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

2.7 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

5.4Parabolas                         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

5.9 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  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

12.5 Double Integrals in Polar Coordinates    749

12.6 Triple Integrals                     757

12.7 Cylindrical and Spherical Coordinates   769

Extra Problems for Chapter 12            783

13 VECTOR CALCULUS                      785

13.1 Directional Derivatives and Gradients   785

13.2 Line Integrals                      793

13.3 Independence of Path               805

13.4  Green’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 Existence and 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

ⅣNatural Logarithms                  A3

Ⅴ Powers and Roots                  A4

 ANSWERS TO SELECTED PROBLEMS          A5

INDEX                              A5

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