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].

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 then the boundary conditions can be used to calculate the coefficients of these expansions. However, this method is either very difficult or impossible, because the vectorial character of the field is cast into the Cartesian coefficients of the series expansions, and these are not compatible with the ellipsoidal geometry of the boundary. The correct approach is to use eigensolutions that carry the vectorial structure of the field, and leave the coefficients of the corresponding expansions in scalar form. This method best fits the needs of any vector problem, but it demands knowledge of a complete set of vector eigenfunctions. It was successfully applied to the spherical case by Hansen in 1935 [172], who introduced vector spherical harmonics in connection with an antenna radiation problem.