(1) To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing. The Laplacian Operator. {\displaystyle \phi } is the azimuthal angle. The Laplacian is. It is nearly ubiquitous. Before going through the Carpal-Tunnel causing calisthenics to calculate its form in cylindrical and spherical coordinates, the results appear here so … Its form is simple and symmetric in Cartesian coordinates. (2) Then the Helmholtz differential equation becomes. The Laplacian Operator is very important in physics. It looks more complicated than in Cartesian coordinates, but solutions in spherical coordinates almost always do not contain cross terms. from Cartesian to Cylindrical to Spherical Coordinates. ChronologyRegionallyMathRandomLinksContact. In spherical coordinates, the scale factors are , , , and the separation functions are , , , giving a Stäckel determinant of . Laplace's Equation--Spherical Coordinates. Laplace's equation in spherical coordinates can then be written out fully like this.