无穷小微积分在阳光下运行

    这是袁萌将于2018年7月2日面呈国家教育部高教司理工处的信函。

中华人民共和国教育部

高教司理工处吴爱华处长：

本文将J.Keisler撰写的微积分教科书的所有章节附后，以便与国内相关教材内容进行对比研究。

                 该微积分教材在第一章就引入超实数系统，个给出现代实无穷小概念，使我国大学00后一年级新生赢在学习微积分的起跑线上，赶超西方发达国家不是梦。

    说明：全书共计14章（含有常微分方程解的存在性证明），全部内容大约需要180学时，两个学期

袁萌  6月29日

 

作者：                      H. Jerome Keisler （1936- ）

书名：Elementary Calculus（无穷小方法）

CONTENTS

INTRODUCTION  xiii

1 REAL AND HVPERREAL NUMBERS 1

1.1 The Real Line  1

1.2 Functions of Real Numbers  6（序偶定义）

1.3 Straight Lines  16

1.4 Slope and Velocity; The Hyperreal Line 21

1.5 Infinitesimal, Finite, and Infinite Numbers 27

1.6 Standard Parts（标准部分） 35

Extra Problems for Chapter I 41

2 DIFFERENTIATION 43

2.1 Derivatives 43

2.2 Differentials and Tangent Lines 53

2.3 Derivatives of Rational Functions 60

2.4 Inverse Functions 70

2.5 Transcendental Functions 78

2.6 Chain Rule 85

2.7 Higher Derivatives 94

2.8 Implicit Functions 97

Extra Problems for Chapter 2  103

3 CONTINUOUS FUNCTIONS 105

3.1 How to Set Up a Problem 105 3

.2 Related Rates 110

3.3 Limits 117

3.4 Continuity 124

3.5 Maxima and Minima 134

3.6 Maxima and Minima - Applications 144

3.7 Derivatives and Curve Sketching 151

3.8 Properties of Continuous Functions 159

Extra Problems for Chapter 3   171

4 INTEGRATION 175

4.1 The Definite Integral 175

4.2 Fundamental Theorem of Calculus 186

4.3 Indefinite Integrals 198

4.4 Integration by Change of Variables 209

4.5 Area between Two Curves 218

4.6 Numerical Integration 224

Extra Problems for Chapter 4   234

 

5 LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS 237

5.1 Infinite Limits 237

5.2 L'Hospital's Rule 242

5.3 Limits and Curve Sketching 248

5.4 Parabolas 256

5.5 Ellipses and Hyperbolas 264

5.6 Second Degree Curves 272

5.7 Rotation of Axes 276

5.8 The e, 8 Condition for Limits 282

5.9 Newton's Method 289

5.10 Derivatives and Increments 294

Extra Problems for Chapter 5   300

6 APPLICATIONS OF THE INTEGRAL 302  

6.1 Infinite Sum Theorem 302

6.2 Volumes of Solids of Revolution 308

6.3 Length of a Curve 319

6.4 Area of a Surface of Revolution 327

6.5 Averages 336

6.6 Some Applications to Physics 341

6.7 Improper Integrals 351

Extra Problems for Chapter 6   362

7 TRIGONOMETRIC FUNCTIONS 365

7.1 Trigonometry 365

7.2 Derivatives of Trigonometric Functions 373

7.3 Inverse Trigonometric Functions 381

7.4 Integration by Parts 391

7.5 Integrals of Powers of Trigonometric Functions 397 7.6 Trigonometric Substitutions 402

7.7 Polar Coordinates 406

7.8 Slopes and Curve Sketching in Polar Coordinates 412

7.9 Area in Polar Coordinates 420

7.10 Length of a Curve in Polar Coordinates 425

Extra Problems for Chapter 7 428

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 431

8.1 Exponential Functions 431

8.2 Logarithmic Functions 436

8.3 Derivatives of Exponential Functions and the Number e 441

8.4 Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454 8.6 Some Differential Equations 461

8.7 Derivatives and Integrals Involving In x 469

8.8 Integration of Rational Functions 474

8.9 Methods of Integration 481

Extra Problems for Chapter 8   489

9 INFINITE SERIES 492

9.1 Sequences 492

9.2 Series 501

9.3 Properties of Infinite Series 507

9.4 Series with Positive Terms 511

9.5 Alternating Series 517

9.6 Absolute and Conditional Convergence 521

9.7 Power Series 528

9.8 Derivatives and Integrals of Power Series 533

9.9 Approximations by Power Series 540

9.10 Taylor's Formula 547

9.11 Taylor Series 554 Extra Problems for Chapter 9 561

10 VECTORS 564

10.1 Vector Algebra 564

10.2 Vectors and Plane Geometry 576

10.3 Vectors and Lines in Space 585

10.4 Products of Vectors 593

10.5 Planes in Space 604

10.6 Vector Valued Functions 615

10.7 Vector Derivatives 620

10.8 Hyperreal Vectors 627

Extra Problems for Chapter I 0   635

11 PARTIAL DIFFERENTIATION 639

II. I Surfaces 639

11.2 Continuous Functions of Two or More Variables 651

11.3 Partial Derivatives 656

11.4 Total Differentials and Tangent Planes 662

11.5 Chain Rule

11.6 Maxima and Minima

11.7 Higher Partial Derivatives

Extra Problems for Chapter II

12 MULTIPLE INTEGRALS

12.1 Double Integrals

12.2 Iterated Integrals

12.3 Infinite Sum

12.4 Theorem and Volume

12.5 Applications to Physics

12.6 Double Integrals in Polar Coordinates

12.7 Cylindrical and Spherical Coordinates

Extra Problems for Chapter 12

13 VECTOR CALCULUS

13.1 Directional Derivatives and Gradients

13.2 Line Integrals

13.3 Independence of Path Green's Theorem

13.4 Surface

13.5 Area and Surface Integrals

13.6 Theorems of Stokes and Gauss

Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS

14.1 Equations with Separable Variables

14.2 First Order Homogeneous Linear Equations

14.3 Existence and Approximation of Solutions

14.4 Complex Numbers

14.5 Second Order Homogeneous Linear Equations

14.6 Second Order Linear Equations

14.7          Second Order Linear Equations

Extra Problems for Chapter 14

EPILOGUE （结束语）

APPENDIX: TABLES I

Trigonometric Functions II

Greek Alphabet III

Exponential Functions IV

Natural Logarithms V

Powers and Roots

ANSWERS TO SELECTED PROBLEMS

INDEX（全书名词索引）