The Comparative Teaching Method of the Local Properties of Multivariable Functions Based on Cases

多元函数的局部性质是高等数学多元函数微积分板块的重要教学内容。本文围绕多元函数局部性质的教学内容，运用对比式的教学方法，结合可视化案例，探讨多元函数局部性质存在性及相互之间的关系，构建多元函数相关性质的知识体系。通过构造和分析若干二元函数特例，并借助数学软件绘制曲面图，将函数的曲面所展示出的不同特性可视化。可视化案例可以帮助学生将抽象知识具象化，同时为数学教学工作者提供课堂教学素材，提升教学效果。The local properties of multivariable functions are important teaching contents in the section of calculus of multivariable functions in advanced mathematics. This paper focuses on the teaching content of the existence of local properties of multivariate functions. By the comparative teaching method and based visual cases, this paper explores the relationships among the local properties of multivariable functions: the existence of limit, continuity, partial derivative and directional derivative of multivariate functions at a certain point, and constructs the knowledge system of the related properties of multivariate functions. By constructing and analyzing several special cases of binary functions, and with the help of mathematical software, this paper draws graphs of functions to show the different characteristics. Visual cases can help students to visualize abstract knowledge, and provide teaching materials for mathematics teaching staff to improve the teaching effect.