Functions Of Several Real Variables

by Martin Moskowitz, Fotios C Paliogiannis

Overview

This book begins with the basics of the geometry and topology of Euclidean space and continues with the main topics in the theory of functions of several real variables including limits, continuity, differentiation and integration. All topics and in particular, differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration such as the Inverse and Implicit Function theorems, Lagrange's multiplier rule, Fubini's theorem, the change of variables formula, Green's, Stokes' and Gauss' theorems are proved in detail and many of them with novel proofs. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results and solve problems in mathematics, the other sciences and economics.Each of the chapters concludes with groups of exercises and problems, many of them with detailed solutions while others with hints or final answers. More advanced topics, such as Morse's lemma, Sard's theorem , the Weierstrass approximation theorem, the Fourier transform, Vector fields on spheres, Brouwer's fixed point theorem, Whitney's embedding theorem, Picard's theorem, and Hermite polynomials are discussed in stared sections.

Product Details

ISBN-13:	9789814299275
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/06/2011
Edition description:	New Edition
Pages:	732
Product dimensions:	5.90(w) x 8.90(h) x 1.50(d)

Table of Contents

Preface and Acknowledgments v

Notations ix

1 Basic Features of Euclidean Space, Rⁿ 1

1.1 Real numbers 1

1.1.1 Convergence of sequences of real numbers 8

1.2 Rⁿ as a vector space 15

1.3 Rⁿ as an inner product space 19

1.3.1 The inner product and norm in Rⁿ 19

1.3.2 Orthogonality 24

1.3.3 The cross product in R³ 32

1.4 Rⁿ as a metric space 38

1.5 Convergence of sequences in Rⁿ 43

1.6 Compactness 48

1.7 Equivalent norms (*) 52

1.8 Solved problems for Chapter 1 54

2 Functions on Euclidean Spaces 61

2.1 Functions from Rⁿ to R^m 61

2.2 Limits of functions 66

2.3 Continuous functions 71

2.4 Linear transformations 80

2.5 Continuous functions on compact sets 85

2.6 Connectedness and convexity 88

2.6.1 Connectedness 88

2.6.2 Path-connectedness 91

2.6.3 Convex sets 95

2.7 Solved problems for Chapter 2 97

3 Differential Calculus in Several Variables 103

3.1 Differentiable functions 103

3.2 Partial and directional derivatives, tangent space 109

3.3 Homogeneous functions and Eulor's equation 129

3.4 The mean value theorem 131

3.5 Higher order derivatives 134

3.5.1 The second derivative 138

3.6 Taylor's theorem 144

3.6.1 Taylor's theorem in one variable 144

3.6.2 Taylor's theorem in several variables 154

3.7 Maxima and minima in several variables 159

3.7.1 Local extrema for functions in several variables 160

3.7.2 Degenerate critical points 172

3.8 The inverse and implicit function theorems 177

3.8.1 The Inverse Function theorem 177

3.8.2 The Implicit Function theorem 185

3.9 Constrained extrema, Lagrange multipliers 194

3.9.1 Applications to economics 205

3.10 Functional dependence 211

3.11 Morse's lemma (*) 217

3.12 Solved problems for Chapter 3 221

4 Integral Calculus in Several Variables 231

4.1 The integral in Rⁿ 231

4.1.1 Darboux sums. Integrability condition 234

4.1.2 The integral over a bounded set 247

4.2 Properties of multiple integrals 250

4.3 Fubini's theorem 260

4.3.1 Center of mass, centroid, moment of inertia 273

4.4 Smooth Urysohn's lemma and partition of unity (*) 280

4.5 Sard's theorem (*) 282

4.6 Solved problems for Chapter 4 285

5 Change of Variables Formula, Improper Multiple Integrals 293

5.1 Change of variables formula 293

5.1.1 Change of variables; linear case 297

5.1.2 Change of variables; the general case 301

5.1.3 Applications, polar and spherical coordinates 308

5.2 Improper multiple integrals 328

5.3 Functions defined by integrals 346

5.3.1 Functions defined by improper integrals 351

5.3.2 Convolution of functions 368

5.4 The Weierstrass approximation theorem (*) 374

5.5 The Fourier transform (*) 376

5.5.1 The Schwartz space 383

5.5.2 The Fourier transform on Rⁿ 389

5.6 Solved problems for Chapter 5 393

6 Line and Surface Integrals 405

6.1 Arc-length and Line integrals 405

6.1.1 Paths and curves 405

6.1.2 Line integrals 409

6.2 Conservative vector fields and Poincare's lemma 417

6.3 Surface area and surface integrals 429

6.3.1 Surface area 429

6.3.2 Surface integrals 441

6.4 Green's theorem and the divergence theorem in R² 450

6.4.1 The divergence theorem in R² 457

6.5 The divergence and curl 462

6.6 Stokes' theorem 466

6.7 The divergence theorem in R³ 474

6.8 Differential forms (*) 483

6.9 Vector fields on spheres and Brouwer fixed point theorem (*) 492

6.9.1 Tangential vector fields on spheres 494

6.9.2 The Brouwer fixed point theorem 497

6.10 Solved problems for Chapter 6 501

7 Elements of Ordinary and Partial Differential Equations 509

7.1 Introduction 509

7.2 First, order differential equations 511

7.2.1 Linear first order ODE 511

7.2.2 Equations with variables separated 517

7.2.3 Homogeneous equations 519

7.2.4 Exact equations 521

7.3 Picard's theorem (*) 525

7.4 Second order differential equations 532

7.4.1 Linear second order ODE with constant coefficients 534

7.4.2 Special types of second order ODE; reduction of order 545

7.5 Higher order ODE and systems of ODE 548

7.6 Some more advanced topics in ODE (*) 564

7.6.1 The method of Frobenius; second order equations with variable coefficients 564

7.6.2 The Hermite equation 568

7.7 Partial differential equations 573

7.8 Second order PDE in two variables 576

7.8.1 Classification and general solutions 576

7.8.2 Boundary value problems for the wave equation 583

7.8.3 Boundary value problems for Laplace's equation 588

7.8.4 Boundary value problems for the heat equation 594

7.8.5 A note on Fourier series 598

7.9 The Fourier transform method (*) 603

7.10 Solved problems for Chapter 7 612

8 An Introduction to the Calculus of Variations 619

8.1 Simple variational problems 620

8.1.1 Some classical problems 627

8.1.2 Sufficient conditions 636

8.2 Generalizations 638

8.2.1 Geodesies on a Riemannian surface 646

8.2.2 The principle of least action 653

8.3 Variational problems with constraints 659

8.4 Multiple integral variational problems 666

8.4.1 Variations of double integrals 667

8.4.2 The case of n variables 673

8.5 Solved problems for Chapter 8 682

Appendix A Countability and Decimal Expansions 691

Appendix B Calculus in One Variable 693

B.1 Differential calculus 693

B.2 Integral calculus 694

B.2.1 Complex-valued functions 695

B.3 Series 695

Appendix C Uniform Convergence 697

C.1 The Stone-Weierstrass theorem 698

Appendix D Linear Algebra 701

Bibliography 705

Index 709