Chapter 5 Infinite Series 1
5.1 Infinite Series 1
5.1.1 The Concept of Infinite Series 1
5.1.2 Conditions for Convergence 3
5.1.3 Properties of Series 5
Exercise 5.1 8
5.2 Tests for Convergence of Positive Series 9
Exercise 5.2 18
5.3 Alternating Series,Absolute Convergence,and Conditional Convergence 19
5.3.1 Alternating Series 19
5.3.2 Absolute Convergence and Conditional Convergence 21
Exercise 5.3 23
5.4 Tests for Improper Integrals 24
5.4.1 Tests for the Improper Integrals:Infinite Limits of Integration 24
5.4.2 Tests for the Improper Integrals:Infinite Integrands 26
5.4.3 The Gamma Function 28
Exercise 5.4 30
5.5 Infinite Series of Functions 31
5.5.1 General Definitions 31
5.5.2 Uniform Convergence of Series 32
5.5.3 Properties of Uniformly Convergent Functional Series 34
Exercise 5.5 36
5.6 Power Series 37
5.6.1 The Radius and Interval of Convergence 37
5.6.2 Properties of Power Series 41
5.6.3 Expanding Functions into Power Series 45
Exercise 5.6 55
5.7 Fourier Series 56
5.7.1 The Concept of Fourier Series 56
5.7.2 Fourier Sine and Cosine Series 62
5.7.3 Expanding Functions with Arbitrary Period 65
Exercise 5.7 68
Review and Exercise 69
Chapter 6 Vectors and Analytic Geometry in Space 72
6.1 Vectors 72
6.1.1 Vectors 72
6.1.2 Linear Operations on Vectors 73
6.1.3 Dot Products and Cross Product 75
Exercise 6.1 79
6.2 Operations on Vectors in Cartesian Coordinates in Three Space 80
6.2.1 Cartesian Coordinates in Three Space 80
6.2.2 Operations on Vectors in Cartesian Coordinates 84
Exercise 6.2 88
6.3 Planes and Lines in Space 89
6.3.1 Equations for Plane 89
6.3.2 Lines 92
6.3.3 Some Problems Related to Lines and Planes 95
Exercise 6.3 100
6.4 Curves and Surfaces in Space 101
6.4.1 Sphere and Cylinder 101
6.4.2 Curves in Space 103
6.4.3 Surfaces of Revolution 105
6.4.4 Quadric Surfaces 106
Exercise 6.4 109
Exercise Review 110
Chapter 7 Multivariable Functions and Partial Derivatives 113
7.1 Functions of Several Variables 113
Exercise 7.1 116
7.2 Limits and Continuity 116
Exercise 7.2 120
7.3 Partial Derivative 121
7.3.1 Partial Derivative 121
7.3.2 Second Order Partial Derivatives 123
Exercise 7.3 126
7.4 Differentials 128
Exercise 7.4 132
7.5 Rules for Finding Partial Derivative 133
7.5.1 The Chain Rule 133
7.5.2 Implicit Differentiation 137
Exercise 7.5 140
7.6 Direction Derivatives,Gradient Vectors 142
7.6.1 Direction Derivatives 142
7.6.2 Gradient Vectors 144
Exercise 7.6 146
7.7 Geometric Applications of Differentiation of Functions of Several Variables 147
7.7.1 Tangent Line and Normal Plan to a Curve 147
7.7.2 Tangent Plane and Normal Line to a Surface 149
Exercise 7.7 152
7.8 Taylor Formula for Functions of Two Variables and Extreme Values 153
7.8.1 Taylor Formula for Functions of Two Variables 153
7.8.2 Extreme Values 155
7.8.3 Absolute Maxima and Minima on Closed Bounded Regions 160
7.8.4 Lagrange Multipliers 161
Exercise 7.8 164
Exercise Review 166
Chapter 8 Multiple Integrals 172
8.1 Concept and Properties of Multiple Integrals 172
8.2 Evaluation of Double Integrals 174
8.2.1 Double Integrals in Rectangular Coordinates 174
8.2.2 Double Integrals in Polar Coordinates 178
8.2.3 Substitutions in Double Integrals 182
Exercise 8.2 185
8.3 Evaluation of Triple Integrals 188
8.3.1 Triple Integrals in Rectangular Coordinates 188
8.3.2 Triple Integrals in Cylindrical and Spherical Coordinates 192
Exercise 8.3 196
8.4 Evaluation of Line Integral with Respect to Arc Length 197
Exercise 8.4 199
8.5 Evaluation of Surface Integrals with Respect to Area 200
8.5.1 Surface Area 200
8.5.2 Evaluation of Surface Integrals with Respect to Area 202
Exercise 8.5 204
8.6 Application for the Integrals 205
Exercise 8.6 208
Review and Exercise 209
Chapter 9 Integration in Vectors Field 213
9.1 Vector Fields 213
Exercise 9.1 215
9.2 Line Integrals of the Second Type 216
9.2.1 The Concept and Properties of the Line Integrals of the Second Type 216
9.2.2 Calculation 218
9.2.3 The Relation between the Two Line Integrals 221
Exercise 9.2 221
9.3 Green Theorem in the Plane 222
9.3.1 Green Theorem 223
9.3.2 Path Independence for the Plane Case 228
Exercise 9.3 232
9.4 The Surface Integral for Flux 234
9.4.1 Orientation 234
9.4.2 The Conception of the Surface Integral for Flux 235
9.4.3 Calculation 237
9.4.4 The Relation between the Two Surface Integrals 240
Exercise 9.4 241
9.5 Gauss Divergence Theorem 242
Exercise 9.5 246
9.6 Stoke Theorem 247
9.6.1 Stoke Theorem 247
9.6.2 Path Independence in Three-space 251
Exercise 9.6 252
Review and Exercise 252
Chapter 10 Complex Analysis 255
10.1 Complex Numbers 255
Exercise 10.1 257
10.2 Complex Functions 259
10.2.1 Complex Valued Functions 259
10.2.2 Limits 259
10.2.3 Continuity 261
Exercise 10.2 263
10.3 Differential Calculus of Complex Functions 264
10.3.1 Derivatives 264
10.3.2 Analytic Functions 268
10.3.3 Elementary Functions 272
Exercise 10.3 276
10.4 Complex Integration 279
10.4.1 Complex Integration 279
10.4.2 Cauchy-Goursat Theorem and Deformation Theorem 282
10.4.3 Cauchy Integral Formula and Cauchy Integral Formula for Derivatives 289
Exercise 10.4 292
10.5 Series Expansion of Complex Function 295
10.5.1 Sequences of Functions 296
10.5.2 Taylor Series 297
10.5.3 Laurent Series 299
Exercise 10.5 304
10.6 Singularities and Residue 307
10.6.1 Singularities and Poles 307
10.6.2 Cauchy Residue Theorem 311
10.6.3 Evaluation of Real Integrals 316
Exercise 10.6 320
Exercise Review 323