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1 Vectors and Functions1

1.1 Preliminaries2

Algebraic Operations2

Order Properties4

Intervals,Disks,and Bounded Sets6

Line Segments and Paths8

1.2 Functions and Their Geometric Properties10

Basic Notions10

Bounded Functions13

Monotonicity and Bimonotonicity14

Functions of Bounded Variation17

Functions of Bounded Bivariation20

Convexity and Concavity25

Local Extrema and Saddle Points26

Intermediate Value Property29

1.3 Cylindrical and Spherical Coordinates30

Cylindrical Coordinates31

Spherical Coordinates32

Notes and Comments33

Exercises34

2 Sequences,Continuity,and Limits43

2.1 Sequences in R243

Subsequences and Cauchy Sequences45

Closure,Boundary,and Interior46

2.2 Continuity48

Composition of Continuous Functions51

Piecing Continuous Functions on Overlapping Subsets53

Characterizations of Continuity55

Continuity and Boundedness56

Continuity and Monotonicity57

Continuity,Bounded Variation,and Bounded Bivariation57

Continuity and Convexity58

Continuity and Intermediate Value Property60

Uniform Continuity61

Implicit Function Theorem63

2.3 Limits67

Limits and Continuity68

Limit from a Quadrant71

Approaching Infinity72

Notes and Comments76

Exercises77

3 Partial and Total Differentiation83

3.1 Partial and Directional Derivatives84

Partial Derivatives84

Directional Derivatives88

Higher-Order Partial Derivatives91

Higher-Order Directional Derivatives99

3.2 Differentiability101

Differentiability and Directional Derivatives109

Implicit Differentiation112

3.3 Taylor's Theorem and Chain Rule116

Bivariate Taylor Theorem116

Chain Rule120

3.4 Monotonicity and Convexity125

Monotonicity and First Partials125

Bimonotonicity and Mixed Partials126

Bounded Variation and Boundedness of First Partials127

Bounded Bivariation and Boundedness of Mixed Partials128

Convexity and Monotonicity of Gradient129

Convexity and Nonnegativity of Hessian133

3.5 Functions of Three Variables138

Extensions and Analogues138

Tangent Planes and Normal Lines to Surfaces143

Convexity and Ternary Quadratic Forms147

Notes and Comments149

Exercises151

4 Applications of Partial Differentiation157

4.1 Absolute Extrema157

Boundary Points and Critical Points158

4.2 Constrained Extrema161

Lagrange Multiplier Method162

Case of Three Variables164

4.3 Local Extrema and Saddle Points167

Discriminant Test170

4.4 Linear and Quadratic Approximations175

Linear Approximation175

Quadratic Approximation178

Notes and Comments180

Exercises181

5 Multiple Integration185

5.1 Double Integrals on Rectangles185

Basic Inequality and Criterion for Integrability193

Domain Additivity on Rectangles197

Integrability of Monotonic and Continuous Functions200

Algebraic and Order Properties202

A Version of the Fundamental Theorem of Calculus208

Fubini's Theorem on Rectangles216

Riemann Double Sums222

5.2 Double Integrals over Bounded Sets226

Fubini's Theorem over Elementary Regions230

Sets of Content Zero232

Concept of Area of a Bounded Subset of R2240

Domain Additivity over Bounded Sets244

5.3 Change of Variables247

Translation Invariance and Area of a Parallelogram247

Case of Affine Transformations251

General Case258

5.4 Triple Integrals267

Triple Integrals over Bounded Sets269

Sets of Three-Dimensional Content Zero273

Concept of Volume of a Bounded Subset of R3273

Change of Variables in Triple Integrals274

Notes and Comments280

Exercises282

6 Applications and Approximations of Multiple Integrals291

6.1 Area and Volume291

Area of a Bounded Subset of R2291

Regions between Polar Curves293

Volume of a Bounded Subset of R3297

Solids between Cylindrical or Spherical Surfaces298

Slicing by Planes and the Washer Method302

Slivering by Cylinders and the Shell Method303

6.2 Surface Area309

Parallelograms in R2 and in R3311

Area of a Smooth Surface313

Surfaces of Revolution319

6.3 Centroids of Surfaces and Solids322

Averages and Weighted Averages323

Centroids of Planar Regions324

Centroids of Surfaces326

Centroids of Solids329

Centroids of Solids of Revolution335

6.4 Cubature Rules338

Product Rules on Rectangles339

Product Rules over Elementary Regions344

Triangular Prism Rules346

Notes and Comments360

Exercises361

7 Double Series and Improper Double Integrals369

7.1 Double Sequences369

Monotonicity and Bimonotonicity373

7.2 Convergence of Double Series376

Telescoping Double Series382

Double Series with Nonnegative Terms383

Absolute Convergence and Conditional Convergence387

Unconditional Convergence390

7.3 Convergence Tests for Double Series392

Tests for Absolute Convergence392

Tests for Conditional Convergence399

7.4 Double Power Series403

Taylor Double Series and Taylor Series411

7.5 Convergence of Improper Double Integrals416

Improper Double Integrals of Mixed Partials420

Improper Double Integrals of Nonnegative Functions421

Absolute Convergence and Conditional Convergence425

7.6 Convergence Tests for Improper Double Integrals428

Tests for Absolute Convergence430

Tests for Conditional Convergence431

7.7 Unconditional Convergence of Improper Double Integrals435

Functions on Unbounded Subsets436

Concept of Area of an Unbounded Subset of R2441

Unbounded Functions on Bounded Subsets443

Notes and Comments447

Exercises449

References463

List of Symbols and Abbreviations467

Index471