高等数学引论

Preface

Translator's note

Introduction

1 Real and complex numbers

1．1 Rational numbers

1．2 The existence of irrational numbers

1．3 A description of real numbers

1．4 Limit

1．5 The Bolzano-Weierstrass theorem

1．6 Definitions for complex numbers and vectors

1．7 Polar coordinates and multiplication

1．8 De Moivre's theorem

1．9 Completeness of the complex numbers

1．10 Introduction to quaternions

Supplement

1．11 Binary arithmetics

1．12 Periodic decimals

1．13 Rational approximations to real numbers

1．14 Error terms

1．15 Solutions to cubic and quartic equations

2 Vector algebra

2．1 Space coordinates and vectors

2．2 Addition of vectors

2．3 The decomposition of a vector

2．4 Inner product （scalar product）

2．5 Vector product （outer product）

2．6 Multiple products

2．7 Change of coordinates

2．8 Planes

2．9 Equation for a line in space

2 Supplement

2．10 Main formulae in spherical trigonometry

2．11 Duality principle

2．12 Right-angled and right-sided triangles

2．13 Forces， systems and equivalent systems

2．14 Combination of parallel forces

2．15 Moments

2．16 Couples

2．17 Standard form for a system

2．18 Equilibrium and its applications

3 Functions and graphs

3．1 Variables

3．2 Functions

3．3 Implicit functions

3．4 Functions represented by graphs and tables

3．5 Several elementary functions

3．6 Functions with simple special properties

3．7 Periodic functions

3．8 Representations for a complex function

3．9 Line of regression

3．10 Lagrange's interpolation formula

3．11 Other interpolation formulae

3．12 Experimental formulae

3．13 Family of curves

4 Limits

4．1 Limits of sequences

4．2 Sequences without limits

4．3 Series

4．4 Conditionally convergent series

4．5 The method of Zu Chongzhi in calculating π

4．6 Archimedes' method for the area of a parabolic region

4．7 Calculating pressure on a boundary

4．8 The number e

4．9 Taking limit in the continuum

4．10 On several important limits

4．11 Some examples

4．12 Orders of infinity

4．13 The symbols ~， O and o

4．14 Continuous functions

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《高等数学引论（第1卷 英文版）》是我国名数学家华罗庚在上世纪60年代编写的教学用书，曾在中国科学技术大学讲授。全书包含了微积分、高等代数、常微分方程、复变函数论等内容。全书反映了作者的“数学是一门有紧密内在联系的学问，应将大学数学系的基础课放在一起来讲”的教学思想，还包括了作者的“要埋有伏笔”、“生书熟讲，熟书生温”等教学技巧，书中还介绍了数学理论的不少应用。