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 (e.g., A·dx) and purely spatial area integrals (e.g., (×A)·da=B·da). 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.

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19.

If one considers sinusoidal time dependence, the magnetic field will be non-zero outside the solenoid and the forms of both the electric and magnetic fields will involve Bessel and Neumann functions (Ref. 3).

20.

Here, the requirement that the linear speed be the same along all line segments is a convenience. However, since one of the applications of our analysis is to the Aharonov-Bohm effect where one wants to eliminate or minimize the external forces on the particle tracing out the spacetime path we take the speed to be constant Of course at the bends in the paths there will be forces but these can be thought of as the bending forces due to crystalline diffraction such as in the experiment in Ref. 11.

21.

Note that the spacetime area integration connected with path 2 runs along dφ. It is not along the linear path 2 which would be an integration along dρ.

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