Paul / Coates / Helgason Huygens' Principle and Hyperbolic Equations

1. Auflage 2014
ISBN: 978-1-4832-6222-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 5, 906 Seiten, Web PDF

Reihe: Perspectives in Mathematics

Huygens' Principle and Hyperbolic Equations is devoted to certain mathematical aspects of wave propagation in curved space-times. The book aims to present special nontrivial Huygens' operators and to describe their individual properties and to characterize these examples of Huygens' operators within certain more or less comprehensive classes of general hyperbolic operators. The materials covered in the book include a treatment of the wave equation for p-forms over a space of constant sectional curvature, the Riesz distributions, the Euler-Poisson-Darboux-equations over a Riemannian manifold, and plane wave manifolds. Physicists will find the book invaluable.

Gunther Paul is an Ergonomist and James Cook University Principal Research Fellow for Occupational Health and Safety at the Australian Institute for Tropical Health and Medicine (AITHM), and the Mackay Institute for Research and Innovation (MIRI). He holds a PhD in Ergonomics and MPhil in Control Engineering from Darmstadt University of Technology. His research focuses on complex work system related issues, such as health systems, respiratory health, human-in-the-loop modelling, or musculoskeletal disorders. Gunther has been the Chief Investigator in 17 research projects. He is the Editor-In-Chief of the International Journal of Human Factors Modelling and Simulation, and a reviewer for over 20 international journals. He chairs the International Ergonomics Association Technical Committee on Human Simulation and Virtual Environments, and is a Member of the Queensland Government Safety Leadership at Work Expert Reference Group, Member of the Commonwealth Department of Employment Research and Evaluation Services Panel, and Member of the Panel of Assessors, Queensland Civil and Administrative Tribunal (QCAT). Gunther is also the Ambassador of the Foundation for Professional Ergonomics in Australia. He has published over 100 journal articles, books and book chapters, and has been regularly presenting and chairing sessions at International conferences over the last 25 years. In his most recent previous employments, Gunther led the Health Safety Environment Discipline in the School of Public Health and Social Work at QUT, and before that he was Director of Ergolab at UniSA. In his 10 year industrial career, he worked as Project Manager for Ford, Daimler, and Faurecia.

Autoren/Hrsg.

Paul, Gunther

Coates, J.

Helgason, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Huygens' Principle and Hyperbolic Equations;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;PREFACE;12
7;ACKNOWLEDGEMENTS;20
8;INTRODUCTION;22
9;CHAPTER I;60
9.1;§1. Normal domains;60
9.2;§2. The causal structure of space-times;67
9.3;§3. Vector bundles;79
9.4;§4. The wave equation for differential forms in non-euclidean spaces;86
9.5;§5. A spinor calculus;123
9.6;Notes and References;160
10;CHAPTER II. RIESZ DISTRIBUTIONS;164
10.1;§1. The Riesz distributions in the Minkowski space;164
10.2;§2. The Riesz distribution in curved space-times;178
10.3;§3. Some generalizations;199
10.4;Notes and References;210
11;CHAPTER III. THE FUNDAMENTAL SOLUTIONS;212
11.1;§1. The Hadamard coefficients;212
11.2;§2. B-Series;223
11.3;§3. The fundamental solutions;233
11.4;§4. Applications of the fundamental solutions;260
11.5;§5. The Cauchv problem;272
11.6;Notes and References;285
12;CHAPTER IV. HUYGENS' OPERATORS;288
12.1;§1. Hadamard's Criterion;288
12.2;§2. Huygens' triples;308
12.3;§3. Diversors. General wave families;325
12.4;§4. Maxwell's equations. Dirac's equations;340
12.5;Notes and References;358
13;CHAPTER V. THE EULER-POISSON-DARBOUX EQUATION;362
13.1;§1. An Application of the Method of Descent;362
13.2;§2. The singular Cauchy problem;375
13.3;§3. Huygens' principle for the EPD-equation;401
13.4;§4. Stellmacher's equations;414
13.5;§5. Elliptic operators with vanishing first Hadamard coefficient;448
13.6;Appendix (Proof of Theorem 5.13 (i));480
13.7;§6. Relations to spectral geometry;493
13.8;Notes and References;513
14;CHAPTER VI. TRANSFORMATION THEORY;518
14.1;§1. The bundle connection associated to an operator P;518
14.2;§2. A property of the Hadamard coefficients;530
14.3;§3. Conformai gauge transformations of an operator P;538
14.4;§4. Tensors with simple transformation law;554
14.5;§5. The moments of a normal hyperbolic operator (even dimension);577
14.6;§6. The moments for Maxwell's equations;592
14.7;Notes and References;611
15;CHAPTER VII. SOME THEOREMS ON HUYGENS' OPERATORS OVER FOUR-DIMENSIONAL SPACE-TIMES;614
15.1;§1. Some preparatory transformations;614
15.2;§2. The moments of order = 3;626
15.3;§3. Applications to Huygens' operators in a four-dimensional space-time;646
15.4;§4. The case of conformally flat metrics;667
15.5;Notes and References;698
16;CHAPTER VIII. PLANE WAVE MANIFOLDS AND HUYGENS' PRINCIPLE;700
16.1;§1. Introduction. Results;700
16.2;§2. pp- and plane wave manifolds;709
16.3;§3. Huygens' principle for plane wave manifolds;729
16.4;§4. A characterization of plane wave manifolds;751
16.5;§5. Some conformally invariant tensors;777
16.6;§6. Testing coefficients by pp-metrics;804
16.7;§7. Testing coefficients by metrics of constant curvature;824
16.8;Notes and References;847
17;TABLE I: Identities for the Weyl tensor;854
18;TABLE II: Moments of order = 4 in four dimensions;858
19;TABLE III: Some formulas for pp-metrics;862
20;TABLE IV: Some formulas for plane wave metrics;864
21;APPENDIX I: METRIC AND CURVATURE IN NORMAL COORDINATES;866
22;APPENDIX II: WEAK HUYGENS' OPERATORS By V. Wünsch;880
23;APPENDIX III: HUYGENS' PRINCIPLE FOR SPIN TENSOR EQUATIONS By V. Wünsch;884
24;INDEX;890
25;BIBLIOGRAPHY;892