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问题在于：无穷小微积分教材的内容是否符合规定的教学大纲？这个问题需要专家研究判定。

迄今为止，J.Keisler精心撰写的“Elementary Calculus”微积分教科书（一学年用）是世界上唯一的基于现代数理逻辑模型论的无穷小微积分教科书，没有第二本。该书内容目录请见本文附件。

袁萌  7月14日

附：无穷小微积分内容目录

CONTENTS（内容目录及其所在页码）

INTRODUCTION（引言） xiii

1 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

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 150

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

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

11.6 Implicit Functions

11.7 Maxima and Minima

11.8 Higher Partial Derivatives

Extra Problems for Chapter II

12 MULTIPLE INTEGRALS

12.1 Double Integrals

12.2 Iterated Integrals

12.3 Infinite Sum Theorem and Volume

12.4 Applications to Physics

12.5 Double Integrals in Polar Coordinates

12.6 Triple Integrals

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

13.4 Green's Theorem

13.5 Surface 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 First Order Linear Equations

14.4 Existence and Approximation of Solutions

14.5 Complex Numbers

14.6 Second Order Homogeneous 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