Ellipsoidal harmonics. Theory and applications (Q2904680)

This book is devoted to ellipsoidal harmonics. After a complete presentation of the theory, applied topics are drawn from geometry, physics, biosciences, and inverse problems. The book contains classical results as well as new material, including ellipsoidal biharmonic functions (Chapter 10), the theory of images in ellipsoidal geometry (Chapter 15), geometrical characteristics of surface perturbations (Chapter 12), and vector surface ellipsoidal harmonics (Chapter 11). Extended appendices provide information needs to solve formally boundary value problems. End-of-chapter problems and tests complement th presented theory and yield to a better understanding of the subject.NEWLINENEWLINEThe book is organized as follows. Chapter 1 contains an introduction to ellipsoidal system and its geometry. The basic differential operators in terms of ellipsoidal coordinates are given in Chapter 2. Chapter 3 covers the analysis of the Lamé equation. Ellipsoidal harmonics are discussed in Chapter 4. The theory of Niven and Cartesian harmonics are exposed in th next chapter. Chapter 6 introduces the analysis of integration techniques used in the anisotropic environment of the ellipsoidal system. The basic theory for solving boundary value problems for the Laplace equation in ellipsoidal domains are given in Chapter 7. Ellipsoidal harmonics are not readily expressed in terms of the classical spherical harmonics of Laplace and Legendre. In Chapter 8 connection between harmonics is described, while Chapter 9 presents a limited introduction to the definitions of the classical elliptic functions and their connection to the theory of Lamé functions and ellipsoidal harmonics. In Chapter 10 ellipsoidal biharmonic functions are introduced and their relation to ellipsoidal harmonics via the Almansi representation theorem are discussed. In Chapter 11 vector surface ellipsoidal harmonics are introduced and an analysis of their orthogonality properties are given.NEWLINENEWLINEThe remaining five chapters are devoted to applications. Chapter 12 is focused on geometrical applications. Applications in physical systems, such as polarization potentials, gravitational potentials, and so on, are included in Chapter 13. Chapter 14 contains a discussion of low-frequency scattering theory from ellipsoidal bodies in acoustics, electromagnetism, and elasticity. Chapter 15 contains some applications to problems of biosciences. Chapter 16 presents some problems on the reconstruction of an ellipsoid from low-frequency scattering data, from tomographic images and inverse EEG problem for a dipole. There are seven appendices (A-G), which contain either complementary or tabulated material. The book can be used as a reference book for applied mathematicians.