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內容簡介

這是一套完整介紹數學分析的教材，內容涉及從實數到流形上的微分形式，其中包括漸近方法、傅立葉分析、拉普拉斯變換、勒讓德變換、橢圓函數以及頻率分布。本書語言通俗，錶達清晰，各章有大量的練習、思考題以及*應用實例。

作者簡介

佐裏奇，是莫斯科國立大學教授。主要從事分析、保角幾何、擬共形映照方麵的研究工作。近期從事熱力學中的數學問題的研究。他解決瞭空間擬共形映照下的球麵同胚問題，並因該研究成果獲得瞭“青年數學傢國傢奬”（National Prize for Young Mathematicians）。同時還擁有一項技

目 錄

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