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., closed integral A . dx) and purely spatial area integrals ( e. g., integral(del x A) . da = integral 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. (C) 2016 American Association of Physics Teachers.
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机译：斯托克斯定理在物理学-电磁学，阿哈罗诺夫-博姆效应和威尔逊循环等许多方面至关重要。但是，这些教学示例和研究工作几乎都只关注那些与时间无关的字段，因此人们只需要处理纯空间线积分（例如，封闭积分A。dx）和纯空间区域积分（例如，积分（ del x A）。da =积分B。da）。在这里，我们通过给出一些明确的例子来解决这一差距，这些例子说明了斯托克斯定理如何在完整的4维时空上下文中与时间相关的字段一起发挥作用。我们还讨论了斯托克斯定理的一些不寻常的特征，它们涉及与规范转换和非简单连接拓扑有关的时变字段。 （C）2016年美国物理教师协会。
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