**微积分公理化的最好范例** 微积分公理化的最好范例

众所周知，进入二十世纪，希尔伯特倡导的数学公理化（公理系统）大行其道，无人可挡，我国除外。

从公理系统的视角来看，微积分公理化的的结果是“包容”初等数学，而不是“高居于”初等数学。换言之，公理化微积分应当定义出全部初等函数。

比如,弧度单位的存在性是公理化微积分的一条定理而不是“想当然”公 理
。

国内微积分教材根本做不到这一点。

请见第七章目录：

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

袁萌 陈启清 11月26日

附件：

CONTENTS

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

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

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