Derivatives of the Gravity Potential with Respect to Rectangular Coordinates

Starting from a complex operator of derivation, we give expressions for derivatives of arbitrary order of the gravity potential with respect to rectangular coordinates. These expressions have a form similar to the original potential expanded in spherical harmonics and are free of singularity at the poles. Computing sets of numerical coefficients once for all, we can compute the derivatives with a very limited work: the same functions are used to compute all derivatives by means of a unique parametrized formula. This is very comfortable for further algebraic manipulations. Numerical tests prove the accuracy and the efficiency of the algorithm derived from our formula to compute the gravity acceleration vector and the gravity gradient tensor.