这是袁萌将于2018年7月2日面呈国家教育部高教司理工处的信函。
中华人民共和国教育部
高教司理工处吴爱华处长：
本文将J.Keisler撰写的微积分教科书的所有章节附后，以便与国内相关教材内容进行对比研究。该微积分教材在第一章就引入超实数系统，个给出现代实无穷小概念，使我国大学00后一年级新生赢在学习微积分的起跑线上，赶超西方发达国家不是梦。
说明：全书共计14章（含有常微分方程解的存在性证明），全部内容大约需要180学时，两个学期
作者：H. Jerome Keisler （1936- ）
书名：Elementary Calculus（无穷小方法）
CONTENTS
INTRODUCTION xiii
1 REAL AND HVPERREAL NUMBERS1
1.1 The Real Line 1
1.2 Functions of Real Numbers 6（序偶定义）
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（标准部分） 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 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 THEINTEGRAL 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 FUNCTIONS365
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 7.6Trigonometric 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 ANDLOGARITHMIC 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 4548.6 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 TaylorSeries 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 DIFFERENTIATION639
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
11.5 Chain Rule
11.6 Maxima and Minima
11.7 Higher Partial Derivatives
Extra Problems for Chapter II
12 MULTIPLE INTEGRALS
12.1 Double Integrals
12.2 Iterated Integrals
12.3 Infinite Sum
12.4 Theorem and Volume
12.5 Applications to Physics
12.6 Double Integrals in Polar Coordinates
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 Green's Theorem
13.4 Surface
13.5 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 Existence and Approximation of Solutions
14.4 Complex Numbers
14.5 Second Order Homogeneous Linear Equations
14.6 Second Order Linear Equations
14.7 Second Order LinearEquations
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（全书名词索引）
中华人民共和国教育部
高教司理工处吴爱华处长：
本文将J.Keisler撰写的微积分教科书的所有章节附后，以便与国内相关教材内容进行对比研究。该微积分教材在第一章就引入超实数系统，个给出现代实无穷小概念，使我国大学00后一年级新生赢在学习微积分的起跑线上，赶超西方发达国家不是梦。
说明：全书共计14章（含有常微分方程解的存在性证明），全部内容大约需要180学时，两个学期
袁萌 6月29日
作者：H. Jerome Keisler （1936- ）
书名：Elementary Calculus（无穷小方法）
CONTENTS
INTRODUCTION xiii
1 REAL AND HVPERREAL NUMBERS1
1.1 The Real Line 1
1.2 Functions of Real Numbers 6（序偶定义）
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（标准部分） 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 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 THEINTEGRAL 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 FUNCTIONS365
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 7.6Trigonometric 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 ANDLOGARITHMIC 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 4548.6 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 TaylorSeries 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 DIFFERENTIATION639
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
11.5 Chain Rule
11.6 Maxima and Minima
11.7 Higher Partial Derivatives
Extra Problems for Chapter II
12 MULTIPLE INTEGRALS
12.1 Double Integrals
12.2 Iterated Integrals
12.3 Infinite Sum
12.4 Theorem and Volume
12.5 Applications to Physics
12.6 Double Integrals in Polar Coordinates
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 Green's Theorem
13.4 Surface
13.5 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 Existence and Approximation of Solutions
14.4 Complex Numbers
14.5 Second Order Homogeneous Linear Equations
14.6 Second Order Linear Equations
14.7 Second Order LinearEquations
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（全书名词索引）