Advanced Calculus (Revised Edition)

by Lynn Harold Loomis, Shlomo Zvi Sternberg

Overview

An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.

Product Details

ISBN-13:	9789814583930
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/12/2014
Edition description:	Revised
Pages:	596
Product dimensions:	6.40(w) x 9.60(h) x 1.30(d)

Table of Contents

Chapter 0 Introduction

1 Logic: quantifiers 1

2 The logical connectives 3

3 Negations of quantifiers 6

4 Sets 6

5 Restricted variables 8

6 Ordered pairs and relations 9

7 Functions and mappings 10

8 Product sets; index notation 12

9 Composition 14

10 Duality 15

11 The Boolean operations 17

12 Partitions and equivalence relations 19

Chapter 1 Vector Spaces

1 Fundamental notions 21

2 Vector spaces and geometry 36

3 Product spaces and Horn(V, W) 43

4 Affine subspaces and quotient spaces 52

5 Direct sums 56

6 Bilinearity 67

Chapter 2 Finite-Dimensional Vector Spaces

1 Bases 71

2 Dimension 77

3 The dual space 81

4 Matrices 88

5 Trace and determinant 99

6 Matrix computations 102

7 The diagonalization of a quadratic form 111

Chapter 3 The Differential Calculus

1 Review in R 117

2 Norms 121

3 Continuity 126

4 Equivalent norms 132

5 Infinitesimals 136

6 The differential 140

7 Directional derivatives; the mean-value theorem 146

8 The differential and product spaces 152

9 The differential and Rⁿ 156

10 Elementary application? 161

11 The implicit-function theorem 164

12 Submanifolds and Lagrange multipliers 172

13 Functional dependence 175

14 Uniform continuity and function-valued mappings 179

15 The calculus of variations 182

16 The second differential and the classification of critical points 186

17 The Taylor formula 191

Chapter 4 Compactness and Completeness

1 Metric spaces; open and closed sets 195

2 Topology 201

3 Sequential convergence 202

4 Sequential compactness 205

5 Compactness and uniformity 210

6 Equicontinuity 215

7 Completeness 216

8 A first look at Banach algebras 223

9 The contraction mapping fixed-point theorem 228

10 The integral of a parametrized arc 236

11 The complex number system 240

12 Weak methods 245

Chapter 5 Scalar Product Spaces

1 Scalar products 248

2 Orthogonal projection 252

3 Sell-adjoint transformations 257

4 Orthogonal transformations 262

5 Compact transformations 264

Chapter 6 Differential Equations

1 The fundamental theorem 266

2 Differentiable dependence on parameters 274

3 The linear equation 276

4 The nth-order linear equation 281

5 Solving the inhomogeneous equation 288

6 The boundary-value problem 294

7 Fourier series 301

Chapter 7 Multilinear Functionals

1 Bilinear functionals 305

2 Multilinear functionals 306

3 Permutations 308

4 The sign of a permutation 309

5 The subspace aⁿ of alternating tensors 310

6 The determinant 312

7 The exterior algebra 316

8 Exterior powers of scalar product spaces 319

9 The star operator 320

Chapter 8 Integration

1 Introduction 321

2 Axioms 322

3 Rectangles and paved sets 324

4 The minimal theory 327

5 The minimal theory (continued) 328

6 Contented sets 331

7 When is a set contented? 333

8 Behavior under linear distortions 335

9 Axioms for integration 336

10 Integration of contented functions 338

11 The change of variables formula 342

12 Successive integration 346

13 Absolutely integrable functions 351

14 Problem set: The Fourier transform 355

Chapter 9 Differentiate Manifolds

1 Atlases 364

2 Functions, convergence 367

3 Differentiable manifolds 369

4 The tangent space 373

5 Flows and vector fields 376

6 Lie derivatives 384

7 Linear differential forms 390

8 Computations with coordinates 393

9 Riemann metrics 397

Chapter 10 The Integral Calculus on Manifolds

1 Compactness 403

2 Partitions of unity 405

3 Densities 408

4 Volume density of a Riemann metric 411

5 Fullback and Lie derivatives of densities 416

6 The divergence theorem 419

7 Afore complicated domains 424

Chapter 11 Exterior Calculus

1 Exterior differential forms 429

2 Oriented manifolds and the integration of exterior differential forms 433

3 The operator d 438

4 Stokes' theorem 442

5 Some illustrations of Stokes' theorem 449

6 The Lie derivative of a differential form 452

Appendix I "Vector analysis" 457

Appendix II Elementary differential geometry of surfaces in E³ 459

Chapter 12 Potential Theory in Eⁿ

1 Solid angle 474

2 Green's formulas 476

3 The maximum principle 477

6 Green's functions 479

5 The Poisson integral formula 482

6 Consequences of the Poisson integral formula 485

7 Harnack's theorem 487

8 Subharmonic functions 489

9 Dirichlet's problem 491

10 Behavior near the boundary 495

11 Dirichlet's principle 499

12 Physical applications 500

13 Problem set: The calculus of residues 503

Chapter 13 Classical Mechanics

1 The tangent and cotangent bundles 511

2 Equations of variation 513

3 The fundamental linear differential form on T*(M) 515

4 The fundamental exterior two-form on T*(M) 517

5 Hamiltonian mechanics 520

6 The central-force problem 523

7 The two-body problem 528

8 Lagrange's equations 530

9 Variational principles 532

10 Geodesic coordinates 537

11 Euler's equations 541

12 Rigid-body motion 544

13 Small oscillations 001

14 Small oscillations (continued) 553

15 Canonical transformations 558

Selected References 569

Notation Index 572

Index 575