UC Berkeley

Editor's Note: Field lines are ubiquitous in introductory physics but mainly as sketches to aid intuition. In this article, the authors explore the surprising and subtle mathematics of field lines. They focus on two questions from electrostatics: (1) When does a set of field lines represent a conservative field? (2) What systems give rise to straight field lines? The canonical examples of Gauss' law—spherical, planar, and line charge distributions—are familiar to physicists, but are there others? To answer these questions, the authors turn to differential geometry. They provide a self-contained introduction to integrability and curvature, starting from vector calculus. The elegant mathematics provides new insights into electrostatics and minimal surfaces, and the material could be used to supplement a wide range of courses. The mathematics will be familiar to students who have taken a junior-level electrodynamics or math methods course, and the methods and results can be applied across the gamut of theoretical physics.