The expressions of spherical functions and their derivatives in a Cartesian coordinate system are obtained. In opposite to the representation of polynomials in a spherical coordinate system, the derived recurrence relations make it possible to use them in the description of physical processes, and the obtained formulae for derivatives of spherical functions within the sphere allow obtaining the solutions to the problems of mathematical physics for spherical bodies in a Cartesian coordinate system. This approach has its advantages precisely in the applied problems. For example, for the determination of the artificial satellites orbits, it is necessary to represent the external potential of gravitation and its derivatives for the GPS systems in a Cartesian coordinate system. Investigation of the internal structure of Earth and astrometric studies of processes in galaxies are associated with the study of internal potential, and, consequently, there is a necessity for its presentation in the Cartesian coordinates.
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