高等数学（上册）Advanced Mathematics (Ⅰ)：英文(潘斌)

本书是根据教育部非数学专业数学基础课教学指导分委员会制定的工科类本科数学基础课程教学基本要求编写的全英文教材，全书分为上、下两册。本书为上册，主要包括函数与极限，一元函数微积分及其应用和微分方程三部分。本书对基本概念的叙述清晰准确，对基本理论的论述简明易懂，例题习题的选配典型多样，强调基本运算能力的培养及理论的实际应用。本书可作为高等理工科院校非数学类专业本科生的教材，也可供其他专业选用和社会读者阅读。

 

The aim of this book is to meet the requirement of bilingual teaching of
advanced mathematics. The selection of the contents is in accordance with the
fundamental requirements of teaching issued by the Ministry of Education of
China.

Base on the property of our university，we select some
examples about petrochemical industry. These examples may help readers to
understand the application of advanced mathematics in petrochemical industry.

This book is divided into two volumes. This volume contains functions and
limits, calculus of functions of a single variable and differential equation.
Basic concepts in this book are clear and accurate. The book introduce the
fundamental theories by a method that is easy for understanding.

This book can be used as a textbook for undergraduate students in the science and
engineering schools whose majors are not mathematics, and may also be suitable
to the readers at the same level.

 

Chapter 1 Functions and limits1

1.1Mappings and functions1

1.1.1Sets1

1.1.2Mappings4

1.1.3Functions5

Exercises 1-1 19

1.2Limits of sequences23

1.2.1Concept of limits of sequences23

1.2.2Properties of convergent sequences27

Exercises 1-2 29

1.3Limits of functions30

1.3.1Definitions of limits of functions30

1.3.2The properties of functional limits33

Exercises 1-3 34

1.4Infinitesimal and infinity quantity36

1.4.1Infinitesimal quantity36

1.4.2Infinity quantity36

Exercises 1-4 38

1.5Rules of limit operations38

Exercises 1-5 43

1.6Principle of limit existence—two
important limits44

Exercises 1-6 49

1.7Comparing with two infinitesimals50

Exercises 1-7 52

1.8Continuity of functions and discontinuous points52

1.8.1Continuity of functions52

1.8.2Discontinuous points of functions54

Exercises 1-8 56

1.9Operations on continuous functions and the continuity of elementary functions57

1.9.1Continuity of the sum，difference，product and quotient of continuous functions57

1.9.2Continuity of inverse functions and composite functions58

1.9.3Continuity of elementary functions59

Exercises 1-9 59

1.10Properties of continuous functions on a closed interval60

1.10.1Boundedness and maximum-minimum theorem60

1.10.2Zero point theorem and intermediate value theorem61

*1.10.3Uniform continuity62

Exercises 1-10 63

Exercises 1 63

Chapter 2 Derivatives and differential66

2.1Concept of derivatives66

2.1.1Examples66

2.1.2Definition of derivatives70

2.1.3Geometric interpretation of derivative77

2.1.4Relationship between derivability and continuity78

Exercises 2-1 79

2.2Fundamental derivation rules81

2.2.1Derivation rules for sum，difference，product and quotient of functions81

2.2.2The rules of derivative of inverse functions83

2.2.3The rules of derivative of composite functions（The
Chain Rule）85

2.2.4Basic derivation rules and derivative formulas89

Exercises 2-2 91

2.3Higher-order derivatives93

Exercises 2-3 95

2.4Derivation of implicit functions and functions defined by parametric
equations97

2.4.1Derivation of implicit functions97

2.4.2Derivation of a function defined by parametric equations101

2.4.3Related rates of change103

Exercises 2-4 103

2.5The Differentials of functions105

2.5.1Concept of the differential105

2.5.2Geometric meaning of the differential107

2.5.3Formulas and rules on differentials108

2.5.4Application of the differential in approximate computation109

Exercises 2-5 110

Exercises 2 111

Chapter 3 Mean value theorems in differential calculus and applications of
derivatives113

3.1Mean value theorems in differential calculus113

Exercises 3-1 120

3.2L’Hospital’s rules121

Exercises 3-2 125

3.3Taylor formula126

Exercises 3-3 130

3.4Monotonicity of functions and convexity of curves131

3.4.1Monotonicity of functions131

3.4.2Convexity of curves and inflection points132

Exercises 3-4 136

3.5Extreme values of functions, maximum and minimum137

3.5.1Extreme values of functions137

3.5.2Maximum and minimum of function140

Exercises 3-5 143

3.6Differentiation of arc and curvature145

3.6.1Differentiation of an arc145

3.6.2curvature146

Exercises 3-6 149

Exercises 3 149

Chapter 4 Indefinite integral151

4.1Concept and property of indefinite integral151

4.1.1Concept of antiderivative and indefinite integral151

4.1.2Table of fundamental indefinite integrals153

4.1.3Properties of the indefinite integral155

Exercises 4-1 157

4.2Integration by substitutions158

4.2.1Integration by substitution of the first kind158

4.2.2Integration by substitution of the second kind163

Exercises 4-2 167

4.3Integration by parts169

Exercises 4-3 173

4.4Integration of rational function173

4.4.1Integration of rational function173

4.4.2Integration which can be transformed into the integration of rational
function175

