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内容简介

这是一套完整介绍数学分析的教材，内容涉及从实数到流形上的微分形式，其中包括渐近方法、傅立叶分析、拉普拉斯变换、勒让德变换、椭圆函数以及频率分布。本书语言通俗，表达清晰，各章有大量的练习、思考题以及*应用实例。

作者简介

佐里奇，是莫斯科国立大学教授。主要从事分析、保角几何、拟共形映照方面的研究工作。近期从事热力学中的数学问题的研究。他解决了空间拟共形映照下的球面同胚问题，并因该研究成果获得了“青年数学家国家奖”（National Prize for Young Mathematicians）。同时还拥有一项技

目 录

1 Some General Mathematical Concepts and Notation. 

1.1 Logical Symbolism 

1.2 Sets and Elementary Operations on them 

1.3 Functions 

1.4 Supplementary Material 

2 The Real Numbers 

2.1 Axioms and Properties of Real Numbers 

2.2 Classes of Real Numbers and Computations 

2.3 Basic Lemmas on Completeness 

2.4 Countable and Uncountable Sets 

3 Limits 

3.1 The Limit of a Sequence 

3.2 The Limit of a Function 

4 Continuous Functions 

4.1 Basic Definitions and Examples 

4.2 Properties of Continuous Functions 

5 Differential Calculus 

5.1 Differentiable Functions 

5.2 The Basic Rules of Differentiation 

5.3 The Basic Theorems of Differential Calculus

5.4 Differential Calculus Used to Study Functions

5.5 Complex Numbers and Elementary Functions

5.6 Examples of Differential Calculus in Natural Science

5.7 Primitives 

6 Integration

6.1 Definition of the Integral

6.2 Linearity, Additivity and Monotonicity of the Iuntegral

6.3 The Integral and the Derivative

6.4 Some Applications of Integration

6.5 Improper Integrals

7 Functions of Several Variables

7.1 The Space Rm and its Subests

7.2 Limits and Continuity of Functions of Several Variables

8 Differential Calculus in Several Variables

8.1 The Linear Structure on Rm

……

Some Problems from the Midterm Examinations

Examination Topics

References

Subject Index

Name Index