摘要
古典微分几何的主旨是通过第一和第二基本形式研究曲面的弯曲程度.目前大部分的教材会在介绍了第二基本形式和主曲率的概念后,给出Weingarten映射的抽象定义,然后才说明主曲率是Weingarten映射的特征值.引入一种新的讲授方法,利用Lagrange乘子法直接求解主曲率,在计算过程中自然得到Weingarten映射的具体表达式,同时表明主曲率是其特征值.这样讲授环环相扣,自然流畅,更有助于学生深入理解Weingarten映射实质上是内积空间上二次型所对应的对称变换.
In classic differential geometry,the first and second fundamental forms are employed to study the curvature of surfaces.Most textbooks,after introducing the second basic form and the concept of principal curvature,will give an abstract definition of Weingarten mapping,and then explain that principal curvature is the eigenvalue of Weingarten mapping.A new teaching method is introduced,Lagrange multiplier method is used to directly solve the principal curvature,the concrete expression of Weingarten mapping is naturally obtained in the calculation process,and it is shown that the principal curvature is its eigenvalue.In this way,the teaching is natural and smooth,which is more helpful for students to understand that Weingarten mapping is essentially a symmetric transformation corresponding to quadratic form in inner product space.
作者
岑正运
邓雪
张玮
CEN Zheng-yun;DENG Xue;ZHANG Wei(School of Mathematics,South China University of Technology,Guangzhou 510641,China)
出处
《高师理科学刊》
2019年第11期79-81,102,共4页
Journal of Science of Teachers＇College and University
基金
广东省高等教育教改项目(Y2192131)
2018中国高等教育学会理科教育专业委员会研究课题(Y1181511)
华南理工大学教改项目(Y1180651,Y1190761)
