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The Helmholtz Decomposition and the Coulomb Gauge

This results in another decomposition of the electric field E which might be different from that of Helmholtz. Here, we explore the relation between these two decompositions. We also recall that the potentials V and A are not unique, but can be redefined in a systematic way such that the fields E and B are invariant under such redefinition. A particular choice of the potentials is called a choice of gauge, and the relations (3)-(4) are said to be gauge invariant. Returning to Helmholtz’ decomposition, we note that he also showed how