The electromagnetic fields have unique solutions as determined by Maxwell’s equations. However, when substituting the scalar and vector potentials into Maxwell’s equations, those potentials typically have an infinite number of solutions — unless additional equations or conditions are applied. For example, in The V Formulation, V is not uniquely defined as V’ = V + C (C is a constant) yields the same electrostatic field; in The A Formulation, A also has infinite solutions since A’ = A + ∇f (f is a scalar field) yields the same magnetic flux density; in The A–V Formulation, A and V have infinite solutions due to Gauge Transformations. Therefore, potentials based formulations of Maxwell’s equations are preliminarily ungauged. An ungauged formulation can only be solved with an iterative solver as the assembled matrix is singular.

In order to have a unique solution for potentials and be solved with a direct solver, a gauge has to be applied. In The V Formulation, V needs at least one additional condition fixing its general level. Most often, this is provided as a boundary condition (ground or applied potential). In The A Formulation and The A–V Formulation, additional constraint on the divergence of A is required. A common choice is the Coulomb gauge ∇ ⋅ A = 0.