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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