The Winding Number and Global Potential Functions

Abstract

Theorem 4.2 of Chapter XV gave us a significant criterion for the existence of a potential function, but falls short of describing completely the nature of global obstructions for its existence if we know that the vector field is locally integrable. The present chapter deals systematically with the obstruction, which will be seen to depend on a single vector field. The same considerations are used in subsequent courses on complex analysis and Cauchy’s theorem. The fundamental result proved in the present chapter is valid more generally, but will constitute perfect preparation for those who will subsequently deal with Cauchy’s theorem. In fact, Emil Artin in the 1940s gave a proof of Cauchy’s theorem basing the topological considerations (called homology) on the winding number (cf. his collected works). I have followed here Artin’s idea, and applied it to locally integrable vector fields in an open set U of R ². A fundamental result, quite independent of analysis, is that if the winding number of a closed rectangular path in U is 0 with respect to every point outside U, then the path is a sum of boundaries of rectangles completely contained in U. See Theorem 3.2. If one knows that for certain vector fields their integrals around boundaries of rectangles are 0, then it immediately follows that their integrals along paths satisfying the above condition is also 0. This is the heart of the proof of the global integrability theorem, and may be viewed as a general theorem on circuits in the plane.