00后大学生：鲁宾逊无穷小微积分教材的详细目录
鲁宾逊微积分的目录与菲氏微积分的目录几乎完全一样，两者都不超纲。
看书，首先要看该书的详细目录，了解书的大意。
请见本文的附录。
附：
H. Jerome Keisler
CONTENTS
INTRODUCTION xiii
REAL AND HVPERREAL NUMBERS 1
1.1 The Real Line 1
1.2 Functions of RealNumbers 6 1.3 Straight Lines 16 1.4 Slope and Velocity; The Hyperreal Line 211.5 Infinitesimal, Finite, and Infinite Numbers 27 1.6 Standard Parts 35 ExtraProblems for Chapter I 41
DIFFERENTIATION 43 2.1Derivatives 43 2.2 Differentials and Tangent Lines 53 2.3 Derivatives ofRational Functions 60 2.4 Inverse Functions 70 2.5 Transcendental Functions 782.6 Chain Rule 85 2.7 Higher Derivatives 94 2.8 Implicit Functions 97 ExtraProblems for Chapter 2 103
CONTINUOUS FUNCTIONS 105 3.1How to Set Up a Problem 105 3.2 Related Rates 110 3.3 Limits 117 3.4 Continuity124 3.5 Maxima and Minima 134 3.6 Maxima and Minima - Applications 144 3.7Derivatives and Curve Sketching 151
vii
viii CONTENTS
3.8 Properties of ContinuousFunctions 159 Extra Problems for Chapter 3 171
4 INTEGRATION 175 4.1 TheDefinite Integral 175 4.2 Fundamental Theorem of Calculus 186 4.3 IndefiniteIntegrals 198 4.4 Integration by Change of Variables 209 4.5 Area between TwoCurves 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 5.3Limits and Curve Sketching 248 5.4 Parabolas 256 5.5 Ellipses and Hyperbolas264 5.6 Second Degree Curves 272 5.7 Rotation of Axes 276 5.8 The e, 8Condition for Limits 282 5.9 Newton's Method 289 5.10 Derivatives andIncrements 294 Extra Problems for Chapter 5 300
6 APPLICATIONS OF THEINTEGRAL 302 6.1 Infinite Sum Theorem 302 6.2 Volumes of Solids of Revolution308 6.3 Length of a Curve 319 6.4 Area of a Surface of Revolution 327 6.5Averages 336 6.6 Some Applications to Physics 341 6.7 Improper Integrals 351Extra Problems for Chapter 6 362
7 TRIGONOMETRIC FUNCTIONS365 7.1 Trigonometry 365 7.2 Derivatives of Trigonometric Functions 373 7.3Inverse Trigonometric Functions 381 7.4 Integration by Parts 391 7.5 Integralsof Powers of Trigonometric Functions 397 7.6 Trigonometric Substitutions 4027.7 Polar Coordinates 406 7.8 Slopes and Curve Sketching in Polar Coordinates412 7.9 Area in Polar Coordinates 420
CONTENTS ix
7.10 Length of a Curve inPolar Coordinates 425 Extra Problems for Chapter 7 428
8 EXPONENTIAL ANDLOGARITHMIC FUNCTIONS 431 8.1 Exponential Functions 431 8.2 LogarithmicFunctions 436 8.3 Derivatives of Exponential Functions and the Number e 441 8.4Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454 8.6 SomeDifferential Equations 461 8.7 Derivatives and Integrals Involving In x 469 8.8Integration of Rational Functions 474 8.9 Methods of Integration 481 ExtraProblems for Chapter 8 489
9 INFINITE SERIES 492 9.1Sequences 492 9.2 Series 501 9.3 Properties of Infinite Series 507 9.4 Serieswith Positive Terms 511 9.5 Alternating Series 517 9.6 Absolute and ConditionalConvergence 521 9.7 Power Series 528 9.8 Derivatives and Integrals of PowerSeries 533 9.9 Approximations by Power Series 540 9.10 Taylor's Formula 5479.11 Taylor Series 554 Extra Problems for Chapter 9 561
