高等数学(下册)ADVANCED MATHEMATICSⅡ(英文)/潘斌

Chapter 8 Vector algebra and analytic geometry of space1

8.1Vectors and their linear operations1

8.1.1The concept of vector1

8.1.2Vector linear operations2

8.1.3Three-dimensional rectangular coordinate system6

8.1.4Component representation of vector linear operations8

8.1.5Length，direction angles and projection of a vector9

Exercises 8-1 12

8.2Multiplicative operations on vectors12

8.2.1The scalar product(dot product，inner product)of two vectors13

8.2.2The vector product(cross product，outer product)of two vectors15

*8.2.3The mixed product of three vectors17

Exercises 8-2 19

8.3Surfaces and their equations19

8.3.1Definition of surface equations19

8.3.2Surfaces of revolution21

8.3.3Cylinders22

8.3.4Quadric surfaces24

Exercises 8-3 26

8.4Space curves and their equations27

8.4.1General form of equations of space curves27

8.4.2Parametric equations of space curves28

*8.4.3Parametric equations of a surface29

8.4.4Projections of space curves on coordinate planes30

Exercises 8-4 31

8.5Plane and its equation32

8.5.1Point-normal form of the equation of a plane32

8.5.2General form of the equation of a plane33

8.5.3The included angle between two planes34

Exercises 8-5 36

8.6Straight line in space and its equation36

8.6.1General form of the equations of a straight line36

8.6.2Parametric equations and symmetric form equations of a straight line37

8.6.3The included angel between two lines38

8.6.4The included angle between a line and a plane38

8.6.5Some examples39

Exercises 8-6 41

Exercises 8 42



Chapter 9 The multivariable differential calculus and its applications44

9.1Basic concepts of multivariable functions44

9.1.1Planar sets n-dimensional space44

9.1.2The concept of a multivariable function47

9.1.3Limits of multivariable functions49

9.1.4Continuity of multivariable functions51

Exercises 9-1 52

9.2Partial derivatives53

9.2.1Definition and computation of partial derivatives53

9.2.2Higher-order partial derivatives57

Exercises 9-2 59

9.3Total differentials60

9.3.1Definition of total differential60

9.3.2Applications of the total differential to approximate computation63

Exercises 9-3 64

9.4Differentiation of multivariable composite functions65

9.4.1Composition of functions of one variable and multivariable functions65

9.4.2Composition of multivariable functions and multivariable functions66

9.4.3Other case66

Exercises 9-4 70

9.5Differentiation of implicit functions71

9.5.1Case of one equation71

9.5.2Case of system of equations73

Exercises 9-5 75

9.6Applications of differential calculus of multivariable functions in geometry76

9.6.1Derivatives and differentials of vector-valued functions of one variable77

9.6.2Tangent line and normal plane to a space curve80

9.6.3Tangent plane and normal line of surfaces82

Exercises 9-6 85

9.7Directlorial derivatives and gradient85

9.7.1Directlorial derivatives85

9.7.2Gradient88

Exercises 9-7 91

9.8Extreme value problems for multivariable functions92

9.8.1Unrestricted extreme values and global maxima and minima92

9.8.2Extreme values with constraints the method of Lagrange multipliers96

Exercises 9-8 99

9.9Taylor formula for functions of two variables100

9.9.1Taylor formula for functions of two variables100

9.9.2Proof of the sufficient condition for extreme values of function of two variables101

Exercises 9-9 102

Exercises 9 102



Chapter 10 Multiple integrals105

10.1The concept and properties of double integrals105

10.1.1The concept of double integrals105

10.1.2Properties of Double Integrals108

Exercises 10-1 109

10.2Computation of double integrals110

10.2.1Computation of double integrals in rectangular coordinates110

10.2.2Computation of double integrals in polar coordinates115

*10.2.3Integration by substitution for double integrals119

Exercises 10-2 123

10.3Triple integrals126

10.3.1Concept of triple integrals126

10.3.2Computation of triple integrals127

Exercises 10-3 132

10.4Application of multiple integrals134

10.4.1Area of a surface134

10.4.2Center of mass136

10.4.3Moment of inertia138

10.4.4Gravitational force139

Exercises 10-4 140

*10.5Integral with parameter142

*Exercises 10-5 145

Exercises 10 146



Chapter 11 Line and surface integrals148

11.1Line integrals with respect to arc lengths148

11.1.1The concept and properties of the line integral with respect to arc lengths148

11.1.2Computation of line integral with respect to arc lengths149

Exercises 11-1 152

11.2Line integrals with respect to coordinates152

11.2.1The concept and properties of the line integrals with respect to coordinates152

11.2.2Computation of line integrals with respect to coordinates155

11.2.3The relationship between the two types of line integral158

Exercises 11-2 158

11.3Green’s formula and the application to fields159

11.3.1Green’s formula159

11.3.2The conditions for a planar line integral to have independence of path163

11.3.3Quadrature problem of the total differential165

Exercises 11-3 169

11.4Surface integrals with respect to acreage170

11.4.1The concept and properties of the surface integral with respect to acreage170

11.4.2Computation of surface integrals with respect to acreage171

Exercises 11-4 173

11.5Surface integrals with respect to coordinates174

11.5.1The concept and properties of the surface integrals with respect to coordinates174

11.5.2Computation of surface integrals with respect to coordinates177

11.5.3The relationship between the two types of surface integral180

Exercises 11-5 181

11.6Gauss’formula181

11.6.1Gauss’formula181

*11.6.2Flux and divergence184

Exercises 11-6 185

11.7Stokes formula186

11.7.1Stokes formula186

11.7.2Circulation and rotation187

Exercises 11-7 188

Exercises 11 188



Chapter 12 Infinite series191

12.1Concepts and properties of series with constant terms191

12.1.1Concepts of series with constant terms191

12.1.2Properties of convergence with series193

*12.1.3Cauchy’s convergence principle195

Exercises 12-1 196

12.2Convergence tests for series with constant terms197

12.2.1Convergence tests for series of positive terms197

12.2.2Alternating series and Leibniz’s test202

12.2.3Absolute and conditional convergence203

Exercises 12-2 204

12.3Power series205

12.3.1Concepts of series of functions205

12.3.2Power series and convergence of power series206

12.3.3Operations on power series211

Exercises 12-3 212

12.4Expansion of functions in power series213

Exercises 12-4 219

12.5Application of expansion of functions in power series219

12.5.1Approximations by power series219

12.5.2Power series solutions of differential equation221

12.5.3Euler formula222

Exercises 12-5 223

12.6Fourier series223

12.6.1Trigonometric series and orthogonality of the system of trigonometric functions223

12.6.2Expand a function into a Fourier series225

12.6.3Expand a function into the sine series and cosine series229

Exercises 12-6 232

12.7The Fourier series of a function of period 2l 233

Exercises 12-7 235

Exercises 12 235



References237