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CHAPTER 5 Vectors and the Geometry of Space 1

5.1 Vectors 1

5.1.1 Concepts of Vectors 1

5.1.2 Linear Operations Involving Vectors 2

5.1.3 Coordinate Systems in Three-Dimensional Space 3

5.1.4 Representing Vectors Using Coordinates 6

5.1.5 Lengths,Direction Angles and Projections of Vectors 8

5.2 Dot Product,Cross Product and Scalar Triple Product 11

5.2.1 The Dot Product 11

5.2.2 The Cross Product 15

5.2.3 Scalar Triple Product 17

5.3 Equations of Planes and Lines 18

5.3.1 Planes 18

5.3.2 Lines 22

5.4 Surfaces In Space 26

5.4.1 Surfaces and Equations 26

5.4.2 Cylinder 27

5.4.3 Surface of Revolution 28

5.4.4 Quadric Surfaces 29

5.5 Curves in Space 30

5.5.1 General Equations of Curves in the Space 30

5.5.2 Parametric Equations of Curves in the Space 31

5.5.3 Parametric Equations of Surfaces in the Space 32

5.5.4 Projections of Curves in the Space 32

5.6 Exercises 34

5.6.1 Vectors 34

5.6.2 Planes and Lines in Space 35

5.6.3 Surfaces and Curves in Space 36

5.6.4 Questions to Guide Your Revision 37

CHAPTER 6 Functions of Several Variables 38

6.1 Functions of Several Variables 38

6.1.1 Definition 38

6.1.2 Limits 41

6.1.3 Continuity 45

6.2 Partial Derivatives 47

6.2.1 Definition 47

6.2.2 Partial Derivative of Higher Order 51

6.3 Total Differential 53

6.3.1 Definition 53

6.3.2 The Total Differential Approximation 55

6.4 The Chain Rule 57

6.5 Implicit Differentiation 62

6.5.1 Functions Defined by a Single Equation 62

6.5.2 Functions Defined Implicitly by System of Equations 64

6.6 Applications of the Differential Calculus 66

6.6.1 Tangent Lines and Normal Planes 66

6.6.2 Tangent Planes and Normal Lines for Surfaces 68

6.7 Directional Derivatives and Gradient Vectors 72

6.8 Maximum and Minimum 76

6.8.1 Extrema of Functions of Several Variables 76

6.8.2 Lagrange Multipliers 81

6.9 Additional Materials 84

6.9.1 Taylor's Theorem for Functions of Two Variables 84

6.9.2 Clairaut 84

6.9.3 Cobb-Douglas Production Function 85

6.10 Exercises 86

6.10.1 Functions of Several Variables 86

6.10.2 Applications of Partial Derivatives 88

6.10.3 Questions to Guide Your Revision 90

CHAPTER 7 Multiple Integrals 92

7.1 Definition and Properties 92

7.2 Iterated Integrals 96

7.2.1 Iterated Integrals in Rectangular Coordinates 96

7.2.2 Change of Variables Formula for Double Integrals 101

7.3 Triple Integrals 104

7.3.1 Triple Integrals in Rectangular Coordinates 104

7.3.2 Change of Variables in Triple Integrals 109

7.4 The Area of a Surface 115

7.5 Additional Materials 117

7.6 Exercises 118

7.6.1 Double Integrals 118

7.6.2 Triple Integrals 120

7.6.3 Applications of Multiple Integrals 120

7.6.4 Questions to Guide Your Revision 121

CHAPTER 8 Line and Surface Integrals 123

8.1 Line Integrals 123

8.1.1 Introduction 123

8.1.2 Definition of the Line Integral with Respect to Arc Length 124

8.1.3 Evaluating Line Integrals,？cf(x,y)ds,in R2 125

8.1.4 Evaluating Line Integrals,？cf(x,y,z)ds,in R3 125

8.2 Vector Fields,Work,and Flows 128

8.2.1 Introduction 128

8.2.2 The Line Integral of a Vector Field Along a Curve C 129

8.2.3 Different Forms of the Line Integral Including ？c？·d？ 131

8.2.4 Examples of Line Integrals 132

8.3 Green's Theorem in R2 135

8.3.1 The Circulation-Curl Form of Green's Theorem 135

8.3.2 The Divergence-Flux Form of Green's Theorem 137

8.3.3 Generalized Green's Theorem 141

8.4 Path Independent Line Integrals and Conservative Fields 142

8.4.1 Introduction 142

8.4.2 Fundamental Results on Path Independent Line Integrals 143

8.5 Surface Integrals 150

8.5.1 Definition of Integration With Respect to Surface Area 150

8.5.2 Evaluation of Surface Integrals 151

8.6 Surface Integrals of Vector Fields 156

8.6.1 Definition and Properties of Flux,？s？·？dS 156

8.6.2 Evaluating？·？dS for a Surfacez=z(x,y) 157

8.7 The Divergence Theorem 163

8.7.1 Introduction 163

8.7.2 Physical interpretation of the Divergence ？·？(x,y,z) 164

8.8 Stoke's Theorem 169

8.9 Additional Materials 173

8.9.1 Green 173

8.9.2 Gauss 174

8.9.3 Stokes 175

8.10 Exercises 176

8.10.1 Line Integrals 176

8.10.2 Surface Integrals 177

8.10.3 Questions to Guide Your Revision 178

CHAPTER 9 Infinite Sequences,Series and Approximations 179

9.1 Infinite Sequences 179

9.2 Infinite Series 181

9.2.1 Definition of Infinite Series 181

9.2.2 Properties of Convergent Series 183

9.3 Tests for Convergence 186

9.3.1 Series with Nonnegative Terms 186

9.3.2 Series with Negative and Positive Terms 194

9.4 Power Series and Taylor Series 198

9.4.1 Power Series 198

9.4.2 Working with Power Series 204

9.4.3 Taylor Series 206

9.4.4 Applications of Power Series 214

9.5 Fourier Series 215

9.5.1 Fourier Series Expansion with Period 2π 216

9.5.2 Fourier Cosine and Sine Series with Period 2π 220

9.5.3 The Fourier Series Expansion with Period 2l 222

9.5.4 Fourier Series with Complex Terms 224

9.6 Additional Materials 225

9.6.1 Fourier 225

9.6.2 Maclaurin 226

9.6.3 Taylor 227

9.7 Exercises 228

9.7.1 Series with Constant Terms 228

9.7.2 Power Series 229

9.7.3 Fourier Series 229

9.7.4 Questions to Guide Your Revision 230

CHAPTER 10 Introduction to Ordinary Differential Equation 231

10.1 Differential Equations and Mathematical Models 231

10.2 Methods for Solving Ordinary Differential Equations 237

10.2.1 Separable Equations 237

10.2.2 Substitution Methods 239

10.2.3 Exact Differential Equations 240

10.2.4 Linear First-Order Differential Equations and Integrating Factors 242

10.2.5 Reducible Second-Order Equations 246

10.2.6 Linear Second-Order Differential Equations 249

10.3 Other Ways of Solving Differential Equations 262

10.3.1 Power Series Method 262

10.3.2 Direction Fields 263

10.3.3 Numerical Approximation:Euler's Method 265

10.4 Additional Materials 269

10.4.1 Euler 269

10.4.2 Bernoulli 270

10.4.3 The Bernoulli Family 270

10.4.4 Development of Calculus 271

10.5 Exercises 271

10.5.1 Introduction to Differential Equations 271

10.5.2 First Order Differential Equation 271

10.5.3 Second Order Differential Equation 272

10.5.4 Questions to Guide Your Revision 272

Answers 274

Reterence Books 284

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