学应用提供了一种有效的数学工具。当时的数学物理学家们虽然在力学中使用了向量,但没有完全认识到向量思想对于力学的真正重要性,没有将其抽象出来作为数学的对象进行深入研究。 二、较详细地论述了与位置几何有关的向量理论。它始于莱布尼兹的位置几何,最主要的工作是格拉斯曼19世纪80年代的扩张论。遗憾的是,由于格拉斯曼的系统内容过于抽象和偏离传统而难为时人理解,它对现代向量理论的创立没有起到应有的作用,但格拉斯曼的工作却对向量的公理化具有较大影响。另外还简单介绍了与几何传统有关的麦比乌斯的重心计算和拜耳拉维提斯的等值计算。 三、详细阐述了源自复数几何表示的向量理论发展的第三条线索,现代向量理论就是在这条线索上发展起来的。这里我们对哈密顿、泰特、麦克斯韦、吉布斯和亥维赛的工作进行了详细论述、比较和总结,梳理出现代意义下向量理论建立的完整线索:复数的几何表示→哈密顿四元数创造→泰特对四元数的发展→麦克斯韦的批判接受→吉布斯、亥维赛向量理论的创立。 四、阐述了向量理论的后继发展。吉布斯和亥维赛的向量理论创立后,随着物理学(特别是电磁学)和数学的进一步发展,物理学家和应用数学家开始大量地把向量语言用在物理学和数学的各个分支。另外还简单介绍了向量的公理化过程。现代数学意义上的向量概念,是作为向量空间的一个元素来定义的,其内涵远比它在创立初期时的含义丰富、深刻得多。 五、详细考察了向量在中国的传播。我们在查阅大量民国时期原始资料的基础上,对向量在中国教学与研究中的传播作了细致的论述和总结。19世纪,随着物理学的东渐,中国学者对西方所谓的有方向的(物理)量有所认识。20世纪初,中国学者通过出国学习和翻译工作,对向量理论在中国的引入和早期传播起到了至关重要的作用。辛亥革命后中国高等教育的蓬勃发展导致了20世纪三、四十年代向量理论在中国传播和发展的繁荣时期。新中国成立后,向量分析渐渐成为数学和物理课程的部分章节内容。20世纪术,向量的部分内容已从大学数学教材下放到高中数学教材。更多还原

【Abstract】 As an indispensable tool in modern physics, the vector theory has many applications in various mathematics branches as well as in other scientific fields. The purpose of this dissertation is to analyze the development of the vector theory according to three clues based on analysis of original materials and investigation of previous researches in this field. The main results gained in this dissertation are as follows:1. The vector theory which originates from parallelogram rule of velocity and force is inspected in detail. Until the first half of the 19th century, the vector theory is merely associated with mechanical application, and the vector appears mainly in form of Cartesian coordinates and at the same time supplies mechanical application with a useful tool. Although the vector was used in mechanics at that time, its exact importance was not realized completely and had not been abstracted and studied in a higher lever.2. The vector theory related to situation geometry is elaborated. This theory stems from Leibniz’s concept of situation geometry. The most important work is Grassmann’s Ausdehnungslehre published in 1880’s. However, Grassmann’s system is so abstract and far away from the tradition as not to be understood. Its influence on the foundation of modern vector analysis is not as it should have impacted although it does affect on the axiomatization of vector theory. Mobius’ barycentric calculus and Bellavitis’ calculus of equipollences related to geometrical tradition are introduced briefly as well.3. The third clue of the development of vector theory is elaborated in view of the geometrical representation of complex numbers, which modern vector theory develops on. We discuss and compare the work of Hamilton, Tait, Maxwell, Gibbs and Heaviside, and sort out the clue : the geometrical representation of complex numbers — Hamilton’s discovery of quaternion — Tait’s development to quaternion — Maxwell’s critical acceptance of quaternion — Gibbs-Heaviside’s creation of modern vector theory.4. The latter development of vector theory is discussed. After created by Gibbs and Heaviside, modern vector theory is applied to branches of physics and mathematics, especially electromagnetics. The process of axiomatization of vector theory is also studied. The modern concept of vector is defined as an element of a vector space, and its connotation is much richer and profounder than when it is originally created.5. The spread of vector theory in China is inspected. On basis of firsthand materials of the Republic of China, the spread of vector theory in teaching and study in China are discussed in detail. In the 19th century, Chinese scholars begin realizing the directed quantities accompanied with Western learning coming to China. At the beginning of the 20th century, Chinese scholars study and translate a great deal of works, which plays important role on the spread and introduction of vector theory inChina. The development of higher education in China after the Revolution of 1911 leads to the golden period of vector theory in the 1930’s and 1940’s. After liberation, vector theory becomes one part of content of mathematics and physics course. At the end of the 20th century, vector theory is set in senior high school courses.更多还原