Table of Contents
Preface vii
1 Basic Concepts 1
1.1 Functions and Limits 1
1.1.1 Limits of Sequences and Functions 1
1.1.2 Continuous Functions 2
1.2 Definite Integrals 7
1.2.1 Calculation of Areas 7
1.2.2 Definition of Definite Integrals 11
1.2.3 Logarithmic Function 18
1.3 Derivatives and Differentials 24
1.3.1 Tangent Lines of a Curve 24
1.3.2 Velocity and Density 25
1.3.3 Definition of Derivatives 27
1.3.4 Differentials 31
1.3.5 Mean-Value Theorem 33
1.4 Fundamental Theorem of Calculus 38
2 Calculations of Derivatives and Integrals 43
2.1 Differentiation 43
2.1.1 Calculations of Derivatives and Differentials 43
2.1.2 Derivatives and Differentials of Higher Order 54
2.1.3 Approximate Calculation by Derivatives 58
2.2 Integration 67
2.2.1 Indefinite Integrals 67
2.2.2 Definite Integrals 86
2.2.3 Approximate Calculations of Definite Integrals 93
3 Some Applications of Differentiation and Integration 101
3.1 Areas, Volumes, Arc Lengths 101
3.1.1 Areas 101
3.1.2 Volumes 103
3.1.3 Are Lengths 106
3.2 Techniques for Graphing functions 111
3.2.1 Increasing and Decreasing Functions 112
3.2.2 Concavity 114
3.2.3 Asymptotes 115
3.2.4 Examples of Graphing Functions 118
3.2.5 Curvatures 120
3.3 Taylor Expansions and Extreme Value Problems 126
3.3.1 Taylor Expansions 126
3.3.2 Extreme Value Problems 131
3.4 Examples in Physics 141
4 Ordinary Differential Equations 149
4.1 First Order Differential Equations 149
4.1.1 Concepts 149
4.1.2 Separation of Variables 152
4.1.3 Linear Differential Equations 161
4.2 Second Order Differential Equations 167
4.2.1 Reducible Differential Equations 167
4.2.2 Second Order Linear Differential Equations 171
4.2.3 Linear Differential Equations with Constant Coefficients 181
4.2.4 Mechanical Vibration 190
4.2.5 General Linear Differential Equations and Systems of Linear Equations 196
5 Vector Algebra and Analytic Geometry in Three-Dimensional Space 209
5.1 Coordinate System of Three-Dimensional Space and Concept of Vectors 209
5.1.1 Rectangular Coordinate System 209
5.1.2 Addition and Scalar Multiplication of Vectors 211
5.2 Products of Vectors 217
5.2.1 Inner Products of Vectors 217
5.2.2 Cross Products of Vectors 220
5.2.3 Scalar Triple Products of Vectors 222
5.3 Planes and Lines 226
5.3.1 Equations of Planes 226
5.3.2 Equations of Lines 229
5.4 Quadric Surfaces 234
5.4.1 Cylindrical Surfaces 234
5.4.2 Surfaces of Revolution 236
5.4.3 Corneal Surfaces 238
5.4.4 Ellipsoid 239
5.4.5 Hyperbolic Paraboloid 241
5.4.6 Hyperboloid of One Sheet 242
5.4.7 Hyperboloid of Two Sheets 242
5.4.8 Elliptic Paraboloid 242
5.5 Transformations of Coordinates 244
5.5.1 Translation of Axes 244
5.5.2 Rotation of Axes 246
6 Multiple Integrals and Partial Derivatives 251
6.1 Multiple Integrals 251
6.1.1 Limits and Continuity of Functions of Several Variables 251
6.1.2 Multiple Integration 253
6.1.3 Calculation of Multiple Integrals 257
6.2 Partial Derivatives 270
6.2.1 Partial Derivatives and Total Differentials 270
6.2.2 Derivatives of Implicit functions 278
6.3 Jacobian Determinants, Area Elements, Volume Elements 295
6.3.1 Properties of Jacobian Determinant 295
6.3.2 Area Elements and Volume Elements 296
7 Line Integrals, Surface Integrals and Exterior Differential Forms 317
7.1 Scalar Fields and Vector Fields 317
7.1.1 Contour Surfaces and Gradient of a Scalar Field 317
7.1.2 Streamlines of Vector Fields 321
7.2 Line Integrals 327
7.2.1 Line Integrals of the First Kind 327
7.2.2 Applications of Line Integrals of the First Kind (Areas of Surfaces of Revolution) 330
7.2.3 Line Integrals of the Second Kind 331
