Laplace’s Equation and Poisson’s Equation

Abstract

In Chapter 3, it was noted that both gravitational and electrostatic potentials satisfy Laplace’s equation ▽² = 0 in any region where the mass and charge densities vanish. In fact, the Laplacian operator ▽² occurs naturally in many applications of mathematics, since it is intimately related to the total outflow rate from a region in which a flux vector is proportional to the gradient of a scalar. Derivation of the important result (

$$ \smallint \smallint _{_{\partial {\cal R}} } \nabla \varphi \cdot ndS = \smallint \smallint \smallint _{\cal R} \nabla ^2 \varphi dV $$

(4.1)

) equating the total flux of the field ▽φ through the bounding surface ∂R of a general three-dimensional region R to the volume integral of ▽²φ over R is a central result in vector calculus (see e.g. Matthews (1998)). Its truth is readily demonstrated, as indicated below, by first considering a general rectangular block, then any combination of adjoining blocks and, finally, by combining this with a result for an infinitesimal tetrahedron.