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Ellipsoidal Harmonics

Book Description

The sphere is what might be called a perfect shape. Unfortunately nature is imperfect and many bodies are better represented by an ellipsoid. The theory of ellipsoidal harmonics, originated in the nineteenth century, could only be seriously applied with the kind of computational power available in recent years. This, therefore, is the first book devoted to ellipsoidal harmonics. Topics are drawn from geometry, physics, biosciences and inverse problems. It contains classical results as well as new material, including ellipsoidal bi-harmonic functions, the theory of images in ellipsoidal geometry and vector surface ellipsoidal harmonics, which exhibit an interesting analytical structure. Extended appendices provide everything one needs to solve formally boundary value problems. End-of-chapter problems complement the theory and test the reader's understanding. The book serves as a comprehensive reference for applied mathematicians, physicists, engineers and for anyone who needs to know the current state of the art in this fascinating subject.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Prologue
  6. 1 The Ellipsoidal System and its Geometry
    1. 1.1 Confocal Families of Second-Degree Surfaces
    2. 1.2 Ellipsoidal Coordinates
    3. 1.3 Analytic Geometry of the Ellipsoidal System
    4. 1.4 Differential Geometry of the Ellipsoidal System
    5. 1.5 Sphero-Conal and Ellipto-Spherical Coordinates
    6. 1.6 The Ellipsoid as a Dyadic
    7. 1.7 Problems
  7. 2 Differential Operators in Ellipsoidal Geometry
    1. 2.1 The Basic Operators in Ellipsoidal Form
    2. 2.2 Ellipsoidal Representations of the Laplacian
    3. 2.3 The Thermometric Parameters of Lamé
    4. 2.4 Spectral Decomposition of the Laplacian
    5. 2.5 Problems
  8. 3 Lamé Functions
    1. 3.1 The Lamé Classes
    2. 3.2 Lamé Functions of Class K
    3. 3.3 Lamé Functions of Classes L and M
    4. 3.4 Lamé Functions of Class N
    5. 3.5 Discussion on the Lamé Classes
    6. 3.6 Lamé Functions of the Second Kind
    7. 3.7 Problems
  9. 4 Ellipsoidal Harmonics
    1. 4.1 Interior Ellipsoidal Harmonics
    2. 4.2 Harmonics of Degree Four
    3. 4.3 Exterior Ellipsoidal Harmonics
    4. 4.4 Surface Ellipsoidal Harmonics
    5. 4.5 Orthogonality Properties
    6. 4.6 Problems
  10. 5 The Theory of Niven and Cartesian Harmonics
    1. 5.1 The Roots of the Lamé Functions
    2. 5.2 The Theory of Niven Harmonics
    3. 5.3 The Characteristic System
    4. 5.4 From Niven Back to Lamé
    5. 5.5 The Klein–Stieltjes Theorem
    6. 5.6 Harmonics of Degree Four Revisited
    7. 5.7 Problems
  11. 6 Integration Techniques
    1. 6.1 Integrals Over an Ellipsoidal Surface
    2. 6.2 The Normalization Constants
    3. 6.3 The Normalization Constants Revisited
    4. 6.4 Problems
  12. 7 Boundary Value Problems in Ellipsoidal Geometry
    1. 7.1 Expansion of the Fundamental Solution
    2. 7.2 Eigensources and Eigenpotentials
    3. 7.3 The Closure Relation
    4. 7.4 Green's Function and its Image System
    5. 7.5 The Neumann Function and its Image System
    6. 7.6 Singularities of Exterior Ellipsoidal Harmonics
    7. 7.7 Problems
  13. 8 Connection Between Harmonics
    1. 8.1 Geometrical Reduction
    2. 8.2 Sphero-Conal Harmonics
    3. 8.3 Differential Formulae for Harmonic Functions
    4. 8.4 Sphero-Conal Expansions of Interior Ellipsoidal Harmonics
