Connectivity and Smoke-Rings: Green's Second Identity in its First Fifty Years

Summary

Introduction

James Clerk Maxwell, in his review of Thomson and Tait's Treatise on Natural Philosophy, noted an important innovation in the authors' approach to mathematics ([8], Vol. 2, p. 777):

The first thing which we observe in the arrangement of the work is the prominence given to kinematics, … and the large space devoted under this heading to what has been hitherto considered part of pure geometry. The theory of curvature of lines and surfaces, for example, has long been recognized as an important branch of geometry, but in treatises on motion it was regarded as lying as much outside of the subject as the four rules of arithmetic or the binomial theorem.

The guiding idea however…is that geometry itself is part of the science of motion, and that it treats, not of the relations between figures already existing in space, but of the process by which these figures are generated by the motion of a point or a line.

This “guiding idea,”which treats geometric entities as physical objects in some sense, had been influential with mathematicians for many years. Countless mathematical problems have their origin in the investigation of the natural world. However, it also happens that the solutions of some problems may be facilitated by attributing physical properties to the mathematical objects under study. In addition, mathematical constructs usually thought of as “purely geometric” may be created by considering such mathematico-physical entities.