黎曼积分的几种定义及其等价关系  

Several Definitions on Riemann Integral and Their Respective Equivalence Relations

作　　者：林庆泽[1] 尚亚东[2] 

机构地区：[1]广州大学物理与电子工程学院,广东广州510006 [2]广州大学数学与信息科学学院,广东广州510006

出　　处：《乐山师范学院学报》2016年第12期19-24,共6页Journal of Leshan Normal University

摘　　要：文章给出了目前国内外高等数学或数学分析教科书中关于黎曼积分(即定积分)的几种或者形象具体、或者较为抽象的定义方式,并且严格证明了这些不同的定义方式之间的等价关系。从不同的角度揭示了定积分的一些本质属性。以一个反例来强调定积分的定义中不能同时地具体指定区间上的划分和函数的黎曼和。This paper gives some definitions on the Riemann integral （definite integral） from text books on higher mathematics and mathematical analysis at home and aboard, of which some are concrete, some abstract. And the equivalence relations between these different definitions are strictly demonstrated. Besides, some essential properties of definite integral are revealed from different perspectives. Finally, a counterexample is taken to emphasize that the definition of definite integral cannot specify the partition of the interval and Riemann sums of functions simultaneously.

关 键 词：极限 黎曼和 黎曼可积 定积分 

分 类 号：O172.2[理学—数学]