1 Vectors and Functions 1
1.1 Preliminaries 2
Algebraic Operations 2
Order Properties 4
Intervals,Disks,and Bounded Sets 6
Line Segments and Paths 8
1.2 Functions and Their Geometric Properties 10
Basic Notions 10
Bounded Functions 13
Monotonicity and Bimonotonicity 14
Functions of Bounded Variation 17
Functions of Bounded Bivariation 20
Convexity and Concavity 25
Local Extrema and Saddle Points 26
Intermediate Value Property 29
1.3 Cylindrical and Spherical Coordinates 30
Cylindrical Coordinates 31
Spherical Coordinates 32
Notes and Comments 33
Exercises 34
2 Sequences,Continuity,and Limits 43
2.1 Sequences in R2 43
Subsequences and Cauchy Sequences 45
Closure,Boundary,and Interior 46
2.2 Continuity 48
Composition of Continuous Functions 51
Piecing Continuous Functions on Overlapping Subsets 53
Characterizations of Continuity 55
Continuity and Boundedness 56
Continuity and Monotonicity 57
Continuity,Bounded Variation,and Bounded Bivariation 57
Continuity and Convexity 58
Continuity and Intermediate Value Property 60
Uniform Continuity 61
Implicit Function Theorem 63
2.3 Limits 67
Limits and Continuity 68
Limit from a Quadrant 71
Approaching Infinity 72
Notes and Comments 76
Exercises 77
3 Partial and Total Differentiation 83
3.1 Partial and Directional Derivatives 84
Partial Derivatives 84
Directional Derivatives 88
Higher-Order Partial Derivatives 91
Higher-Order Directional Derivatives 99
3.2 Differentiability 101
Differentiability and Directional Derivatives 109
Implicit Differentiation 112
3.3 Taylor's Theorem and Chain Rule 116
Bivariate Taylor Theorem 116
Chain Rule 120
3.4 Monotonicity and Convexity 125
Monotonicity and First Partials 125
Bimonotonicity and Mixed Partials 126
Bounded Variation and Boundedness of First Partials 127
Bounded Bivariation and Boundedness of Mixed Partials 128
Convexity and Monotonicity of Gradient 129
Convexity and Nonnegativity of Hessian 133
3.5 Functions of Three Variables 138
Extensions and Analogues 138
Tangent Planes and Normal Lines to Surfaces 143
Convexity and Ternary Quadratic Forms 147
Notes and Comments 149
Exercises 151
4 Applications of Partial Differentiation 157
4.1 Absolute Extrema 157
Boundary Points and Critical Points 158
4.2 Constrained Extrema 161
Lagrange Multiplier Method 162
Case of Three Variables 164
4.3 Local Extrema and Saddle Points 167
Discriminant Test 170
4.4 Linear and Quadratic Approximations 175
Linear Approximation 175
Quadratic Approximation 178
Notes and Comments 180
Exercises 181
5 Multiple Integration 185
5.1 Double Integrals on Rectangles 185
Basic Inequality and Criterion for Integrability 193
Domain Additivity on Rectangles 197
Integrability of Monotonic and Continuous Functions 200
Algebraic and Order Properties 202
A Version of the Fundamental Theorem of Calculus 208
Fubini's Theorem on Rectangles 216
Riemann Double Sums 222
5.2 Double Integrals over Bounded Sets 226
Fubini's Theorem over Elementary Regions 230
Sets of Content Zero 232
Concept of Area of a Bounded Subset of R2 240
Domain Additivity over Bounded Sets 244
5.3 Change of Variables 247
Translation Invariance and Area of a Parallelogram 247
Case of Affine Transformations 251
General Case 258
5.4 Triple Integrals 267
Triple Integrals over Bounded Sets 269
Sets of Three-Dimensional Content Zero 273
Concept of Volume of a Bounded Subset of R3 273
Change of Variables in Triple Integrals 274
Notes and Comments 280
Exercises 282
6 Applications and Approximations of Multiple Integrals 291
6.1 Area and Volume 291
Area of a Bounded Subset of R2 291
Regions between Polar Curves 293
Volume of a Bounded Subset of R3 297
Solids between Cylindrical or Spherical Surfaces 298
Slicing by Planes and the Washer Method 302
Slivering by Cylinders and the Shell Method 303
6.2 Surface Area 309
Parallelograms in R2 and in R3 311
Area of a Smooth Surface 313
Surfaces of Revolution 319
6.3 Centroids of Surfaces and Solids 322
Averages and Weighted Averages 323
Centroids of Planar Regions 324
Centroids of Surfaces 326
Centroids of Solids 329
Centroids of Solids of Revolution 335
6.4 Cubature Rules 338
Product Rules on Rectangles 339
Product Rules over Elementary Regions 344
Triangular Prism Rules 346
Notes and Comments 360
Exercises 361
7 Double Series and Improper Double Integrals 369
7.1 Double Sequences 369
Monotonicity and Bimonotonicity 373
7.2 Convergence of Double Series 376
Telescoping Double Series 382
Double Series with Nonnegative Terms 383
Absolute Convergence and Conditional Convergence 387
Unconditional Convergence 390
7.3 Convergence Tests for Double Series 392
Tests for Absolute Convergence 392
Tests for Conditional Convergence 399
7.4 Double Power Series 403
Taylor Double Series and Taylor Series 411
7.5 Convergence of Improper Double Integrals 416
Improper Double Integrals of Mixed Partials 420
Improper Double Integrals of Nonnegative Functions 421
Absolute Convergence and Conditional Convergence 425
7.6 Convergence Tests for Improper Double Integrals 428
Tests for Absolute Convergence 430
Tests for Conditional Convergence 431
7.7 Unconditional Convergence of Improper Double Integrals 435
Functions on Unbounded Subsets 436
Concept of Area of an Unbounded Subset of R2 441
Unbounded Functions on Bounded Subsets 443
Notes and Comments 447
Exercises 449
References 463
List of Symbols and Abbreviations 467
Index 471