The mathematical tools of natural sciences have faithfully served the research and engineering communities in their pursuit of technological advance. Nevertheless, a review of those tools is warranted in light of the availability of modern computing resources and their capabilities. (Most mathematical tools were developed for the use by humans who prefer to do fewer, more complex operations to doing many simpler ones, and who prefer derivatives to integrals, to name just two cases. The computers exhibit the opposite preferences and casting of the problem for computer evaluation calls for new algorithmic approaches.) The potential theory is one such example, and we will examine it from the computational point of view. Starting from its underlying principles, we will show how a familiar equivalent to the potential theory can be converted from a static into a dynamic mathematical tool suitable for computer evaluation. A similar approach applied to Maxwell's equations opens the door for development of computer algorithms that effectively address complex processes in conductors and semiconductors, radiation from accelerated electrons, and other dynamic phenomena associated with moving electric charges.