Principles of Mathematical Analysis, 3/e (2023 Edition)

Description
This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first-year students who study mathematics.

NEW to This Edition

• The material on functions of several variables in Chapter 9 is significantly rewritten, with many details filled in, and with more examples and more motivation. The proof of the inverse function theorem is simplified by means of the fixed point theorem about contraction mappings. Differential forms are discussed in much greater detail. Several applications of Stokes' theorem are included.
• The Riemann-Stieltjes integral in Chapter 6 has been trimmed a bit.
• A short do-it-yourself section on the gamma function has been added to Chapter 8.
• There is a large number of new exercises throughout the book, most of them with fairly detailed hints.
• Several references to articles appearing in the American Mathematical Monthly and in Mathematics Magazine are included for students to develop the habit of looking into the journal literature.

描述

本書旨在作為高年級本科生或研究數學的一年級學生修讀的分析課程的教材。



本版新增內容

• 第9章關於多變數函數的內容進行了重寫，填補了許多細節，並增加了更多的例子和動機。通過關於收縮映射的不動點定理，簡化了反函數定理的證明。對微分形式進行了更詳細的討論。包括了幾個斯托克斯定理的應用。


• 第6章中的黎曼-斯蒂爾吉積分進行了一些修剪。


• 在第8章中增加了一個關於伽馬函數的簡短自學部分。


• 全書中增加了大量的新習題，其中大部分都有相當詳細的提示。


• 包括了幾篇刊登在《美國數學月刊》和《數學雜誌》上的文章的參考資料，以培養學生查閱期刊文獻的習慣。

Table of Contents
Chapter 1: The Real and Complex Number Systems 
Chapter 2: Basic Topology 
Chapter 3: Numerical Sequences and Series 
Chapter 4: Continuity 
Chapter 5: Differentiation 
Chapter 6: The Riemann-Stieltjes Integral 
Chapter 7: Sequences and Series of Functions 
Chapter 8: Some Special Functions 
Chapter 9: Functions of Several Variables 
Chapter 10: Integration of Differential Forms 
Chapter 11: The Lebesgue Theory

目錄

第一章：實數和複數系統

第二章：基本拓撲學

第三章：數列和級數

第四章：連續性

第五章：微分

第六章：黎曼-斯蒂爾吉積分

第七章：函數的數列和級數

第八章：一些特殊函數

第九章：多變數函數

第十章：微分形式的積分

第十一章：勒貝格理論