7.2.4 Calculation of Line Integrals of the Second Kind 335
7.2.5 Relation Between Line Integrals of the First Kind and the Second Kind 338
7.2.6 Circulations of Vector Fields and Line Integrals of Vectors 339
7.3 Surface Integrals 345
7.3.1 Surface integrals of the First Kind 345
7.3.2 Flux of Vector Fields, Surface Integrals of the Second Kind (Integral with respect to the projections of the area element) 348
7.3.3 Calculation of Surface Integrals of the Second Kind 350
7.4 Stokes Theorem 357
7.4.1 Green's Theorem 357
7.4.2 Gauss's Theorem, Divergence 361
7.4.3 Stokes' Theorem, and The Curl of a Vector Field 367
7.5 Total Differentials and Line Integrals 377
7.5.1 Line Integrals that are Independent of Paths 377
7.5.2 Potential Fields 381
7.5.3 Solenoidal Vector Fields 383
7.6 Exterior Differential Forms 387
7.6.1 Exterior Products, Exterior Differential Forms 387
7.6.2 Exterior Differentiation, Poincaré Lemma and its Inverse 394
7.6.3 Mathematical Meaning of Gradient, Divergence and Curl 399
7.6.4 Fundamental Theorem of Calculus in Several Variables (Stokes' Theorem) 401
8 Sonic Applications of Calculus in Several Variables 405
8.1 Taylor Expansions and Extremal Problems 405
8.1.1 Taylor Expansions of Functions in Several Variables 405
8.1.2 Extremal Problems of Functions in Several Variables 406
8.1.3 Conditional Extremum Problems 411
8.2 Examples of Applications in Physics 417
8.2.1 Barycenter, Moment of Inertia and Gravitational Force 417
8.2.2 Complete System of Equations of Fluid Dynamics 423
8.2.3 Propagation of Sound 426
8.2.4 Heat Exchange 427
9 The ε-δ Definitions of Limits 433
9.1 The ε-δ Definition of Limits of Number Sequences 433
9.1.1 Definition of Limits of Number Sequences 433
9.1.2 Properties of Limits of Number Sequences 435
9.1.3 Criteria for the Existence of Limits 438
9.2 The ε-δ Definition of Continuity of Functions 447
9.2.1 Limits of Functions 447
9.2.2 Definition of Continuous Functions 454
9.2.3 Properties of Continuous Functions 457
9.2.4 Uniform Continuity of Functions 460
9.3 Existence of Definite Integrals 466
9.3.1 Darboux Sums 466
9.3.2 Integrability of Continuous Functions 468
9.3.3 Generalization of the Concept of Definite Integrals (Improper Integrals) 475
10 Infinite Series and Infinite Integrals 485
10.1 Number Series 485
10.1.1 Basic Concepts 485
10.1.2 Some Convergence Criteria 487
10.1.3 Conditionally Convergent Series 493
10.2 Function Series 502
10.2.1 Infinite Sums 502
10.2.2 Uniformly Convergent, Sequences of Functions 504
10.2.3 Uniformly Convergent Function Series 508
10.2.4 Existence Theorem of Implicit Functions 512
10.2.5 Existence and Uniqueness Theorem of the Solution of Ordinary Differential Equations 516
10.3 Power Series and Taylor Series 524
10.3.1 Convergence Radius of Power Series 524
10.3.2 Properties of Power Series 527
10.3.3 Taylor Series 532
10.3.4 Applications of Power Series 539
10.4 Infinite Integrals and Integrals with Parameters 553
10.4.1 Convergence Criteria for Infinite Integrals 553
10.4.2 Integrals with Parameters 565
10.4.3 Infinite Integrals with Parameters 570
10.4.4 Several Important Infinite Integrals 583
11 Fourier Series and Fourier Integrals 597
11.1 Fourier Series 597
11.1.1 Orthogonality of the System of Trigonometric Functions 597
11.1.2 Bessel Inequality 607
11.1.3 Convergence Criterion for Fourier Series 611
11.2 Fourier Integrals 616
11.2.1 Fourier Integrals 616
11.2.2 Fourier Transforms 619
11.2.3 Applications of Fourier Transforms 624
11.2.4 Higher-Dimensional Fourier Transforms 625
Answers 627
Index 671