Exercises 4-4 177

Exercises 4 178

Chapter 5 Definite integrals180

5.1Concept and properties of definite integrals180

5.1.1Examples of definite integral problems180

5.1.2The definition of define integral182

5.1.3Properties of definite integrals184

Exercises 5-1 186

5.2Fundamental formula of calculus188

5.2.1The relationship between the displacement and the velocity188

5.2.2A function of upper limit of integral188

5.2.3Newton-Leibniz formula189

Exercises 5-2 192

5.3Integration by substitution and parts for definite integrals194

5.3.1Integration by substitution for definite integrals194

5.3.2Integration by parts for definite integral198

Exercises 5-3 199

5.4Improper integrals201

5.4.1Improper integrals on an infinite interval201

5.4.2Improper integrals of unbounded functions203

Exercises 5-4 205

5.5Tests for convergence of improper integrals 
Γ-function206

5.5.1Test for convergence of infinite integral206

5.5.2Test for convergence of improper integrals of unbounded functions209

5.5.3 Γ-function209

Exercises 5-5 211

Exercises 5 212

Chapter 6 Applications of definite integrals214

6.1Method of elements for definite integrals214

6.2The applications of the definite integral in geometry215

6.2.1Areas of plane figures215

6.2.2The volumes of solid219

6.2.3Length of plane curves222

Exercises 6-2 224

6.3The applications of the definite Integral in physics227

6.3.1Work done by variable force227

6.3.2Force by a liquid228

6.3.3Gravity229

Exercises 6-3 230

Exercises 6 230

Chapter 7 Differential equations232

7.1Differential equations and their solutions232

Exercises 7-1 236

7.2Separable equations237

Exercises 7-2 240

7.3Homogeneous equations241

7.3.1Homogeneous equations241

7.3.2Reduction to homogeneous equation243

Exercises 7-3 245

7.4A first-order linear differential equations245

7.4.1Linear equations245

7.4.2Bernoulli’s equation248

Exercises 7-4 249

7.5Reducible second-order equations250

Exercises 7-5 254

7.6Second-order linear equations254

7.6.1Construction of solutions of second-order linear equation254

7.6.2The method of variation of parameters257

Exercises 7-6 259

7.7Homogeneous linear differential equation with constant coefficients259

Exercises 7-7 263

7.8Nonhomogeneous linear differential equation with constant coefficients264

Exercises 7-8 270

7.9Euler’s differential equation270

Exercises 7-9 271

Exercises 7 271

Appendix273

References280

 

English is the most important language in international
academia. In order to strengthen academic exchange with western countries, many
universities in China pay more and more attention to the bilingual teaching in
classrooms in recent years. Considering the importance of advanced mathematics
and scarcity of bilingual mathematics textbook, we have written this book.

The main subject of this book is calculus. Besides, it also includes
differential equation, space analytic geometry, vector algebra and infinite
series. This book is divided into two volumes. The first volume contains
calculus of functions of a single variable and differential equation. The
second volume contains space analytic geometry and vector algebra, calculus of
multivariate function, curve integral and surface integral, infinite series.

We have attempted to give this book the following characteristics.

① The main users of this
book are those foreign students studied in China. There are lots of differences
in their mathematical ability because there are no standard admission test for
enrolling foreign students. Through the teaching experience in the past 5
years, in this edition, we begin with pretests to
assess the necessary mathematical ability.

② The contents of this book are based on the Chinese
textbook “Advanced Mathematics (seventh edition)” which is written by department of mathematics of Tongji University.
The readers may read this book and use the Chinese textbook “Advanced Mathematics” as a reference. It may
help readers to understand the mathematical contents and to improve the level
of their English.

③ In order to train the mathematical quality and
ability of the students, we use some modern ideas, language and methods of
mathematics. We also bring in some mathematical and logical symbol.

④ We pay more attention to the application of
mathematics in practical problems. We have added some additional examples and
exercises in physics, chemistry, economies and even daily life.

⑤ Considering the different teaching requirements in
different university, we mark some difficult sections and exercises by the
symbol “*”. Teachers and
students may choose suitable contents as required.

In this volume, Chapter 1 to Chapter 3 are written by Bin Pan, Chapter 4 is
written by Jinqiu Li, Chapter 5 is written by Jingxian Yu, Chapter 6 is written
by Hong Niu, Chapter 7 is written by Li Chen. All the chapters are checked and
revised by Bin Pan.

We hope this book can bring readers some help in the studying and teaching of
bilingual mathematics. Due to the limit of our ability, it is impossible to
avoid some unclear explanations. We would appreciate any constructive
criticisms and corrections from readers.

Authors

2019-5