高等数学 下 英文版PDF电子书下载

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