Poisson’s differential equation

The relation between the spatial distribution of the electrostatic potential /(jc) and the spatial distribution of charge density Qy(x) can be stated, generally, in terms of Poisson s differential equation. 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields... 

The property of the surface configuration to reach its equilibrium value, with minimum area, in seconds is used to obtain the solution of mathematical minimization problems in the following two chapters, and in Chapter 6 it is applied to the solution of Laplace s and Poisson s differential equation. 

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