In classical electromagnetism, the importance of potentials can not be overemphasized. Although gradient (or closed) additions to the potentials do not effect the E and B fields, there do exist physical effects which are "gauge" dependent. In fact, the concepts of classical electromagnetism appear to be formulated in terms of two topological constraints on the domain of independent variables. The two topological constraints can be formulated as
F-dA= 0 and J-dG=0.
The 2-form of F (field intensities) and the 3-form J (charge currents) are exact
by definition of the classical electromagnetic system. Domains of support for F
cannot be compact without boundary except for the two special cases of the torus
and the Klein bottle. Similar ideas apply to the domains of support for the
charge currents, J.
There exist two other electormagnetic 3-forms (in 4 dimensions), the Torsion
vector, A^F, and the Spin current, A^G.
The divergence of these 3-forms create the deformable integral invariants called
the Poincare invariants of classical systems.