正交变换在多元函数积分学中的应用及其数学思想方法

Application of Orthogonal Transformation in Integral Calculus of Multivariate Function and Its Mathematical Thinking Method

摘要: 多元函数积分学是在一元函数积分基本思想的发展和应用中形成的，是数学分析的一个重要教学模块。其计算往往存在着一定的难度和技巧，这困扰了不少学生。正交变换是高等代数的核心内容之一，也是解题的重要方法。很多学者对正交变换在积分学中的应用也进行了相关研究并取得了一定的成果，基于此，本文主要研究当积分区域为矩形或长方体时，如何巧用正交变换计算重积分；并挖掘了正交变换在解决这些问题时所蕴含的数学思想方法，这对学生思维观和方法论的形成都非常重要。

Abstract: Integral calculus of multivariate function is formed in the development and application of the basic idea of single variable integral calculus, which is an important teaching module of mathematical analysis. Its calculation often has certain difficulty and skill, which troubles many students. Or-thogonal transformation is one of the core contents of advanced algebra and it is also an important method to solve problems. Many scholars have also carried out related research on the application of orthogonal transformation in integral and achieved certain results. Based on this, the paper mainly studies how to skillfully use orthogonal transformation to calculate multiple integral when the integral region is rectangular or cuboid. It also explores the mathematical thinking method of orthogonal transformation in solving these problems, which is very important to the formation of students’ thinking concept and methodology.