Ellipsoidal Harmonics

Theory and Applications

George Dassios (University of Patras, Greece)

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English

Cambridge University Press
12 July 2012

Differential calculus & equations; Integral calculus & equations; Geometry

Series: Encyclopedia of Mathematics and its Applications

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.

By: George Dassios (University of Patras Greece)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 146
Dimensions: Height: 240mm, Width: 160mm, Spine: 28mm
Weight: 860g
ISBN: 9780521113090
ISBN 10: 0521113091
Series: Encyclopedia of Mathematics and its Applications
Pages: 474
Publication Date: 12 July 2012
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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.

George Dassios is Professor of Applied Mathematics at the University of Patras, Greece and at ICE-FT/FORTH (a research institute in Greece).