二重积分及其应用  

Double integrals and their applications

作　　者：赵振刚[1] ZHAO Zhengang(School of Mathematical Sciences,Capital Normal University,Beijing 100048)

机构地区：[1]首都师范大学数学科学学院,北京100048

出　　处：《首都师范大学学报（自然科学版）》2024年第1期68-76,共9页Journal of Capital Normal University：Natural Science Edition

基　　金：首都师范大学本科教学建设与改革项目。

摘　　要：本文首先定义了矩形区域上的二重积分,并使用平行于x轴和y轴的直线网将矩形区域划分。对于相关性质,给出了详细、完整的证明,证明了在矩形区域上定义的有界函数可积的充要条件是该函数在矩形区域上的不连续点集是零测集。假设函数在矩形区域上黎曼可积,成功地解决了固定变量x后,作为y的函数在相应闭区间上的黎曼积分未必存在的问题。其次,建立了有界函数在有界集上的积分,并将有界集上的积分转化为矩形区域上的积分。再次,介绍了二重积分的应用,证明了在开集上定义的二元函数的2个混合偏导数如果连续,则必然相等。最后,还使用二重积分证明了级数求和公式^(∞)∑=11/^(2)=π^(2)/6。This paper begins by defining the double integral over a rectangular region,which is divided into subsections by using a grid of lines parallel to the x-axis and y-axis.We provide detailed and complete proofs for the related properties of double integrals.Furthermore,we demonstrate that a necessary and sufficient condition for a bounded function to be integrable over a rectangular region is that the set of its points of discontinuity within the region forms a set of measure zero.Assuming that a function is Riemann integrable over a rectangular region,we successfully overcome the problem where the Riemann integral may not exist when fixing variable x and considering the function as that of y over the corresponding closed interval.Subsequently,we establish the integration of a bounded function over a bounded set and convert the integration over a bounded set into an integration over a rectangular area.Then,we discuss the applications of the double integral,prove that if two mixed partial derivatives of a binary function defined on an open set are continuous,they must be equal.Lastly,the double integral is employed to validate the summation formula for the series^(∞)∑=11/^(2)=π^(2)/6.

关 键 词：上积分 下积分 零测集 

分 类 号：O173.1[理学—数学]