Potential Theory

Some basic equations of potential theory are presented in this chapter, together with terminology and definitions needed to understand the theory. An especially important result is Green’s representation formula, which gives the potential at a point \(\boldsymbol{\xi }\) inside a region V, but not on its boundary S, in terms of single and double layer potentials with density functions equal to (a) the normal derivative of the potential on S and (b) the potential on the same boundary. Expressions are also given for the gradient and normal derivative of the potential \(\phi \) at points \(\boldsymbol{\xi }\) inside region V, as well as limit values of all expressions for the case that \(\boldsymbol{\xi }\rightarrow S\) .