10 VECTORS 564 10.1 VectorAlgebra 564 10.2 Vectors and Plane Geometry 576 10.3 Vectors and Lines in Space585 10.4 Products of Vectors 593 10.5 Planes in Space 604 10.6 Vector ValuedFunctions 615 10.7 Vector Derivatives 620 10.8 Hyperreal Vectors 627 ExtraProblems for Chapter I 0 635
11 PARTIAL DIFFERENTIATION639 II. I Surfaces 639 11.2 Continuous Functions of Two or More Variables 65111.3 Partial Derivatives 656 11.4 Total Differentials and Tangent Planes 662
X CONTENTS
11.5 11.6 11.7 11.8
Chain Rule ImplicitFunctions Maxima and Minima Higher Partial Derivatives Extra Problems forChapter II
12 MULTIPLE INTEGRALS 12.112.2 12.3 12.4 12.5 12.6 12.7 Double Integrals Iterated Integrals Infinite SumTheorem and Volume Applications to Physics Double Integrals in PolarCoordinates Triple Integrals Cylindrical and Spherical Coordinates ExtraProblems for Chapter 12
13 VECTOR CALCULUS 13.1 13.213.3 13.4 13.5 13.6 Directional Derivatives and Gradients Line IntegralsIndependence of Path Green's Theorem Surface Area and Surface Integrals Theoremsof Stokes and Gauss Extra Problems for Chapter 13
14 DIFFERENTIAL EQUATIONS14.1 14.2 14.3 14.4 14.5 14.6 14.7 Equations with Separable Variables FirstOrder Homogeneous Linear Equations First Order Linear Equations Existence andApproximation of Solutions Complex Numbers Second Order Homogeneous LinearEquations Second Order Linear Equations Extra Problems for Chapter 14
EPILOGUE
APPENDIX: TABLES ITrigonometric Functions II Greek Alphabet III Exponential Functions IV NaturalLogarithms V Po[BY1]
鲁宾逊微积分的目录与菲氏微积分的目录几乎完全一样，两者都不超纲。
看书，首先要看该书的详细目录，了解书的大意。
请见本文的附录。
袁萌 6月13日
附：
H. Jerome Keisler
CONTENTS
INTRODUCTION xiii
REAL AND HVPERREAL NUMBERS 1
1.1 The Real Line 1
1.2 Functions of RealNumbers 6 1.3 Straight Lines 16 1.4 Slope and Velocity; The Hyperreal Line 211.5 Infinitesimal, Finite, and Infinite Numbers 27 1.6 Standard Parts 35 ExtraProblems for Chapter I 41
DIFFERENTIATION 43 2.1Derivatives 43 2.2 Differentials and Tangent Lines 53 2.3 Derivatives ofRational Functions 60 2.4 Inverse Functions 70 2.5 Transcendental Functions 782.6 Chain Rule 85 2.7 Higher Derivatives 94 2.8 Implicit Functions 97 ExtraProblems for Chapter 2 103
CONTINUOUS FUNCTIONS 105 3.1How to Set Up a Problem 105 3.2 Related Rates 110 3.3 Limits 117 3.4 Continuity124 3.5 Maxima and Minima 134 3.6 Maxima and Minima - Applications 144 3.7Derivatives and Curve Sketching 151
vii
viii CONTENTS
3.8 Properties of ContinuousFunctions 159 Extra Problems for Chapter 3 171
4 INTEGRATION 175 4.1 TheDefinite Integral 175 4.2 Fundamental Theorem of Calculus 186 4.3 IndefiniteIntegrals 198 4.4 Integration by Change of Variables 209 4.5 Area between TwoCurves 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 5.3Limits and Curve Sketching 248 5.4 Parabolas 256 5.5 Ellipses and Hyperbolas264 5.6 Second Degree Curves 272 5.7 Rotation of Axes 276 5.8 The e, 8Condition for Limits 282 5.9 Newton's Method 289 5.10 Derivatives andIncrements 294 Extra Problems for Chapter 5 300
