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Ellipsoidal Harmonics by George Dassios

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11Vector ellipsoidal harmonics

In the present chapter we introduce vector ellipsoidal harmonics and discuss their peculiarities as well as the limitations that prevented their appearance for many years. In fact, it was not until 2009 that these functions were first introduced [121] and their complete understanding and effectiveness are still open for further investigation [82, 83].

11.1 Vector ellipsoidal harmonics

Without any knowledge of vectorial harmonics, all vector boundary value problems, governed by the Laplace equation in ellipsoidal domains, can be solved in the framework of a combined Cartesian-ellipsoidal treatment. That is, each Cartesian component of the vector solution we seek can be expanded in scalar ellipsoidal harmonics and ...

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