Time dependent electromagnetic fields and 4-dimensional Stokes' theorem

Abstract

Stokes' theorem is central to many aspects of physics -- electromagnetism, the Aharonov-Bohm effect, and Wilson loops to name a few. However, the pedagogical examples and research work almost exclusively focus on situations where the fields are time-independent so that one need only deal with purely spatial line integrals ({\it e.g.} $\oint {\bf A} \cdot d{\bf x}$) and purely spatial area integrals ({\it e.g.} $\int (\nabla \times {\bf A}) \cdot d{\bf a} = \int {\bf B} \cdot d{\bf a}$). Here we address this gap by giving some explicit examples of how Stokes' theorem plays out with time-dependent fields in a full 4-dimensional spacetime context. We also discuss some unusual features of Stokes' theorem with time-dependent fields related to gauge transformations and non-simply connected topology.