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Published in:
2014 | OriginalPaper | Chapter
Vector Ellipsoidal Harmonics Structure Peculiarities and Limitations
Author : George Dassios
Published in: Mathematical Methods in Engineering
Publisher: Springer Netherlands
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Abstract
The theory of scalar ellipsoidal harmonics was introduced by Lamé in 1837, more than half a century after Laplace introduced his theory of spherical harmonics in 1782. It is amazing that the relative theory of vector spherical harmonics was demonstrated as late as 1935 by Hansen.The appearance of a corresponding theory for vector ellipsoidal harmonics was resisting until 2009, for the very simple reason that such a theory can not exist, at least not in such a nice form as the theory of their spherical counterparts. The intrinsic difficulties of the ellipsoidal coordinate system, the fine and symmetric structure that is encoded in the anisotropic character of the ellipsoidal geometry, as well as the necessary generalizations and limitations that are needed are discussed in this presentation. Furthermore, the new analytical techniques that are suggested through the introduction of vector ellipsoidal harmonics are also demonstrated via special examples.