Table of Contents
Introduction ix
1 Vectors and the geometry of space 1
1.1 Vectors 1
1.1.1 Concepts of vectors 1
1.1.2 Linear operations involving vectors 2
1.1.3 Coordinate systems in three-dimensional space 3
1.1.4 Representing vectors using coordinates 5
1.1.5 Lengths, direction angles 7
1.2 Dot product, cross product, and triple product 9
1.2.1 The dot product 9
1.2.2 Projections 12
1.2.3 The cross product 13
1.2.4 Scalar triple product 17
1.3 Equations of tines and planes 18
1.3.1 Lines 18
1.3.2 Planes 23
1.4 Curves and vector-valued functions 30
1.5 Calculus of vector-valued functions 32
1.5.1 Limits, derivatives, and tangent vectors 32
1.5.2 Antiderivatives and definite integrals 35
1.5.3 Length of curves, curvatures, TNB frame 37
1.6 Surfaces in space 42
1.6.1 Graph of an equation F(x, y, z) = 0 42
1.6.2 Cylinder 44
1.6.3 Quadric surfaces 46
1.6.4 Surface of revolution 46
1.7 Parameterized surfaces 49
1.8 Intersecting surfaces and projection curves 50
1.9 Regions bounded by surfaces 56
1.10 Review 57
1.11 Exercises 59
1.11.1 Vectors 59
1.11.2 Lines and planes in space 60
1.11.3 Curves and surfaces in space 61
2 Functions of multiple variables 65
2.1 Functions of multiple variables 65
2.1.1 Definitions 65
2.1.2 Graphs and level curves 67
2.1.3 Functions of more than two variables 69
2.1.4 Limits 70
2.1.5 Continuity 75
2.2 Partial derivatives 76
2.2.1 Definition 76
2.2.2 Interpretations of partial derivatives 80
2.2.3 Partial derivatives of higher order 82
2.3 Total differential 83
2.3.1 Linearization and differentiability 83
2.3.2 The total differential 89
2.3.3 The linear/differential approximation 90
2.4 The chain rule 92
2.4.1 The chain rule with one independent variable 92
2.4.2 The chain rule with more than one independent variable 94
2.4.3 Partial derivatives for abstract functions 97
2.5 The Taylor expansion 98
2.6 Implicit differentiation 101
2.6.1 Functions implicitly denned by a single equation 101
2.6.2 Functions defined implicitly by systems of equations 103
2.7 Tangent lines and tangent planes 106
2.7.1 Tangent lines and normal planes to a curve 106
2.7.2 Tangent planes and normal lines to a surface 109
2.8 Directional derivatives and gradient vectors 113
2.9 Maximum and minimum values 122
2.9.1 Extrema of functions of two variables 122
2.9.2 Lagrange multipliers 130
2.10 Review 136
2.11 Exercises 138
2.11.1 Functions of two variables 138
2.11.2 Partial derivatives and differentiability 139
2.11.3 Chain rules and implicit differentiation 140
2.11.4 Tangent lines/planes, directional derivatives 141
2.11.5 Maximum/minimum problems 142
3 Multiple integrals 145
3.1 Definition and properties 145
3.2 Double integrals in rectangular coordinates 150
3.3 Double integral in polar coordinates 157
3.4 Change of variables formula for double integrals 161
3.5 Triple integrals 165
3.5.1 Triple integrals in rectangular coordinates 165
3.5.2 Cylindrical and spherical coordinates 175
3.6 Change of variables in triple integrals 179
3.7 Other applications of multiple integrals 181
3.7.1 Surface area 181
3.7.2 Center of mass, moment of inertia 187
3.8 Review 188
3.9 Exercises 191
3.9.1 Double integrals 191
3.9.2 Triple integrals 192
3.9.3 Other applications of multiple integrals 193
4 Line and surface integrals 195
4.1 Line integral with respect to arc length 195
4.1.1 Definition and properties 196
4.1.2 Evaluating a line integral, fcf(x,y)ds, in R2 197
4.1.3 Line integrals fcf(x, y, z)ds in R3 199
4.2 Line integral of a vector field 201
4.2.1 Vector fields 201
4.2.2 The line Integral of a vector field along a curve C 202
4.3 The fundamental theorem of line integrals 208
4.4 Green's theorem: circulation-curl form 216
4.4.1 Positive oriented simple curve and simply connected region 216
4.4.2 Circulation around a closed curve 217
4.4.3 Circulation density 217
4.4.4 Green's theorem: circulation-curl form 219
4.4.5 Applications of Green's theorem in circulation-curt form 222
4.5 Green's theorem: flux-divergence form 231
4.5.1 Flux 231
4.5.2 Flux density - divergence 232
4.5.3 The diverge nee-flux form of Green's theorem 233
4.6 Source-free vector fields 235
4.7 Surface integral with respect to surface area 237
4.8 Surface integrals of vector fields 241
4.8.1 Orientable surfaces 241
4.8.2 Flux integral ∫∫s(F · N)dS 242
4.9 Divergence theorem 248
4.9.1 Divergence of a three-dimensional vector field 248
4.9.2 Divergence theorem 250
4.10 Stokes theorem 256
4.10.1 The curl of a three-dimensional vector field 256
4.10.2 Stokes theorem 258
4.11 Review 265
4.12 Exercises 268
4.12.1 Line integrals 268
4.12.2 Surface integrals 269
5 Introduction to ordinary differential equations 273
5.1 Introduction 273
5.2 First-order ODEs 275
5.2.1 General and particular solutions and direction fields 275
5.2.2 Separable differential equations 277
5.2.3 Substitution methods 279
5.2.4 Exact differential equations 281
5.2.5 First-order linear differential equations 283
5.3 Second-order ODEs 287
5.3.1 Reducible second-order equations 287
5.3.2 Second-order linear differential equations 291
5.3.3 Variation of parameters 307
5.4 Other ways of solving differential equations 308
5.4.1 Power series method 309
5.4.2 Numerical approximation: Euler's method 310
5.5 Review 313
5.6 Exercises 315
5.6.1 Introduction to differential equations 315
5.6.2 First-order differential equations 315
5.6.3 Second-order differential equations 316
Further reading 319
Index 321
