无穷小微积分严谨的概念体系与独特的理论架构   

上世纪著名的大数学家Tarsky（1901-1983）的高徒、模型论专家J.Keisler（1936.12.3 - ），多年潜心研究A.Robinson(1918-1974)关于实无穷小的理论。在此基础上，经过4年多的教学实践，积累经验，在其讲稿的基础上，进过多次反复修改，精心编排、写作《基础微积分》（Elementary Calculus），全球发行，影响巨大，意义深远。

现附上《基础微积分》的章节目录。该目录电子文件相当珍贵（转录不易），值得保存、研究。

说明：这份目录大纲就是无穷小放飞互联网的基本依据。现在，我们亮出了“底牌”，完全不同于国内的微积分体制（仿原苏联微积分体系）。

认真学习无穷小微积分，站得高，望的远。

袁萌  9月26日

        

CONTENTS  

INTRODUCTION                 xiii

1 REAL AND HYPERREAL NUMBERS            1

1.1 The Real Line

1.2 Functions of Real Numbers           6

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                    35

Extra Problems for Chapter 1            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

2.7Higher 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

3.5Maxima and Minima             134

3.6Maxima and Minima Applications  144

3.7Derivatives and Curve Sketching   151

3.8Properties of Continuous Functions  159

ExtraProblemsforChapter3            171

4 INTEGRATION                         175

4.1 The Definite Integral              175

4.2 Fundamental Theorem of Calculus   186

4.3Indefinite Integrals                198

4.4Integration by Change of Variables    209

4.5Area 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

5.4Parabolas                       256

5.5 Ellipses and Hyperbolas           264

5.6 Second Degree Curves            272

5.7 Rotation of Axes                 276

5.8 The ε, δ Condition for Limits        282

5.9Newton’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 Chapter7          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  series                        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 Absoluteand 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 10         635

11 PARTIAL DIFFERENTIATION             639

11.1 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                     671

11.6 Implicit Functions               678

11.7 Maxima and Minima            688

11.8 Higher Partial Derivatives          702

Extra Problems for Chapter 11              708

12  MULTIPLE  INTEGRALS                711

12.1 Double Integrals                711

12.2 Iterated Integrals               724

12.3 Infinite Sum Theorem and Volume  736

12.4 Applications to Physics           743

1 2.5 Double Integrals in Polar Coordinates  749

12.6 Triple Integrals                  757

12.7 Cylindrical and Spherical Coordinates769

Extra Problems for Chapter 12          783

13 VECTOR CALCULUS                    785

13.1 Directional Derivatives and Gradients785

13.2Line Integrals                    793

13.3Independence of Path            805

13.4Green’s Theorem               815

13.5 Surface Area and Surface Integrals  824

13.6 Theorems of Stokes and Gauss       832

Extra Problems for Chapter 13        842

14 DIFFERENTIAL  EQUATIONS           846

14.1 Equations with Separable Variables    846

14.2 First Order Homogeneous Linear Equations 852

14.3 First Order Linear Equations     857

14.4 Existenceand Approximation of Solutions 864

14.5 Complex Numbers             874

14.6 Second Order Homogeneous Linear Equations 881

14.7 Second Order Linear Equations    892

Extra Problems for Chapter 14        900

EPILOGUE（后记）                    902

APPENDIX: TABLES                     A1

I Trigonometric Functions           A1

П Greek                         A2

Ⅲ   Exponential Functions          A3

Ⅳ Natural Logarithms             A3

Ⅴ Powers and Roots              A4

ANSWERS TO SELECTED PROBLEMS       A5

INDEX                          A5

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