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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