An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism is a fundamental publication by George Green in 1828, where he extends previous work of Siméon Denis Poisson on electricity and magnetism. The work in mathematical analysis, notably including what is now universally known as Green's theorem, is of the greatest importance in all branches of mathematical physics. It contains the first exposition of the theory of potential. In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
It is in this essay that the term 'potential function' first occurs. Herein also his remarkable theorem in pure mathematics, since universally known as Green's theorem, and probably the most important instrument of investigation in the whole range of mathematical physics, made its appearance. We are all now able to understand, in a general way at least, the importance of Green's work, and the progress made since the publication of his essay in 1828. But to fully appreciate his work and subsequent progress one needs to know the outlook for the mathematico-physical sciences as it appeared to Green at this time and to realize his refined sensitiveness in promulgating his discoveries.
Overview
Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George Green. Green's treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point: to this function Green gave the name potential, by which it has always since been known.
In 1828, Green published the paper which is the essay he is most famous for today. When Green published his Essay, it was sold on a subscription basis to 51 people, most of whom were friends and probably could not understand it. The wealthy landowner and mathematician Edward Bromhead bought a copy and encouraged Green to do further work in mathematics. Not believing the offer was sincere, Green did not contact Bromhead for two years.
Upon publishing the work, he first introduced the term 'potential' to denote the result obtained by adding the masses of all the particles of a system, each divided by its distance from a given point; and the properties of this function are first considered and applied to the theories of magnetism and electricity. This was followed by two papers communicated by Sir Bromhead to the Cambridge Philosophical Society: (1)' On the Laws of the Equilibrium of Fluids analogous to the Electric Fluid ' (12 Nov. 1832); (2)' On the Determination of the Attractions of Ellipsoids of Variable Densities ' (6 May 1833). Both papers display great analytical power, but are rather curious than practically interesting. Green's 1828 essay was neglected by mathematicians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson J. It did influence the work of Lord Kelvin and James Clerk Maxwell.
The self-taught mathematician's essay was one of the greatest advances that were made in the mathematical theory of electricity up to his time. "His researches," as Sir William Thomson has observed, "have led to the elementary proposition which must constitute the legitimate foundation of every perfect mathematical structure that is to be made from the materials furnished in the experimental laws of Coulomb. Not only do they afford a natural and complete explanation of the beautiful quantitative experiments which has been so interesting at all times to practical electricians, but they suggest to the mathematician the simplest and most powerful methods of dealing with problems which, if attacked by the mere force of the old analysis, must have remained forever unsolved."