Gauge freedom in magnetostatics and the effect on helicity in toroidal volumes

Magnetostatics defines a class of boundary value problems in which the topology domain plays subtle role. For example, representability divergence-free field as curl vector potential comes about because homological considerations. With this mind, we study gauge freedom magnetostatics and its effect on comparison between magnetic configurations through key quantities such helicity. this, apply Hodge decomposition k-forms compact orientable Riemaniann manifolds with smooth boundary, well de Rham cohomology, to representation fields one-forms toroidal volumes. An advantage approach is recovery classical results without explicit coordinates assumptions exterior domain. In particular, detailed construction minimal gauges formal proof relative helicity formulas are presented.