The basic mathematical concepts: differential manifolds, vector fields, Lie derivatives and distributions are explained successively, which are the foundation for developing the differential geometric control theory of nonlinear control systems.

By analyzing several specific examples,the paper points out the serious polysemy of basic mathematical concepts existing in the mathematical teaching material and other mathematical literatures in higher school and prove that such phenomenon is unreasonable and It's necessary and urgent for us to standardize it.

No matter for teaching material of electric circuit theory or for teaching in classroom,many teachers pay more attention to the explanations of physical concepts,and it is short of the employing basic mathematical concepts to explain electric circuit principles.

Non-thickness" is an important and basic mathematical concept, as well as a key word for understanding other related concepts. The author refers widely to the predecessors' works on this topic, putting forward some new explanations for the term "nonthickness" and other words. In the light of the philosophic thought of Mo Zi, non-thickness means something no one could see or give any measurement, and it seems that Mo Zi didn't firmly believe any of its measurements to be zero, for there was no concept...

Non-thickness" is an important and basic mathematical concept, as well as a key word for understanding other related concepts. The author refers widely to the predecessors' works on this topic, putting forward some new explanations for the term "nonthickness" and other words. In the light of the philosophic thought of Mo Zi, non-thickness means something no one could see or give any measurement, and it seems that Mo Zi didn't firmly believe any of its measurements to be zero, for there was no concept of zero in his time yet. However, according to Hut Shi who was more than 100 years later than Mo Zi, non-thickness means something whose thickness equals zero, so he said: "That which has no thickness cannot be piled up; yet it is 1000li in dimension". In addition, some new explanations for duan (端), ti (体), jian (兼) and other words are also given. In the second half of this paper, the author discusses the proposition fei ban (非半) in Mo Jing and the proposition chi chui (尺棰) in the work of Zhuang Zi, and points out that they used the same method which essentially was dichotomy, but their conclusions were quite different, and the latter represented a challenge to the former. The paper ends with a set-theoretical analysis of the proposition chi chui.

The sources that lead to wrong mathematical solutions are deeply analysed, and it is Pointed out that the countermeasures to avoid them lies in deepening the basic mathematical concept and basic ways.

深刻剖析了数学问题错解的根源，指出预防错误的具体对策是深化数学基本概念和基本方法．

New mathematical courss reform has changed much from reason and contents to application. With the development of times,especially wide application of math,mathematical coures offered and application should look closely at basic knowledge and form new "double bases" with the demands of times. The basic content of sourse increases plane vector. Plane vector is one of the most important and basic mathematical concept in modern mathematics and it has rich practical background. Plane...

New mathematical courss reform has changed much from reason and contents to application. With the development of times,especially wide application of math,mathematical coures offered and application should look closely at basic knowledge and form new "double bases" with the demands of times. The basic content of sourse increases plane vector. Plane vector is one of the most important and basic mathematical concept in modern mathematics and it has rich practical background. Plane vector is a tool of dealing with geometric and physical problems.