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Chapter 7 Differential Equations1

7.1 Basic Concepts of Differential Equations1

7.1.1 Examples of Differential Equations1

7.1.2 Basic Concepts3

7.1.3 Geometric Interpretation of the First-Order Differential Equation4

Exercises 7.15

7.2 First-Order Differential Equations6

7.2.1 First-Order Separable Differential Equation6

7.2.2 Homogeneous First-Order Equations8

7.2.3 Linear First-Order Equations10

7.2.4 Bernoulli's Equation13

7.2.5 Some Other Examples that can be Reduced to Linear First-Order Equations14

Exercises 7.216

7.3 Reducible Second-Order Differential Equations17

Exercises 7.321

7.4 Higher-Order Linear Differential Equations22

7.4.1 Some Examples of Linear Differential Equation of Higher-Order22

7.4.2 Structure of Solutions of Linear Differential Equations24

Exercises 7.428

7.5 Higher-Order Linear Equations with Constant Coefficients29

7.5.1 Higher-Order Homogeneous Linear Equations with Constant Coefficients29

7.5.2 Higher-Order Nonhomogeneous Linear Equations with Constant Coefficients33

Exercises 7.539

7.6 Euler's Differential Equation40

Exercises 7.642

7.7 Applications of Differential Equations42

Exercises 7.746

Chapter 8 Vectors and Solid Analytic Geometry54

8.1 Vectors in Plane and in Space54

8.1.1 Vectors54

8.1.2 Operations on Vectors56

8.1.3 Vectors in Plane59

8.1.4 Rectangular Coordinate System61

8.1.5 Vectors in Space63

Exercises 8.166

Part A66

Part B67

8.2 Products of Vectors68

8.2.1 Scalar Product of two Vectors68

8.2.2 Vector Product of two Vectors72

8.2.3 Triple Scalar Product of three Vectors77

8.2.4 Applications of Products of Vectors80

Exercises 8.283

Part A83

Part B84

8.3 Planes and Lines in Space85

8.3.1 Equations of Planes85

8.3.2 Equations of Lines in Space90

Exercises 8.396

Part A96

Part B98

8.4 Surfaces and Space Curves99

8.4.1 Cylinders99

8.4.2 Cones101

8.4.3 Surfaces of Revolution102

8.4.4 Quadric Surfaces104

8.4.5 Space Curves110

8.4.6 Cylindrical Coordinate System114

8.4.7 Spherical Coordinate System115

Exercises 8.4117

Part A117

Part B119

Chapter 9 The Differential Calculus for Multi-variable Functions120

9.1 Definition of Multi-variable Functions and their Basic Properties120

9.1.1 Space R2 and Rn120

9.1.2 Multi-variable Functions128

9.1.3 Visualization of Multi-variable Functions130

9.1.4 Limits and Continuity of Multi-variable Functions135

Exercises 9.1142

Part A142

Part B143

9.2 Partial Derivatives and Total Differentials of Multi-variable Functions143

9.2.1 Partial Derivatives144

9.2.2 Total Differentials149

9.2.3 Higher-Order Partial Derivatives157

9.2.4 Directional Derivatives and the Gradient159

Exercises 9.2166

Part A166

Part B169

9.3 Differentiation of Multi-variable Composite and Implicit Functions169

9.3.1 Partial Derivatives and total Differentials of Multi-variable Composite Functions170

9.3.2 Differentiation of Implicit Functions176

9.3.3 Differentiation of Implicit Functions determined by Equation Systems178

Exercises 9.3181

Part A181

Part B183

Chapter 10 Applications of Multi-variable Functions184

10.1 Approximate Function Values by total Differential184

10.2 Extreme Values of Multi-variable Functions187

10.2.1 Unrestricted Extreme Values187

10.2.2 Global Maxima and Minima190

10.2.3 The Method of Least Squares192

10.2.4 Constrained Extreme Values194

10.2.5 The Method of Lagrange Multipliers196

Exercises 10.2199

Part A199

Part B200

10.3 Applications in Geometry200

10.3.1 Arc Length along a Curve200

10.3.2 Tangent Line and Normal Plane of a Space Curve204

10.3.3 Tangent Planes and Normal Lines to a Surface209

10.3.4 Curvature for Plane Curves213

Exercises 10.3214

Part A214

Part B217

Synthetic exercises217

Chapter 11 Multiple Integrals219

11.1 Concept and Properties of Double Integrals219

11.1.1 Concept of Double Integrals219

11.1.2 Properties of Double Integrals222

Exercises 11.1223

11.2 Evaluation of Double Integrals224

11.2.1 Geometric Meaning of Double Integrals225

11.2.2 Double Integrals in Rectangular Coordinates226

11.2.3 Double Integrals in Polar Coordinates233

11.2.4 Integration by Substitution for Double Integrals in General241

Exercises 11.2246

Part A246

Part B250

11.3 Triple Integrals251

11.3.1 Concept and Properties of Triple Integrals251

11.3.2 Triple Integrals in Rectangular Coordinates252

11.3.3 Triple Integrals in Cylindrical and Spherical Coordinates257

11.3.4 Integration by Substitution for Triple Integrals in General265

Exercises 11.3266

Part A266

Part B269

11.4 Applications of Multiple Integrals270

11.4.1 Surface Area271

11.4.2 The Center of Gravity273

11.4.3 The Moment of Inertia275

Exercises 11.4276

Part A276

Part B276

Chapter 12 Line Integrals and Surface Integrals278

12.1 Line Integrals278

12.1.1 Line Integrals with respect to Arc Length278

12.1.2 Line Integrals with respect to Coordinates284

12.1.3 Relations between two Types of Line Integrals289

Exercises 12.1289

Part A289

Part B292

12.2 Green's Formula and its Applications294

12.2.1 Green's Formula294

12.2.2 Conditions for Path Independence of Line Integrals299

Exercises 12.2307

Part A307

Part B309

12.3 Surface Integrals311

12.3.1 Surface Integrals with respect to Surface Area311

12.3.2 Surface Integrals with respect to Coordinates315

Exercises 12.3323

Part A323

Part B325

12.4 Gauss'Formula326

Exercises 12.4331

Part A331

Part B332

12.5 Stokes'Formula332

12.5.1 Stokes'Formula332

12.5.2 Conditions for Path Independence of Space Line Integrals335

Exercises 12.5337

Bibliography339