Chapter 6 Differential Equations 1
6.1 Basic Concepts of Differential Equations 1
6.1.1 Examples of Differential Equations 1
6.1.2 Basic Concepts 2
Exercise 6.1 3
6.2 First-order Differential Equations 3
6.2.1 First-order Separable Differential Equation 3
6.2.2 First-order Homogeneous Equation 4
6.2.3 First-order Linear Differential Equation 5
6.2.4 Bernoulli's Equation 7
Exercise 6.2 8
6.3 Order-reducible Differential Equations 9
6.3.1 The Equation of the Form y(n)=f(x) 9
6.3.2 The Equation of the Form y"=f(x,y') 10
6.3.3 The Equation of the Form y"=f(y,y') 11
Exercise 6.3 12
6.4 Second-order Linear Differential Equations 12
Exercise 6.4 14
6.5 Higher-order Linear Equations with Constant Coefficients 14
6.5.1 Second-order Homogeneous Linear Equations with Constant Coefficients 14
6.5.2 Second-order Linear Non-homogeneous Equations with Constant Coefficients 16
Exercise 6.5 19
第6章 微分方程 21
6.1 微分方程的基本概念 21
6.1.1 微分方程的例子 21
6.1.2 基本概念 22
习题6.1 22
6.2 一阶微分方程 23
6.2.1 一阶可分离变量型微分方程 23
6.2.2 一阶齐次方程 24
6.2.3 一阶线性微分方程 25
6.2.4 伯努利方程 27
习题6.2 28
6.3 可降阶的二阶方程 29
6.3.1 形如y(n)=f(x）的微分方程 29
6.3.2 形如y"=f(x,y'）的微分方程 29
6.3.3 形如y"=f(y,y'）的微分方程 30
习题6.3 31
6.4 二阶线性微分方程 31
习题6.4 33
6.5 高阶常系数齐次线性微分方程 33
6.5.1 二阶常系数齐次线性微分方程 33
6.5.2 二阶常系数非齐次线性微分方程 34
习题6.5 38
Chapter 7 Infinite Series 39
7.1 Concepts and Properties of Series with Constant Terms 39
7.1.1 Concepts of Series with Constant Terms 39
7.1.2 Properties of Series with Constant Terms 41
Exercise 7.1 43
7.2 Convergence Tests for Series with Positive Constant Terms 44
Exercise 7.2 47
7.3 Alternating Series 48
Exercise 7.3 50
7.4 Absolute Convergence and the Ratio and Root Tests 50
Exercise 7.4 55
7.5 Power Series 55
7.5.1 Power Series and Its Convergence 55
7.5.2 Operations of Power Series 61
Exercise 7.5 62
7.6 Taylor and Maclaurin Series 63
Exercise 7.6 68
7.7 The Binomial Series 68
Exercise 7.7 71
第7章 无穷级数 72
7.1 常数项级数的概念和性质 72
7.1.1 常数项级数的概念 72
7.1.2 常数项级数的性质 74
习题7.1 75
7.2 正项级数收敛性判别法 76
习题7.2 79
7.3 交错级数 80
习题7.3 81
7.4 绝对收敛性和比值判别法、根式判别法 82
习题7.4 86
7.5 幂级数 87
7.5.1 幂级数及其敛散性 87
7.5.2 幂级数的运算 92
习题7.5 93
7.6 泰勒级数和麦克劳林级数 93
习题7.6 98
7.7 二项式级数 99
习题7.7 101
Chapter 8 Vectors and the Geometry of Space 102
8.1 Three-dimensional Rectangular Coordinate Systems 102
Exercise 8.1 105
8.2 Vectors 105
8.2.1 Combining Vectors 106
8.2.2 Components 107
Exercise 8.2 111
8.3 The Dot Ptroduct 111
8.3.1 Work and the Dot Product 111
8.3.2 Dot Product in Component Form 112
Exercise 8.3 114
8.4 The Cross Product 115
8.4.1 Torque and the Cross Product 115
8.4.2 The Cross Product in Component Form 117
Exercise 8.4 118
8.5 Planes and Lines in Space 119
8.5.1 Equations of Planes 119
8.5.2 Equations of Lines in Space 122
Exercise 8.5 127
8.6 Functions and Surfaces 128
8.6.1 Functions of Two Variables 128
8.6.2 Cylinders 129
8.6.3 Cones 130
8.6.4 Quadric Surfaces 131
8.6.5 Surfaces of Revolution 134
Exercise 8.6 135
8.7 Space Curves 136
Exercise 8.7 138
第8章 向量与空间解析几何 139
8.1 三维直角坐标系 139
习题8.1 142
8.2 向量 142
8.2.1 向量运算 142
8.2.2 分量表示 144
习题8.2 147
8.3 点积 147
8.3.1 功和点积 147
8.3.2 分量形式的点积 148
习题8.3 149
8.4 叉积 150
8.4.1 扭矩和叉积 150
8.4.2 叉积的分量表示形式 152
习题8.4 153
8.5 空间平面和直线 154
8.5.1 平面方程 154
