ABSTRACT
An electric field describes the rate of change of electrostatic potential at a point of space. That is, it is the negative gradient of the potential function, and, when multiplied by a particle's charge, gives the force acting on that particle. Since electrostatic potential can be computed in a linear scaling fashion, finding the force through differentiation of the electrostatic potential is the easiest method for systems containing large numbers of particles. Differentiation of the electrostatic potential expanded in a multipole series involves simply finding the derivative of the regular multipole moments. If these are defined in spherical polar coordinates, then the final derivatives in Cartesian coordinates can be generated directly using the chain rule. This chapter explains all the steps involved in this differentiation and ends with the derivation of the electrostatic force in Cartesian coordinates in a form suitable for use in the Fast Multipole Method.