    5. 8.5 Integral Formulae for Harmonic Functions
    6. 8.6 Sphero-Conal Expansions of Exterior Ellipsoidal Harmonics
    7. 8.7 Problems
  14. 9 The Elliptic Functions Approach
    1. 9.1 The Weierstrass Approach
    2. 9.2 The Jacobi Approach
    3. 9.3 The Weierstrass–Jacobi Connection
    4. 9.4 Integral Equations for Lamé Functions
    5. 9.5 Integral Representations for Ellipsoidal Harmonics
    6. 9.6 Problems
  15. 10 Ellipsoidal Biharmonic Functions
    1. 10.1 Eigensolutions of the Ellipsoidal Biharmonic Equation
    2. 10.2 Re-Orthogonalization of Surface Harmonics
    3. 10.3 The Leading Biharmonics
    4. 10.4 Problems
  16. 11  Vector Ellipsoidal Harmonics
    1. 11.1 Vector Ellipsoidal Harmonics
    2. 11.2 Orthogonality
    3. 11.3 The Expansion Theorem
    4. 11.4 Problems
  17. 12 Applications to Geometry
    1. 12.1 Perturbation of the First Fundamental Form
    2. 12.2 Perturbation of the Unit Normal
    3. 12.3 Perturbation of the Second Fundamental Form
    4. 12.4 Perturbation of the Curvatures
    5. 12.5 The Ellipsoidal Stereographic Projection
    6. 12.6 The Surface Area of an Ellipsoid
    7. 12.7 Problems
  18. 13 Applications to Physics
    1. 13.1 Thermal Equilibrium
    2. 13.2 The Gravitational Potential
    3. 13.3 The Conductor Potential
    4. 13.4 The Polarization Potential
    5. 13.5 The Virtual Mass Potential
    6. 13.6 The Generalized Polarization Potentials
    7. 13.7 Reduction to Spheroids, Asymptotic Degeneracies, and Spheres
    8. 13.8 Problems
  19. 14 Applications to Low-Frequency Scattering Theory
    1. 14.1 Acoustic Scattering
    2. 14.2 Electromagnetic Scattering
    3. 14.3 Elastic Scattering
    4. 14.4 Problems
  20. 15 Applications to Bioscience
    1. 15.1 Electromagnetic Activity of the Brain
    2. 15.2 Electroencephalography
    3. 15.3 Magnetoencephalography
    4. 15.4 The Magnetic Potential of the Ellipsoid
    5. 15.5 Tumor Growth
    6. 15.6 The Nutrient Concentration Field
    7. 15.7 The Pressure Field
    8. 15.8 Evolution of the Boundary
    9. 15.9 The Spherical Tumor
    10. 15.10  Problems
  21. 16 Applications to Inverse Problems
    1. 16.1 Inversion of Low-Frequency Scattering Data
    2. 16.2 Inversion of Scattering Data in the Time Domain
    3. 16.3 Inversion of Tomographic Data
    4. 16.4 The Inverse EEG Problem for a Dipole
    5. 16.5 Problems
  22. Epilogue
  23. Appendix A Background Material
    1. A.1 The Fundamental Solution
    2. A.2 Kelvin's Theorem
    3. A.3 Surface Curvatures
    4. A.4 Elliptic Integrals
  24. Appendix B Elements of Dyadic Analysis
  25. Appendix C Legendre Functions and Spherical Harmonics
  26. Appendix D The Fundamental Polyadic Integral
  27. Appendix E Forms of the Lamé Equation
  28. Appendix F Table of Formulae
    1. F.1 Explicit Form of Lamé Functions
    2. F.2 Explicit Form of Ellipsoidal Harmonics
    3. F.3 Explicit Form of Vector Ellipsoidal Harmonics
    4. F.4 The Normalization Constants
  29. Appendix G Miscellaneous Relations
    1. G.1 Relations Among Constants
    2. G.2 Relations Among Elliptic Integrals
    3. G.3 Relations Among Interior Ellipsoidal Harmonics
    4. G.4 Relations Among Exterior Ellipsoidal Harmonics
    5. G.5 Ellipsoidal Representations of Cartesian Expressions
    6. G.6 Gradients of Ellipsoidal Harmonics
    7. G.7 General Vector and Dyadic Relations
    8. G.8 Particular Integrals
  30. Bibliography
  31. Index