一元函数的极值及其奇异性

Extreme Values and Singularities of Smooth Functions in One Variable

针对一元函数的极值理论,为学生介绍这一理论的进一步发展即奇点理论的一些基本概念和理论.指出二者之间的密切联系.通过对平面曲线的高斯映射以及奇点的介绍,向学生展示高斯曲率与函数的二阶导数以及高斯映射的奇点与函数的拐点之间的奇妙关系,并指出它们的几何意义.同时强调在学习高等数学的相关内容时,应该养成思考它们的几何背景及意义的习惯.这对大学生学好高等数学是十分有益的.

Aiming at the extreme value theory of functions in one variable, this paper introduces its further development, called singularity theory, and some basic concepts. The relationship of the two theories is studied. By introducing Gauss maps of plane curves and their singular points, we discuss the relationships between Gaussian curvature and the second derivative, and between the singular points of Gauss map and the inflection points of functions. Geometrical meanings are also considered. We believe that it is important for freshmen to develop a good habit of looking for the geometric background and meanings of the related mathematics.