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### Integral formulas for computing a third-order gravitational tensor from volumetric mass density, disturbing gravitational potential, gravity anomaly and gravity disturbance

 Citace: ŠPRLÁK, M., NOVÁK, P. Integral formulas for computing a third-order gravitational tensor from volumetric mass density, disturbing gravitational potential, gravity anomaly and gravity disturbance. JOURNAL OF GEODESY, 2015, roč. 89, č. 2, s. 141-157. ISSN: 0949-7714 ČLÁNEK eng Integral formulas for computing a third-order gravitational tensor from volumetric mass density, disturbing gravitational potential, gravity anomaly and gravity disturbance 2015 Ing. Michal Šprlák Ph.D. , Prof. Ing. Pavel Novák Ph.D. , Nový matematický aparát pro výpočet gravitačního tenzoru 3. řádu je diskutován v tomto příspěvku. Prvně byly sestaveny diefernciální operátory pro složky gravitačního tenzoru 3. řádu v lokálním rámci LNOF. Je ukázáno, že diferenciální operátory mohou být rozloženy do dvou komponent: azimutální a izotropní. Diferenciální operátory jsou zjednodušeny pro jistou třídu izotropních jader. A new mathematical model for evaluation of the third-order (disturbing) gravitational tensor is formulated in this article. Firstly, we construct corresponding differential operators for the components of the third-order (disturbing) gravitational tensor in a spherical local north-oriented frame. We show that the differential operators may efficiently be decomposed into an azimuthal and an isotropic part. The differential operators are even more simplified for a certain class of isotropic kernels. Secondly, the differential operators are applied to the well-known integrals of Newton, Abel-Poisson, Pizzetti and Hotine. In this way, 40 new integral formulas are derived. The new integral formulas allow for evaluation of the components of the third-order (disturbing) gravitational tensor from density distribution, disturbing gravitational potential, gravity anomalies and gravity disturbances. Thirdly, we investigate the behaviour of the corresponding integral kernels in the spatial domain. The new mathematical formulas extend the theoretical apparatus of geodesy, i.e. the well-known Meissl scheme, and reveal important properties of the third-order gravitational tensor. They may be exploited in geophysical studies, continuation of gravitational field quantities and analysing the gradiometric-geodynamic boundary value problem.

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