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所在平台: Udemy |
课程主页: https://www.udemy.com/course/real-analysis-part-4/
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**Coursera 课程总结：实分析（第四部分）- 黎曼-斯蒂尔切斯积分** 本课程是实分析系列的第四部分，深入探讨了黎曼-斯蒂尔切斯积分（Riemann-Stieltjes Integral）。黎曼-斯蒂尔切斯积分是黎曼积分的一种推广，以伯恩哈特·黎曼和托马斯·约翰内斯·斯蒂尔切斯的名字命名。它不仅是理解勒贝格积分（Lebesgue Integral）的重要基础，也是统一处理离散和连续概率统计定理的宝贵工具。 课程的核心内容围绕黎曼-斯蒂尔切斯积分的定义和存在性展开。在介绍积分定义时，课程会用到区间 $[a,b]$ 的划分序列 $P$，并通过划分范数（最长子区间长度）的极限来定义积分。 **课程涵盖的主要概念包括：** * **划分的共同精细（Common Refinement of a partition）** * **黎曼上和与黎曼下和（Upper Riemann and Lower Riemann sum）** * **黎曼和（Riemann Sum）** * **黎曼-斯蒂尔切斯积分的性质（Properties of Riemann Stieltjes Integral）** * **积分与微分（Integration and Differentiation）** * **微积分基本定理（The Fundamental Theorem Of Calculus）** * **向量值函数的积分（Integration Of Vector Valued Functions）** * **可求长曲线（Rectifiable Curves）** 此外，课程还将详细讲解上述概念相关的预期定理和解题示例，并描述黎曼-斯蒂尔切斯积分的各种恒等式。 总体而言，本课程内容详实，涵盖了丰富的概念，并配有大量的解题示例、课后作业和练习题。几乎所有预期的定理都得到了详细阐述。完成课程后，您将获得课程结业证书。 **如有任何课程相关疑问，欢迎通过私信联系，随时为您提供帮助。**
Riemann Stieltjes Integral is a generalization of the Riemann Integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. It serves as an instructive and useful precursor of the Lebesgue Integral. and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability. The definition of Riemann Stieltjes Integral uses a sequence of partitions P of the given interval [a,b].The integral, the, is defined to be the limit, as the norm (the length of the longest sub interval) of the partitions The course introduces the definition and existence of the Riemann Stieltjes integral including the concepts of the following:Common Refinement of a partition , Upper Riemann and Lower Riemann sum, Riemann Sum, Properties of Riemann Stieltjes Integral, Integration and Differentiation, The Fundamental Theorem Of Calculus,Integration Of Vector Valued Functions,Rectifiable Curves,that also covers the Expected Theorems and Solved Examples based above contents.This course also describes the various identities of Riemann Stieltjes integral. Overall , every thing has been covered with lots of concepts and solved examples , assignments, exercises and almost all the expected theorems.For any queries related to the course, I would be happy to assist you. Just ping me via Inbox. You will get a course completion certificate after finishing the course.