8.5.2 空间直线方程 157
习题8.5 161
8.6 二元函数与曲面 162
8.6.1 二元函数 162
8.6.2 柱面 163
8.6.3 锥面 164
8.6.4 二次曲面 165
8.6.5 旋转曲面 167
习题8.6 168
8.7 空间曲线 169
习题8.7 171
Chapter 9 The Differential Calculus for Multi-variable Functions 172
9.1 Multi-variable Functions 172
9.1.1 Some Related Concepts 173
9.1.2 Limits of Multi-variable Functions 174
9.1.3 Continuity of Functions of Two Variables 175
Exercise 9.1 176
9.2 Partial Derivatives and Higher Order Partial Derivatives 177
9.2.1 Partial Derivatives 178
9.2.2 Higher Order Partial Derivatives 182
Exercise 9.2 184
9.3 Total Differentials of Multi-variable Functions 185
9.3.1 Total Differentials 185
9.3.2 Applications of Total Differential to Approximate Computation 188
Exercise 9.3 189
9.4 Differentiation of Multivariable Composite Functions 189
Exercise 9.4 195
9.5 Implicit Partial Differentiation 196
9.5.1 Differentiation of Implicit Functions Defined by One Equation 196
9.5.2 Differentiation of Implicit Functions Defined by A System of Equations 199
Exercise 9.5 201
9.6 Applications of Differential Calculus of Multivariable Functions in Geometry 202
9.6.1 Tangent Line and Normal Plane to A Space Curve 202
9.6.2 Tangent Plane and Normal Line of Surfaces 207
Exercise 9.6 210
9.7 Directional Derivatives and Gradient 211
9.7.1 Directional Derivatives 211
9.7.2 Gradient 214
Exercise 9.7 216
9.8 Extreme Value Problems for Multivariable Functions 217
9.8.1 Unrestricted Extreme Values 217
9.8.2 Global Maxima and Minima 220
9.8.3 Extreme Values with Constraints:the Method of Lagrange Multipliers 222
Exercise 9.8 228
第9章 多元函数微分学 229
9.1 多元函数 229
9.1.1 一些相关概念 229
9.1.2 多元函数的极限 230
9.1.3 二元函数的连续性 231
习题9.1 232
9.2 偏导数和高阶导数 233
9.2.1 偏导数 233
9.2.2 高阶偏导数 237
习题9.2 239
9.3 多元函数的全微分 239
9.3.1 全微分 239
9.3.2 全微分在近似计算中的应用 243
习题9.3 243
9.4 多元复合函数的微分法 243
习题9.4 249
9.5 多元隐函数的偏导数 250
9.5.1 由一个方程所确定的隐函数的求导法 250
9.5.2 由方程组所确定的隐函数的求导法 252
习题9.5 254
9.6 多元函数的微分法在几何上的应用 255
9.6.1 曲线的切线和法平面 255
9.6.2 曲面的切平面和法线 259
习题9.6 262
9.7 方向导数和梯度 262
9.7.1 方向导数 262
9.7.2 梯度 266
习题9.7 268
9.8 多元函数的极值 268
9.8.1 无条件极值 268
9.8.2 最大值与最小值 271
9.8.3 有条件极值：拉格朗日乘数法 273
习题9.8 277
Chapter 10 Multiple Integrals 278
10.1 The Concept and Properties of Double Integrals 278
10.1.1 The Concept of Double Integrals 278
10.1.2 Properties of Double Integrals 282
Exercise 10.1 284
10.2 Computation of Double Integrals in Rectangular Coordinate System 285
Exercise 10.2 294
10.3 Computation of Double Integrals in Polar Coordinates 296
Exercise 10.3 302
10.4 Triple Integrals 303
10.4.1 Concept of Triple Integrals 303
10.4.2 Properties of Triple Integrals 305
10.4.3 Computation of Triple Integrals in Rectangular Coordinates 305
Exercise 10.4 311
10.5 Triple Integrals in Some Other Coordinates 312
10.5.1 Triple Integrals in Cylindrical Coordinates 312
10.5.2 Triple Integrals in Spherical Coordinates 314
Exercise 10.5 318
第10章 多重积分 319
10.1 二重积分的概念和性质 319
10.1.1 二重积分的概念 319
10.1.2 二重积分的性质 322
习题10.1 324
10.2 直角坐标系下的二重积分的计算 325
习题10.2 334
10.3 极坐标系下的二重积分的计算 335
习题10.3 340
10.4 三重积分 342
10.4.1 三重积分的概念 342
10.4.2 三重积分的性质 343
10.4.3 直角坐标系下的三重积分的计算 344
习题10.4 349
10.5 在其他坐标系下的三重积分 350
10.5.1 柱坐标下的三重积分 350
10.5.2 球坐标系下的三重积分 352
习题10.5 355
参考文献 357