利用Green公式计算二重积分  

Calculation of double integral by Green's theorem

作　　者：巩星田 杨树伟[2] GONG Xingtian;YANG Shuwei(Teaching Department of Mathematics,Lanzhou Technology and Business College,Lanzhou 730101,China;School of Science,Lanzhou University of Technology,Lanzhou 730050,China)

机构地区：[1]兰州工商学院数学教学部,甘肃兰州730101 [2]兰州理工大学理学院,甘肃兰州730050

出　　处：《渤海大学学报（自然科学版）》2023年第4期338-342,共5页Journal of Bohai University:Natural Science Edition

基　　金：甘肃省教育厅高等学校创新基金项目(No:2023B-403).

摘　　要：二重积分的计算一直都是高等数学教学的核心内容之一.基于被积函数齐次性的特点,结合Green公式,给出了将闭区域上二重积分转化为边界上曲线积分的计算公式以及其特殊情形.将二重积分转化为一重积分,降维使得计算更加地便捷简化;另一方面,该结论又提供了一种计算二重积分的新途径,特别是可以用来求解部分利用常规方法难于求解或计算繁杂的二重积分,这使得该方法具有更大的优势.最后,通过一些典型性算例证实理论的有效性.The calculation of double integrals has always been one of the core contents of higher mathematics teaching.Based on the homogeneity of the integrand,combined with Green's theorem,the calculation formula for converting double integrals on a closed domain into curve integrals on the boundary and its special cases are provided.Transforming double integrals into single integrals,dimensionality reduction makes calculations more convenient and simplified;On the other hand,this conclusion provides a new way to calculate double integrals,especially for solving some difficult or complex double integrals that are difficult to solve using conventional methods,which makes this method more advantageous.Finally,the effectiveness of the theory is confirmed through some typical examples.

关 键 词：二重积分 Green公式 曲线积分 齐次函数 

分 类 号：O172.2[理学—数学]