魯賓遜非標準微積分全面覆蓋國內高等數學課程內容

  此時此刻，“預告：電子版微積分投放安排，陽光事業在陽光下進行”，2018年08月05日發表， 11:58:31 yuanmeng001 此刻閱讀數：10101。這裡的“10101”是一個好數字，不拐彎抹角，有話直說。

我們向全國普通高校投放魯賓遜非標準微積分教科書電子版，有什麼價值？也就是說，魯賓遜非標準微積分的內容是否全面覆蓋國內高等數學課程內容？是與不是，讓資料；來說話。

訪問“無窮小微積分”網站，下載魯賓遜非標準微積分，從中提取內容目錄，與國內高等數學教科書進行對比研究即可。請讀者見本文附件。      

袁萌 陳啟清  10月26日

附：魯賓遜非標準微積分內容目錄

CONTENT

INTRODUCTION   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

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 Chain Rule

                                

11.6 Implicit Functions

 

11.7 Implicit Functions

 

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

 

12.7 Applications to Physics

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

Equations with

14.1 Complex Numbers

14.2 Separable Variables

14.3 First Order

14.4 Homogeneous Linear Equations

14.5 First Order Linear Equations

14.6 Existence and

14.7 Approximation of Solutions

Complex Numbers

Second Order Homogeneous Linear Equations

Second Order Linear Equations（全文完）