本书是分析学的第一门课程，全书共九章内容，包含实数、微分法、积分法、函数序列、度量空间与欧几里得空间、高维微分法、高维积分法、曲线与曲面、微分形式等内容。本书试图用一种从一个问题开始，并进行逐步分析的阐述形式，最终回答这个问题，并引入相关的定义、论据、猜想和例子。本书适合高等院校师生、研究人员及数学爱好者参考阅读。
Preface
1 The Real Numbers
1.1 Sets and Functions
1.2 The Real Numbers
1.3 Convergence
1.4 Series
1.5 Countable and Uncountable Sets
1.6 Open Sets and Closed Sets
1.7 Continuous Functions
1.8 Trigonometric Functions
2 Differentiation
2.1 Limits
2,2 The Derivative
2.3 The Sign of the Derivative
2.4 Critical Points
2.5 Some Applications
3 Integration
3.1 The Riemann Integral
3.2 The Fundamental Theorem of Calculus
3.3 The Logarithm and Exponential Functions
3.4 Improper Integrals
3.5 Sets of Measure Zero and Integrability
3.6 The Riemann-Stieltjes Integral
4 Sequences of Functions
4.1 Uniform Convergence
4.2 Power Series
5 Metric and Euclidean Spaces
5.1 Definitions and Examples
5.2 Sequences and Completeness
5.3 Open and Closed Sets
5.4 Continuity
5.5 Compactness
5.6 Connectedness
5.7 The Space of Continuous Functions
6 Differentiation in Higher Dimensions
6.1 Vector-valued Functions
6.2 Differentiable Functions, Part 1
6.3 0rthogonality
6.4 Linear Transformations
6.5 Differentiable Functions, Part 2
6.6 Critical Points
6.7 Tangent Planes
6.8 Inverse Function Theorem
6.9 Implicit Function Theorem
6.10 Lagrange Multipliers
7 Integration in Higher Dimensions
7.1 Integration of Vector-valued Functions
7.2 The Riemann Integral
7.3 Iterated Integration
7.4 Change of Variables
7.5 Differentiation under the Integral Sign
8 Curves and Surfaces
8.1 Curves
8.2 Green's Theorem
8.3 Surfaces
8.4 Integration on Surfaces
8.5 The Theorems of Gauss and Stokes
9 Differential Forms
9.1 Introduction
9.2 Change of Variables for Forms
9.3 Simplexes and Chains
9.4 Oriented Boundaries
9.5 Stokes's Theorem
9.6 Closed and Exact Forms
9.7 Denouement
Bibliography
Index of Terms
Index of Symbols
编辑手记
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