Chapter 7 Differential Equations 1
7.1 Basic Concepts of Differential Equations 1
7.1.1 Examples of Differential Equations 1
7.1.2 Basic Concepts 3
7.1.3 Geometric Interpretation of the First-Order Differential Equation 4
Exercises 7.1 5
7.2 First-Order Differential Equations 6
7.2.1 First-Order Separable Differential Equation 6
7.2.2 Homogeneous First-Order Equations 8
7.2.3 Linear First-Order Equations 10
7.2.4 Bernoulli's Equation 13
7.2.5 Some Other Examples that can be Reduced to Linear First-Order Equations 14
Exercises 7.2 16
7.3 Reducible Second-Order Differential Equations 17
Exercises 7.3 21
7.4 Higher-Order Linear Differential Equations 22
7.4.1 Some Examples of Linear Differential Equation of Higher-Order 22
7.4.2 Structure of Solutions of Linear Differential Equations 24
Exercises 7.4 28
7.5 Higher-Order Linear Equations with Constant Coefficients 29
7.5.1 Higher-Order Homogeneous Linear Equations with Constant Coefficients 29
7.5.2 Higher-Order Nonhomogeneous Linear Equations with Constant Coefficients 33
Exercises 7.5 39
7.6 Euler's Differential Equation 40
Exercises 7.6 42
7.7 Applications of Differential Equations 42
Exercises 7.7 46
Chapter 8 Vectors and Solid Analytic Geometry 54
8.1 Vectors in Plane and in Space 54
8.1.1 Vectors 54
8.1.2 Operations on Vectors 56
8.1.3 Vectors in Plane 59
8.1.4 Rectangular Coordinate System 61
8.1.5 Vectors in Space 63
Exercises 8.1 66
Part A 66
Part B 67
8.2 Products of Vectors 68
8.2.1 Scalar Product of two Vectors 68
8.2.2 Vector Product of two Vectors 72
8.2.3 Triple Scalar Product of three Vectors 77
8.2.4 Applications of Products of Vectors 80
Exercises 8.2 83
Part A 83
Part B 84
8.3 Planes and Lines in Space 85
8.3.1 Equations of Planes 85
8.3.2 Equations of Lines in Space 90
Exercises 8.3 96
Part A 96
Part B 98
8.4 Surfaces and Space Curves 99
8.4.1 Cylinders 99
8.4.2 Cones 101
8.4.3 Surfaces of Revolution 102
8.4.4 Quadric Surfaces 104
8.4.5 Space Curves 110
8.4.6 Cylindrical Coordinate System 114
8.4.7 Spherical Coordinate System 115
Exercises 8.4 117
Part A 117
Part B 119
Chapter 9 The Differential Calculus for Multi-variable Functions 120
9.1 Definition of Multi-variable Functions and their Basic Properties 120
9.1.1 Space R2 and Rn 120
9.1.2 Multi-variable Functions 128
9.1.3 Visualization of Multi-variable Functions 130
9.1.4 Limits and Continuity of Multi-variable Functions 135
Exercises 9.1 142
Part A 142
Part B 143
9.2 Partial Derivatives and Total Differentials of Multi-variable Functions 143
9.2.1 Partial Derivatives 144
9.2.2 Total Differentials 149
9.2.3 Higher-Order Partial Derivatives 157
9.2.4 Directional Derivatives and the Gradient 159
Exercises 9.2 166
Part A 166
Part B 169
9.3 Differentiation of Multi-variable Composite and Implicit Functions 169
9.3.1 Partial Derivatives and total Differentials of Multi-variable Composite Functions 170
9.3.2 Differentiation of Implicit Functions 176
9.3.3 Differentiation of Implicit Functions determined by Equation Systems 178
Exercises 9.3 181
Part A 181
Part B 183
Chapter 10 Applications of Multi-variable Functions 184
10.1 Approximate Function Values by total Differential 184
10.2 Extreme Values of Multi-variable Functions 187
10.2.1 Unrestricted Extreme Values 187
10.2.2 Global Maxima and Minima 190
10.2.3 The Method of Least Squares 192
10.2.4 Constrained Extreme Values 194
10.2.5 The Method of Lagrange Multipliers 196
Exercises 10.2 199
Part A 199
Part B 200
10.3 Applications in Geometry 200
10.3.1 Arc Length along a Curve 200
10.3.2 Tangent Line and Normal Plane of a Space Curve 204
10.3.3 Tangent Planes and Normal Lines to a Surface 209
10.3.4 Curvature for Plane Curves 213
Exercises 10.3 214
Part A 214
Part B 217
Synthetic exercises 217
Chapter 11 Multiple Integrals 219
11.1 Concept and Properties of Double Integrals 219
11.1.1 Concept of Double Integrals 219
11.1.2 Properties of Double Integrals 222
Exercises 11.1 223
11.2 Evaluation of Double Integrals 224
11.2.1 Geometric Meaning of Double Integrals 225
11.2.2 Double Integrals in Rectangular Coordinates 226
11.2.3 Double Integrals in Polar Coordinates 233
11.2.4 Integration by Substitution for Double Integrals in General 241
Exercises 11.2 246
Part A 246
Part B 250
11.3 Triple Integrals 251
11.3.1 Concept and Properties of Triple Integrals 251
11.3.2 Triple Integrals in Rectangular Coordinates 252
11.3.3 Triple Integrals in Cylindrical and Spherical Coordinates 257
11.3.4 Integration by Substitution for Triple Integrals in General 265
Exercises 11.3 266
Part A 266
Part B 269
11.4 Applications of Multiple Integrals 270
11.4.1 Surface Area 271
11.4.2 The Center of Gravity 273
11.4.3 The Moment of Inertia 275
Exercises 11.4 276
Part A 276
Part B 276
Chapter 12 Line Integrals and Surface Integrals 278
12.1 Line Integrals 278
12.1.1 Line Integrals with respect to Arc Length 278
12.1.2 Line Integrals with respect to Coordinates 284
12.1.3 Relations between two Types of Line Integrals 289
Exercises 12.1 289
Part A 289
Part B 292
12.2 Green's Formula and its Applications 294
12.2.1 Green's Formula 294
12.2.2 Conditions for Path Independence of Line Integrals 299
Exercises 12.2 307
Part A 307
Part B 309
12.3 Surface Integrals 311
12.3.1 Surface Integrals with respect to Surface Area 311
12.3.2 Surface Integrals with respect to Coordinates 315
Exercises 12.3 323
Part A 323
Part B 325
12.4 Gauss'Formula 326
Exercises 12.4 331
Part A 331
Part B 332
12.5 Stokes'Formula 332
12.5.1 Stokes'Formula 332
12.5.2 Conditions for Path Independence of Space Line Integrals 335
Exercises 12.5 337
Bibliography 339