Calculus 2【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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CHAPTER 5 Vectors and the Geometry of Space1

5.1 Vectors1

5.1.1 Concepts of Vectors1

5.1.2 Linear Operations Involving Vectors2

5.1.3 Coordinate Systems in Three-Dimensional Space3

5.1.4 Representing Vectors Using Coordinates6

5.1.5 Lengths,Direction Angles and Projections of Vectors8

5.2 Dot Product,Cross Product and Scalar Triple Product11

5.2.1 The Dot Product11

5.2.2 The Cross Product15

5.2.3 Scalar Triple Product17

5.3 Equations of Planes and Lines18

5.3.1 Planes18

5.3.2 Lines22

5.4 Surfaces In Space26

5.4.1 Surfaces and Equations26

5.4.2 Cylinder27

5.4.3 Surface of Revolution28

5.4.4 Quadric Surfaces29

5.5 Curves in Space30

5.5.1 General Equations of Curves in the Space30

5.5.2 Parametric Equations of Curves in the Space31

5.5.3 Parametric Equations of Surfaces in the Space32

5.5.4 Projections of Curves in the Space32

5.6 Exercises34

5.6.1 Vectors34

5.6.2 Planes and Lines in Space35

5.6.3 Surfaces and Curves in Space36

5.6.4 Questions to Guide Your Revision37

CHAPTER 6 Functions of Several Variables38

6.1 Functions of Several Variables38

6.1.1 Definition38

6.1.2 Limits41

6.1.3 Continuity45

6.2 Partial Derivatives47

6.2.1 Definition47

6.2.2 Partial Derivative of Higher Order51

6.3 Total Differential53

6.3.1 Definition53

6.3.2 The Total Differential Approximation55

6.4 The Chain Rule57

6.5 Implicit Differentiation62

6.5.1 Functions Defined by a Single Equation62

6.5.2 Functions Defined Implicitly by System of Equations64

6.6 Applications of the Differential Calculus66

6.6.1 Tangent Lines and Normal Planes66

6.6.2 Tangent Planes and Normal Lines for Surfaces68

6.7 Directional Derivatives and Gradient Vectors72

6.8 Maximum and Minimum76

6.8.1 Extrema of Functions of Several Variables76

6.8.2 Lagrange Multipliers81

6.9 Additional Materials84

6.9.1 Taylor's Theorem for Functions of Two Variables84

6.9.2 Clairaut84

6.9.3 Cobb-Douglas Production Function85

6.10 Exercises86

6.10.1 Functions of Several Variables86

6.10.2 Applications of Partial Derivatives88

6.10.3 Questions to Guide Your Revision90

CHAPTER 7 Multiple Integrals92

7.1 Definition and Properties92

7.2 Iterated Integrals96

7.2.1 Iterated Integrals in Rectangular Coordinates96

7.2.2 Change of Variables Formula for Double Integrals101

7.3 Triple Integrals104

7.3.1 Triple Integrals in Rectangular Coordinates104

7.3.2 Change of Variables in Triple Integrals109

7.4 The Area of a Surface115

7.5 Additional Materials117

7.6 Exercises118

7.6.1 Double Integrals118

7.6.2 Triple Integrals120

7.6.3 Applications of Multiple Integrals120

7.6.4 Questions to Guide Your Revision121

CHAPTER 8 Line and Surface Integrals123

8.1 Line Integrals123

8.1.1 Introduction123

8.1.2 Definition of the Line Integral with Respect to Arc Length124

8.1.3 Evaluating Line Integrals,？cf(x,y)ds,in R2125

8.1.4 Evaluating Line Integrals,？cf(x,y,z)ds,in R3125

8.2 Vector Fields,Work,and Flows128

8.2.1 Introduction128

8.2.2 The Line Integral of a Vector Field Along a Curve C129

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

8.2.4 Examples of Line Integrals132

8.3 Green's Theorem in R2135

8.3.1 The Circulation-Curl Form of Green's Theorem135

8.3.2 The Divergence-Flux Form of Green's Theorem137

8.3.3 Generalized Green's Theorem141

8.4 Path Independent Line Integrals and Conservative Fields142

8.4.1 Introduction142

8.4.2 Fundamental Results on Path Independent Line Integrals143

8.5 Surface Integrals150

8.5.1 Definition of Integration With Respect to Surface Area150

8.5.2 Evaluation of Surface Integrals151

8.6 Surface Integrals of Vector Fields156

8.6.1 Definition and Properties of Flux,？s？·？dS156

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

8.7 The Divergence Theorem163

8.7.1 Introduction163

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

8.8 Stoke's Theorem169

8.9 Additional Materials173

8.9.1 Green173

8.9.2 Gauss174

8.9.3 Stokes175

8.10 Exercises176

8.10.1 Line Integrals176

8.10.2 Surface Integrals177

8.10.3 Questions to Guide Your Revision178

CHAPTER 9 Infinite Sequences,Series and Approximations179

9.1 Infinite Sequences179

9.2 Infinite Series181

9.2.1 Definition of Infinite Series181

9.2.2 Properties of Convergent Series183

9.3 Tests for Convergence186

9.3.1 Series with Nonnegative Terms186

9.3.2 Series with Negative and Positive Terms194

9.4 Power Series and Taylor Series198

9.4.1 Power Series198

9.4.2 Working with Power Series204

9.4.3 Taylor Series206

9.4.4 Applications of Power Series214

9.5 Fourier Series215

9.5.1 Fourier Series Expansion with Period 2π216

9.5.2 Fourier Cosine and Sine Series with Period 2π220

9.5.3 The Fourier Series Expansion with Period 2l222

9.5.4 Fourier Series with Complex Terms224

9.6 Additional Materials225

9.6.1 Fourier225

9.6.2 Maclaurin226

9.6.3 Taylor227

9.7 Exercises228

9.7.1 Series with Constant Terms228

9.7.2 Power Series229

9.7.3 Fourier Series229

9.7.4 Questions to Guide Your Revision230

CHAPTER 10 Introduction to Ordinary Differential Equation231

10.1 Differential Equations and Mathematical Models231

10.2 Methods for Solving Ordinary Differential Equations237

10.2.1 Separable Equations237

10.2.2 Substitution Methods239

10.2.3 Exact Differential Equations240

10.2.4 Linear First-Order Differential Equations and Integrating Factors242

10.2.5 Reducible Second-Order Equations246

10.2.6 Linear Second-Order Differential Equations249

10.3 Other Ways of Solving Differential Equations262

10.3.1 Power Series Method262

10.3.2 Direction Fields263

10.3.3 Numerical Approximation:Euler's Method265

10.4 Additional Materials269

10.4.1 Euler269

10.4.2 Bernoulli270

10.4.3 The Bernoulli Family270

10.4.4 Development of Calculus271

10.5 Exercises271

10.5.1 Introduction to Differential Equations271

10.5.2 First Order Differential Equation271

10.5.3 Second Order Differential Equation272

10.5.4 Questions to Guide Your Revision272

Answers274

Reterence Books284