Conversion of Laplace equation from the spherical coordinate description to Cartesian coordinate system

Abstract

In this article, we show how the solution of the Laplace's equation in spherical coordinate and its coefficients can be rewritten in Cartesian coordinates, and we show the necessary normalization of these coefficients to ensure the identity of these solutions with the pure Cartesian solution. The solution of the Laplace equation in spherical coordinates is obtained by separating the longitude, colatitude and radius variables. An elementary solution – a basis function - is the product of the negative integer exponent of the radius coordinate, the associated Legendre function of the colatitude coordinate, and the harmonic function of the longitude coordinate. Using symbolic computations in MATLAB©, we generated these functions. The associated Legendre functions were created with the Rodrigues formula. Within the symbolic module we were able to simplify the functions with trigonometric transformations and convert the spherical coordinates into Cartesian coordinates. As a result, the obtained formulas that could be directly compared to the pure Cartesian solution based on the 3D Taylor series, and this enabled us to calculate the normalization coefficients. The coefficients we obtained were the Gauss normalization coefficients, instead of Schmidt normalization coefficients, regularly used in geoid calculations.