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Stokes' theorem is central to many aspects of physics -- electromagnetism,the Aharonov-Bohm effect, and Wilson loops to name a few. However, thepedagogical examples and research work almost exclusively focus on situationswhere the fields are time-independent so that one need only deal with purelyspatial line integrals ({\it e.g.} $\oint {\bf A} \cdot d{\bf x}$) and purelyspatial 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 someexplicit examples of how Stokes' theorem plays out with time-dependent fieldsin a full 4-dimensional spacetime context. We also discuss some unusualfeatures of Stokes' theorem with time-dependent fields related to gaugetransformations and non-simply connected topology.

Time dependent electromagnetic fields and 4-dimensional Stokes' theorem