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Chapter 7 Differential Equations 1

7.1 Basic Concepts of Differential Equations 1

7.1.1 Examples of Differential Equations 1

7.1.2 Basic Concepts 3

7.1.3 Geometric Interpretation of the First-Order Differential Equation 4

Exercises 7.1 5

7.2 First-Order Differential Equations 6

7.2.1 First-Order Separable Differential Equation 6

7.2.2 Homogeneous First-Order Equations 8

7.2.3 Linear First-Order Equations 10

7.2.4 Bernoulli's Equation 13

7.2.5 Some Other Examples that can be Reduced to Linear First-Order Equations 14

Exercises 7.2 16

7.3 Reducible Second-Order Differential Equations 17

Exercises 7.3 21

7.4 Higher-Order Linear Differential Equations 22

7.4.1 Some Examples of Linear Differential Equation of Higher-Order 22

7.4.2 Structure of Solutions of Linear Differential Equations 24

Exercises 7.4 28

7.5 Higher-Order Linear Equations with Constant Coefficients 29

7.5.1 Higher-Order Homogeneous Linear Equations with Constant Coefficients 29

7.5.2 Higher-Order Nonhomogeneous Linear Equations with Constant Coefficients 33

Exercises 7.5 39

7.6 Euler's Differential Equation 40

Exercises 7.6 42

7.7 Applications of Differential Equations 42

Exercises 7.7 46

Chapter 8 Vectors and Solid Analytic Geometry 54

8.1 Vectors in Plane and in Space 54

8.1.1 Vectors 54

8.1.2 Operations on Vectors 56

8.1.3 Vectors in Plane 59

8.1.4 Rectangular Coordinate System 61

8.1.5 Vectors in Space 63

Exercises 8.1 66

Part A 66

Part B 67

8.2 Products of Vectors 68

8.2.1 Scalar Product of two Vectors 68

8.2.2 Vector Product of two Vectors 72

8.2.3 Triple Scalar Product of three Vectors 77

8.2.4 Applications of Products of Vectors 80

Exercises 8.2 83

Part A 83

Part B 84

8.3 Planes and Lines in Space 85

8.3.1 Equations of Planes 85

8.3.2 Equations of Lines in Space 90

Exercises 8.3 96

Part A 96

Part B 98

8.4 Surfaces and Space Curves 99

8.4.1 Cylinders 99

8.4.2 Cones 101

8.4.3 Surfaces of Revolution 102

8.4.4 Quadric Surfaces 104

8.4.5 Space Curves 110

8.4.6 Cylindrical Coordinate System 114

8.4.7 Spherical Coordinate System 115

Exercises 8.4 117

Part A 117

Part B 119

Chapter 9 The Differential Calculus for Multi-variable Functions 120

9.1 Definition of Multi-variable Functions and their Basic Properties 120

9.1.1 Space R2 and Rn 120

9.1.2 Multi-variable Functions 128

9.1.3 Visualization of Multi-variable Functions 130

9.1.4 Limits and Continuity of Multi-variable Functions 135

Exercises 9.1 142

Part A 142

Part B 143

9.2 Partial Derivatives and Total Differentials of Multi-variable Functions 143

9.2.1 Partial Derivatives 144

9.2.2 Total Differentials 149

9.2.3 Higher-Order Partial Derivatives 157

9.2.4 Directional Derivatives and the Gradient 159

Exercises 9.2 166

Part A 166

Part B 169

9.3 Differentiation of Multi-variable Composite and Implicit Functions 169

9.3.1 Partial Derivatives and total Differentials of Multi-variable Composite Functions 170

9.3.2 Differentiation of Implicit Functions 176

9.3.3 Differentiation of Implicit Functions determined by Equation Systems 178

Exercises 9.3 181

Part A 181

Part B 183

Chapter 10 Applications of Multi-variable Functions 184

10.1 Approximate Function Values by total Differential 184

10.2 Extreme Values of Multi-variable Functions 187

10.2.1 Unrestricted Extreme Values 187

10.2.2 Global Maxima and Minima 190

10.2.3 The Method of Least Squares 192

10.2.4 Constrained Extreme Values 194

10.2.5 The Method of Lagrange Multipliers 196

Exercises 10.2 199

Part A 199

Part B 200

10.3 Applications in Geometry 200

10.3.1 Arc Length along a Curve 200

10.3.2 Tangent Line and Normal Plane of a Space Curve 204

10.3.3 Tangent Planes and Normal Lines to a Surface 209

10.3.4 Curvature for Plane Curves 213

Exercises 10.3 214

Part A 214

Part B 217

Synthetic exercises 217

Chapter 11 Multiple Integrals 219

11.1 Concept and Properties of Double Integrals 219

11.1.1 Concept of Double Integrals 219

11.1.2 Properties of Double Integrals 222

Exercises 11.1 223

11.2 Evaluation of Double Integrals 224

11.2.1 Geometric Meaning of Double Integrals 225

11.2.2 Double Integrals in Rectangular Coordinates 226

11.2.3 Double Integrals in Polar Coordinates 233

11.2.4 Integration by Substitution for Double Integrals in General 241

Exercises 11.2 246

Part A 246

Part B 250

11.3 Triple Integrals 251

11.3.1 Concept and Properties of Triple Integrals 251

11.3.2 Triple Integrals in Rectangular Coordinates 252

11.3.3 Triple Integrals in Cylindrical and Spherical Coordinates 257

11.3.4 Integration by Substitution for Triple Integrals in General 265

Exercises 11.3 266

Part A 266

Part B 269

11.4 Applications of Multiple Integrals 270

11.4.1 Surface Area 271

11.4.2 The Center of Gravity 273

11.4.3 The Moment of Inertia 275

Exercises 11.4 276

Part A 276

Part B 276

Chapter 12 Line Integrals and Surface Integrals 278

12.1 Line Integrals 278

12.1.1 Line Integrals with respect to Arc Length 278

12.1.2 Line Integrals with respect to Coordinates 284

12.1.3 Relations between two Types of Line Integrals 289

Exercises 12.1 289

Part A 289

Part B 292

12.2 Green's Formula and its Applications 294

12.2.1 Green's Formula 294

12.2.2 Conditions for Path Independence of Line Integrals 299

Exercises 12.2 307

Part A 307

Part B 309

12.3 Surface Integrals 311

12.3.1 Surface Integrals with respect to Surface Area 311

12.3.2 Surface Integrals with respect to Coordinates 315

Exercises 12.3 323

Part A 323

Part B 325

12.4 Gauss'Formula 326

Exercises 12.4 331

Part A 331

Part B 332

12.5 Stokes'Formula 332

12.5.1 Stokes'Formula 332

12.5.2 Conditions for Path Independence of Space Line Integrals 335

Exercises 12.5 337

Bibliography 339