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Principles of Mathematical Analysis: 두 판 사이의 차이

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9번째 줄: 9번째 줄:


=== Chapter 1. The Real and Complex Number Systems ===
=== Chapter 1. The Real and Complex Number Systems ===
:* Introduction
* Introduction
:* Ordered Sets
* Ordered Sets
:* Fields
* Fields
:* The Real Field
* The Real Field
:* The Extended Real Number System
* The Extended Real Number System
:* The Complex Field
* The Complex Field
:* Euclidean Spaces
* Euclidean Spaces
=== Chapter 2. Basic Topology ===
=== Chapter 2. Basic Topology ===
:* Finite, Countable, and Uncountable Sets
* Finite, Countable, and Uncountable Sets
:* Metric Spaces
* Metric Spaces
:* Compact Sets
* Compact Sets
:* Perfect Sets
* Perfect Sets
:* Connected Sets
* Connected Sets
=== Chapter 3. Numerical Sequences and Series ===
=== Chapter 3. Numerical Sequences and Series ===
:* Convergent Sequences
* Convergent Sequences
:* Subsequences
* Subsequences
:* Cauchy Sequences
* Cauchy Sequences
:* Upper and Lower Limits
* Upper and Lower Limits
:* Some Special Sequences
* Some Special Sequences
:* Series
* Series
:* Series of Nonnegative Terms
* Series of Nonnegative Terms
:* The Number ''e''
* The Number ''e''
:* The Root and Ratio Tests
* The Root and Ratio Tests
:* Power Series
* Power Series
:* Summation by Parts
* Summation by Parts
:* Absolute Convergence
* Absolute Convergence
:* Addition and Multiplication of Series
* Addition and Multiplication of Series
:* Rearrangements
* Rearrangements
=== Chapter 4. Continuity ===
=== Chapter 4. Continuity ===
:* Limit of Functions
* Limit of Functions
:* Continuous Functions
* Continuous Functions
:* Continuity and Compactness
* Continuity and Compactness
:* Continuity and Connectedness
* Continuity and Connectedness
:* Discontinuities
* Discontinuities
:* Monotonic Functions
* Monotonic Functions
:* Infinite Limits and Limits at Infinity
* Infinite Limits and Limits at Infinity
=== Chapter 5. Differentiation ===
=== Chapter 5. Differentiation ===
:* The Derivative of a Real Function
* The Derivative of a Real Function
:* Mean Value Theorems
* Mean Value Theorems
:* The Continuity of Derivatives
* The Continuity of Derivatives
:* L'Hospital's Rule
* L'Hospital's Rule
:* Derivatives of Higher Order
* Derivatives of Higher Order
:* Taylor's Theorem
* Taylor's Theorem
:* Differentiation of Vector-valued Functions
* Differentiation of Vector-valued Functions
=== Chapter 6. The Riemann-Stieltjes Integral ===
=== Chapter 6. The Riemann-Stieltjes Integral ===
* Definition and Existence of the Integral
* Definition and Existence of the Integral
70번째 줄: 70번째 줄:


=== Chapter 8. Some Special Functions ===
=== Chapter 8. Some Special Functions ===
 
* Power Series
* The Exponential and Logarithmic Functions
* The Trigonometric Functions
* The Algebraic Completeness of the Complex Field
* Fourier Series
* The Gamma Function


=== Chapter 9. Functions of Several Variables ===
=== Chapter 9. Functions of Several Variables ===

2015년 7월 19일 (일) 18:45 판

Walter Rudin 3종 세트: PMA, RCA, FA

목숨이 아깝다면 이 책을 멀리 하라!

Principles of Mathematical Analysis(PMA)는 해석학 교재로 가장 많이 쓰이는 책 중 하나이다. 해석학 교재 하면 대부분의 사람이 PMA를 추천한다Chapter 9~11는 보지 않는 것이 정신건강에 좋다. 물론 Rudin의 책답게 영 좋지 않은 서술이 되어 있다. 출판사는 McGraw Hill, Inc.이다. 참고로 아마존 평점은 4.2 out of 5 stars로, 높은 편이다.[1]

목차

이하 Appendix(부록)와 Exercises(예제)는 생략한다.

Chapter 1. The Real and Complex Number Systems

  • Introduction
  • Ordered Sets
  • Fields
  • The Real Field
  • The Extended Real Number System
  • The Complex Field
  • Euclidean Spaces

Chapter 2. Basic Topology

  • Finite, Countable, and Uncountable Sets
  • Metric Spaces
  • Compact Sets
  • Perfect Sets
  • Connected Sets

Chapter 3. Numerical Sequences and Series

  • Convergent Sequences
  • Subsequences
  • Cauchy Sequences
  • Upper and Lower Limits
  • Some Special Sequences
  • Series
  • Series of Nonnegative Terms
  • The Number e
  • The Root and Ratio Tests
  • Power Series
  • Summation by Parts
  • Absolute Convergence
  • Addition and Multiplication of Series
  • Rearrangements

Chapter 4. Continuity

  • Limit of Functions
  • Continuous Functions
  • Continuity and Compactness
  • Continuity and Connectedness
  • Discontinuities
  • Monotonic Functions
  • Infinite Limits and Limits at Infinity

Chapter 5. Differentiation

  • The Derivative of a Real Function
  • Mean Value Theorems
  • The Continuity of Derivatives
  • L'Hospital's Rule
  • Derivatives of Higher Order
  • Taylor's Theorem
  • Differentiation of Vector-valued Functions

Chapter 6. The Riemann-Stieltjes Integral

  • Definition and Existence of the Integral
  • Properties of the Integral
  • Integration and Differentiation
  • Integration of Vector-valued Functions
  • Rectifiable Curves

Chapter 7. Sequences and Series of Functions

  • Discussion of Main Problem
  • Uniform Convergence
  • Uniform Convergence and Continuity
  • Uniform Convergence and Integration
  • Uniform Convergence and Differentiation
  • Equicontinuous Families of Functions
  • The Stone-Weierstrass Theorem

Chapter 8. Some Special Functions

  • Power Series
  • The Exponential and Logarithmic Functions
  • The Trigonometric Functions
  • The Algebraic Completeness of the Complex Field
  • Fourier Series
  • The Gamma Function

Chapter 9. Functions of Several Variables

이하 보지 마세요

Chapter 10. Integration of Differential Forms

Chapter 11. The Lebesgue Theory

각주

  1. ↑ http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X