Huygens' Principle and Hyperbolic Equations

Huygens' Principle and Hyperbolic Equations

1st Edition - January 1, 1988

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  • Author: Paul Günther
  • eBook ISBN: 9781483262222

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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.

Table of Contents

  • Preface



    Chapter I

    §1 Normal Domains

    §2 The Causal Structure of Space-Times

    §3 Vector Bundles

    §4 The Wave Equations for Differential Forms in Non-Euclidean Spaces

    §5 A Spinor Calculus

    Chapter II Riesz Distributions

    §1 The Riesz Distributions in the Minkowski Space

    §2 The Riesz Distributions in Curved Space-Times

    §3 Some Generalizations

    Chapter III The Fundamental Solutions

    §1 The Hadamard Coefficients

    §2 B-Series

    §3 The Fundamental Solutions

    §4 Applications of the Fundamental Solutions

    §5 The Cauchy Problem

    Chapter IV Huygens' Operators

    §1 Hadamard's Criterion

    §2 Huygens' Triples

    §3 Diversors. General Wave Families

    §4 Maxwell's Equations. Dirac's Equations

    Chapter V The Euler-Poisson-Darboux Equation

    §1 An Application of the Method of Descent

    §2 The Singular Cauchy Problem

    §3 Huygens' Principle for the EPD-Equation

    §4 Stellmacher's Equations

    §5 Elliptic Operators with Vanishing First Hadamard Coefficient


    §6 Relations to Spectral Geometry

    Chapter VI Transformation Theory

    §1 The Bundle Connection Associated to an Operator P

    §2 A Property of the Hadamard Coefficients

    §3 Conformal Gauge Transformations of an Operator P

    §4 Tensors with Simple Transformation Law

    §5 The Moments of a Normal Hyperbolic Operator (Even Dimension)

    §6 The Moments for Maxwell's Equations

    Chapter VII Some Theorems on Huygens' Operators Over Four-Dimensional Space-Times

    §1 Some Preparatory Transformations

    §2 The Moments of Order ≤ 3

    §3 Applications to Huygens' Operators in a Four-Dimensional Space-Time

    §4 The Case of Conformally Flat Metrics

    Chapter VIII Plane Wave Manifolds and Huygens' Principle

    §1 Introduction. Results

    §2 pp- and Plane Wave Manifolds

    §3 Huygens' Principle for Plane Wave Manifolds

    §4 A Characterization of Plane Wave Manifolds

    §5 Some Conformally Invariant Tensors

    §6 Testing Coefficients by pp-Metrics

    §7 Testing Coefficients by Metrics of Constant Curvature

    Table I Identities for the Weyl Tensor

    Table II Moments of Order ≤ 4 in Four Dimensions

    Table III Some Formulas for pp-Metrics

    Table IV Some Formulas for Plane Wave Metrics

    Appendix I Metric and Curvature in Normal Coordinates

    Appendix II Weak Huygens' Operators by V. Wünsch

    Appendix III Huygens' Principle for Spin Tensor Equations by V. Wünsch



Product details

  • No. of pages: 906
  • Language: English
  • Copyright: © Academic Press 1988
  • Published: January 1, 1988
  • Imprint: Academic Press
  • eBook ISBN: 9781483262222

About the Author

Paul Günther

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.

Affiliations and Expertise

Principal Research Fellow, Australian Institute for Tropical Health and Medicine (AITHM), Australia

About the Editors

J. Coates

S. Helgason

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