The Problem of the Restoration of the Magnetic Potential along the Direction Field of Its Gradient in the Exterior of the Unit Ball

The magnetic fields of the internal source in the bowels of the Earth, observed on the surface of the planet or from the satellite orbit are the subjects of the research. We study the space of all harmonic functions g in the outer ball region, which are specified as rows in the standard basis. Functions decrease at infinity and on the boundary sphere their gradients are orthogonal to a given field of directions, the field of directions is itself given by the gradient of one or another basis function f. The results are aimed at solving a general problem when it is not assumed that f and g on the boundary sphere depend on two spherical coordinates. Specific calculations were obtained only under the assumption that neither g nor f is independent of the longitude coordinate. The known results are refined when f is a dipole or quadrupole harmonic and a new result is obtained if f is an octupole harmonic. The results are likely to be useful in the task of restoring the Sun’s magnetic field.