Poincaré gauge in electrodynamics

The gauge presented here, which we call the Poincaré gauge, is a generalization of the well-known expressions φ = −r⋅E0 and A = 1/2 B0×r for the scalar and vector potentials which describe static, uniform electric and magnetic fields. This gauge provides a direct method for calculating a vector potential for any given static or dynamic magnetic field. After we establish the validity and generality of this gauge, we use it to produce a simple and unambiguous method of computing the flux linking an arbitrary knotted and twisted closed circuit. The magnetic flux linking the curve bounding a Möbius band is computed as a simple example. Arguments are then presented that physics students should have the opportunity of learning early in their curriculum modern geometric approaches to physics. (The language of exterior calculus may be as important to future physics as vector calculus was to the past.) Finally, an appendix illustrates how the Poincaré gauge (and others) may be derived from Poincaré’s lemma relating exact and closed exterior differential forms.