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Chapter 5 Infinite Series1

5.1 Infinite Series1

5.1.1 The Concept of Infinite Series1

5.1.2 Conditions for Convergence3

5.1.3 Properties of Series5

Exercise 5.18

5.2 Tests for Convergence of Positive Series9

Exercise 5.218

5.3 Alternating Series,Absolute Convergence,and Conditional Convergence19

5.3.1 Alternating Series19

5.3.2 Absolute Convergence and Conditional Convergence21

Exercise 5.323

5.4 Tests for Improper Integrals24

5.4.1 Tests for the Improper Integrals:Infinite Limits of Integration24

5.4.2 Tests for the Improper Integrals:Infinite Integrands26

5.4.3 The Gamma Function28

Exercise 5.430

5.5 Infinite Series of Functions31

5.5.1 General Definitions31

5.5.2 Uniform Convergence of Series32

5.5.3 Properties of Uniformly Convergent Functional Series34

Exercise 5.536

5.6 Power Series37

5.6.1 The Radius and Interval of Convergence37

5.6.2 Properties of Power Series41

5.6.3 Expanding Functions into Power Series45

Exercise 5.655

5.7 Fourier Series56

5.7.1 The Concept of Fourier Series56

5.7.2 Fourier Sine and Cosine Series62

5.7.3 Expanding Functions with Arbitrary Period65

Exercise 5.768

Review and Exercise69

Chapter 6 Vectors and Analytic Geometry in Space72

6.1 Vectors72

6.1.1 Vectors72

6.1.2 Linear Operations on Vectors73

6.1.3 Dot Products and Cross Product75

Exercise 6.179

6.2 Operations on Vectors in Cartesian Coordinates in Three Space80

6.2.1 Cartesian Coordinates in Three Space80

6.2.2 Operations on Vectors in Cartesian Coordinates84

Exercise 6.288

6.3 Planes and Lines in Space89

6.3.1 Equations for Plane89

6.3.2 Lines92

6.3.3 Some Problems Related to Lines and Planes95

Exercise 6.3100

6.4 Curves and Surfaces in Space101

6.4.1 Sphere and Cylinder101

6.4.2 Curves in Space103

6.4.3 Surfaces of Revolution105

6.4.4 Quadric Surfaces106

Exercise 6.4109

Exercise Review110

Chapter 7 Multivariable Functions and Partial Derivatives113

7.1 Functions of Several Variables113

Exercise 7.1116

7.2 Limits and Continuity116

Exercise 7.2120

7.3 Partial Derivative121

7.3.1 Partial Derivative121

7.3.2 Second Order Partial Derivatives123

Exercise 7.3126

7.4 Differentials128

Exercise 7.4132

7.5 Rules for Finding Partial Derivative133

7.5.1 The Chain Rule133

7.5.2 Implicit Differentiation137

Exercise 7.5140

7.6 Direction Derivatives,Gradient Vectors142

7.6.1 Direction Derivatives142

7.6.2 Gradient Vectors144

Exercise 7.6146

7.7 Geometric Applications of Differentiation of Functions of Several Variables147

7.7.1 Tangent Line and Normal Plan to a Curve147

7.7.2 Tangent Plane and Normal Line to a Surface149

Exercise 7.7152

7.8 Taylor Formula for Functions of Two Variables and Extreme Values153

7.8.1 Taylor Formula for Functions of Two Variables153

7.8.2 Extreme Values155

7.8.3 Absolute Maxima and Minima on Closed Bounded Regions160

7.8.4 Lagrange Multipliers161

Exercise 7.8164

Exercise Review166

Chapter 8 Multiple Integrals172

8.1 Concept and Properties of Multiple Integrals172

8.2 Evaluation of Double Integrals174

8.2.1 Double Integrals in Rectangular Coordinates174

8.2.2 Double Integrals in Polar Coordinates178

8.2.3 Substitutions in Double Integrals182

Exercise 8.2185

8.3 Evaluation of Triple Integrals188

8.3.1 Triple Integrals in Rectangular Coordinates188

8.3.2 Triple Integrals in Cylindrical and Spherical Coordinates192

Exercise 8.3196

8.4 Evaluation of Line Integral with Respect to Arc Length197

Exercise 8.4199

8.5 Evaluation of Surface Integrals with Respect to Area200

8.5.1 Surface Area200

8.5.2 Evaluation of Surface Integrals with Respect to Area202

Exercise 8.5204

8.6 Application for the Integrals205

Exercise 8.6208

Review and Exercise209

Chapter 9 Integration in Vectors Field213

9.1 Vector Fields213

Exercise 9.1215

9.2 Line Integrals of the Second Type216

9.2.1 The Concept and Properties of the Line Integrals of the Second Type216

9.2.2 Calculation218

9.2.3 The Relation between the Two Line Integrals221

Exercise 9.2221

9.3 Green Theorem in the Plane222

9.3.1 Green Theorem223

9.3.2 Path Independence for the Plane Case228

Exercise 9.3232

9.4 The Surface Integral for Flux234

9.4.1 Orientation234

9.4.2 The Conception of the Surface Integral for Flux235

9.4.3 Calculation237

9.4.4 The Relation between the Two Surface Integrals240

Exercise 9.4241

9.5 Gauss Divergence Theorem242

Exercise 9.5246

9.6 Stoke Theorem247

9.6.1 Stoke Theorem247

9.6.2 Path Independence in Three-space251

Exercise 9.6252

Review and Exercise252

Chapter 10 Complex Analysis255

10.1 Complex Numbers255

Exercise 10.1257

10.2 Complex Functions259

10.2.1 Complex Valued Functions259

10.2.2 Limits259

10.2.3 Continuity261

Exercise 10.2263

10.3 Differential Calculus of Complex Functions264

10.3.1 Derivatives264

10.3.2 Analytic Functions268

10.3.3 Elementary Functions272

Exercise 10.3276

10.4 Complex Integration279

10.4.1 Complex Integration279

10.4.2 Cauchy-Goursat Theorem and Deformation Theorem282

10.4.3 Cauchy Integral Formula and Cauchy Integral Formula for Derivatives289

Exercise 10.4292

10.5 Series Expansion of Complex Function295

10.5.1 Sequences of Functions296

10.5.2 Taylor Series297

10.5.3 Laurent Series299

Exercise 10.5304

10.6 Singularities and Residue307

10.6.1 Singularities and Poles307

10.6.2 Cauchy Residue Theorem311

10.6.3 Evaluation of Real Integrals316

Exercise 10.6320

Exercise Review323