Ellipsoidal Harmonics
Theory and Applications

The first book devoted to ellipsoidal harmonics presents the state of the art in this fascinating subject.

George Dassios (Author)

9780521113090, Cambridge University Press

Hardback, published 12 July 2012

474 pages, 32 b/w illus.
24 x 16 x 2.8 cm, 0.86 kg

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.

Prologue
1. The ellipsoidal system and its geometry
2. Differential operators in ellipsoidal geometry
3. Lamé functions
4. Ellipsoidal harmonics
5. The theory of Niven and Cartesian harmonics
6. Integration techniques
7. Boundary value problems in ellipsoidal geometry
8. Connection between sphero-conal and ellipsoidal harmonics
9. The elliptic functions approach
10. Ellipsoidal bi-harmonic functions
11. Vector ellipsoidal harmonics
12. Applications to geometry
13. Applications to physics
14. Applications to low-frequency scattering theory
15. Applications to bioscience
16. Applications to inverse problems
Epilogue
Appendix A. Background material
Appendix B. Elements of dyadic analysis
Appendix C. Legendre functions and spherical harmonics
Appendix D. The fundamental polyadic integral
Appendix E. Forms of the Lamé equation
Appendix F. Table of formulae
Appendix G. Miscellaneous relations
Bibliography
Index.

Subject Areas: Mathematical modelling [PBWH], Integral calculus & equations [PBKL], Differential calculus & equations [PBKJ]