无穷小微积分摆在数学评论的聚光灯下

    近日，按照国家教育部的安排，无穷小微积分将摆在数学评论的聚光灯下，以便找出一个可接受普的说法。

    我们相信，这是无穷小微积分的大好机会。

    为达此目的，首先请见本文附件。

 袁萌  陈启清   11月22日

附件：

无穷小微积分课程内容目录

CONTENTS

INTRODUCTION xiii

REAL AND HVPERREAL NUMBERS 1 1.1 The Real Line 1

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 I 41

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

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

vii

viii CONTENTS

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

CONTENTS ix

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

X CONTENTS

11.5 11.6 11.7 11.8

Chain Rule Implicit Functions Maxima and Minima Higher Partial Derivatives Extra Problems for Chapter II

12 MULTIPLE INTEGRALS 12.1 12.2 12.3 12.4 12.5 12.6 12.7 Double Integrals Iterated Integrals Infinite Sum Theorem and Volume Applications to Physics Double Integrals in Polar Coordinates Triple Integrals Cylindrical and Spherical Coordinates Extra Problems for Chapter 12

13 VECTOR CALCULUS 13.1 13.2 13.3 13.4 13.5 13.6 Directional Derivatives and Gradients Line Integrals Independence of Path Green's Theorem Surface Area and Surface Integrals Theorems of Stokes and Gauss Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS 14.1 14.2 14.3 14.4 14.5 14.6 14.7 Equations with Separable Variables First Order Homogeneous Linear Equations First Order Linear Equations Existence and Approximation of Solutions Complex Numbers Second Order Homogeneous Linear Equations Second Order Linear Equations Extra Problems for Chapter 14

EPILOGUE

APPENDIX: TABLES I Trigonometric Functions II Greek Alphabet III Exponential Functions IV Natural Logarithms V Powers and Roots

ANSWERS