线性变换在重积分中的应用

Application of Linear Transformation in Multiple Integral

摘要: 线性变换是高等代数教学中的核心内容，也是学习数学必备的基本思想方法。最大优点在于它能够化繁为简，化多为少，因此有着广泛的应用。重积分是多元函数积分学的重要组成部分，其计算往往需要学生放眼全局，运用所学知识，统筹分析积分区域和被积函数，进而确定合适的方法，这也是很多学生在处理重积分问题时所面临的最大难点。鉴于此，本文详细地分析如何根据被积函数的结构特征和积分区域的形状确定合适的线性变换，为重积分的计算提供一种行之有效的方法；同时这种分析问题、解决问题的过程有助于培养学生的科学思维方法和自主探究能力。

Abstract: Linear transformation is the core content in advanced algebra, and it is also the necessary basic thinking method to study mathematics. The biggest advantage lies in that it can reduce complex problems to simple problems, reduce more to less, so it has a wide range of applications. Multiple integral is an important part of integral calculus of multivariate function. The calculation requires student to look at the big picture, using the knowledge learned, overall analyze the integral region and the integrand function, and then determine an appropriate method. This is the biggest difficulty which many students face when calculating the multiple integral. In view of this, the paper analyzes in detail how to determine the appropriate linear transformation according to the struc-ture characteristics of integrand function and the shape of integral domain, so as to provide an ef-fective method for the calculation of multiple integral. At the same time, this process of analyzing problem and solving problem helps to cultivate students’ scientific thinking methods and ability of independent inquiry.