6 APPLICATIONS OF THEINTEGRAL 302 6.1 Infinite Sum Theorem 302 6.2 Volumes of Solids of Revolution308 6.3 Length of a Curve 319 6.4 Area of a Surface of Revolution 327 6.5Averages 336 6.6 Some Applications to Physics 341 6.7 Improper Integrals 351Extra Problems for Chapter 6 362
7 TRIGONOMETRIC FUNCTIONS365 7.1 Trigonometry 365 7.2 Derivatives of Trigonometric Functions 373 7.3Inverse Trigonometric Functions 381 7.4 Integration by Parts 391 7.5 Integralsof Powers of Trigonometric Functions 397 7.6 Trigonometric Substitutions 4027.7 Polar Coordinates 406 7.8 Slopes and Curve Sketching in Polar Coordinates412 7.9 Area in Polar Coordinates 420
CONTENTS ix
7.10 Length of a Curve inPolar Coordinates 425 Extra Problems for Chapter 7 428
8 EXPONENTIAL ANDLOGARITHMIC FUNCTIONS 431 8.1 Exponential Functions 431 8.2 LogarithmicFunctions 436 8.3 Derivatives of Exponential Functions and the Number e 441 8.4Some Uses of Exponential Functions 449 8.5 Natural Logarithms 454 8.6 SomeDifferential Equations 461 8.7 Derivatives and Integrals Involving In x 469 8.8Integration of Rational Functions 474 8.9 Methods of Integration 481 ExtraProblems for Chapter 8 489
9 INFINITE SERIES 492 9.1Sequences 492 9.2 Series 501 9.3 Properties of Infinite Series 507 9.4 Serieswith Positive Terms 511 9.5 Alternating Series 517 9.6 Absolute and ConditionalConvergence 521 9.7 Power Series 528 9.8 Derivatives and Integrals of PowerSeries 533 9.9 Approximations by Power Series 540 9.10 Taylor's Formula 5479.11 Taylor Series 554 Extra Problems for Chapter 9 561
10 VECTORS 564 10.1 VectorAlgebra 564 10.2 Vectors and Plane Geometry 576 10.3 Vectors and Lines in Space585 10.4 Products of Vectors 593 10.5 Planes in Space 604 10.6 Vector ValuedFunctions 615 10.7 Vector Derivatives 620 10.8 Hyperreal Vectors 627 ExtraProblems for Chapter I 0 635
11 PARTIAL DIFFERENTIATION639 II. I Surfaces 639 11.2 Continuous Functions of Two or More Variables 65111.3 Partial Derivatives 656 11.4 Total Differentials and Tangent Planes 662
X CONTENTS
11.5 11.6 11.7 11.8
Chain Rule ImplicitFunctions Maxima and Minima Higher Partial Derivatives Extra Problems forChapter II
12 MULTIPLE INTEGRALS 12.112.2 12.3 12.4 12.5 12.6 12.7 Double Integrals Iterated Integrals Infinite SumTheorem and Volume Applications to Physics Double Integrals in PolarCoordinates Triple Integrals Cylindrical and Spherical Coordinates ExtraProblems for Chapter 12
13 VECTOR CALCULUS 13.1 13.213.3 13.4 13.5 13.6 Directional Derivatives and Gradients Line IntegralsIndependence of Path Green's Theorem Surface Area and Surface Integrals Theoremsof Stokes and Gauss Extra Problems for Chapter 13
14 DIFFERENTIAL EQUATIONS14.1 14.2 14.3 14.4 14.5 14.6 14.7 Equations with Separable Variables FirstOrder Homogeneous Linear Equations First Order Linear Equations Existence andApproximation of Solutions Complex Numbers Second Order Homogeneous LinearEquations Second Order Linear Equations Extra Problems for Chapter 14
EPILOGUE
APPENDIX: TABLES ITrigonometric Functions II Greek Alphabet III Exponential Functions IV NaturalLogarithms V Po[BY1]