高斯公式在几个等式证明中的妙用——教学与科研相长的实践

Magical Effect of Gauss’s Law in Proving Some Identities—The Practice of Teaching and Research

摘要: 高斯公式是高等数学的一个重要知识点，在计算数学的科学研究中也有许多应用。本文给出了一个教学与科研相互促进的实例。在有限元基函数存在唯一性的科学研究中，提炼出一个多面体中带权有向面积等式，利用高斯公式给出简化的证明，并把该等式应用到高斯公式的教学中。

Abstract: Gauss’s formula is important in higher mathematics and has many applications in the scientific research of computational mathematics. This paper provides an example of the mutual promotion between the teaching and the scientific research. In the research of the existence and uniqueness of the finite element basis function, an equation related to a weighted oriented area in polyhedrons is proposed. A simplified proof is given by using Gauss’s law, and the equation is applied to the teaching of Gauss’s formula in the course of higher mathematics.