Electromagnetism uniqueness theorem

The electromagnetism uniqueness theorem states the uniqueness (but not necessarily the existence) of a solution to Maxwell's equations, if the boundary conditions provided satisfy the following requirements: At t = 0 {\displaystyle t=0} , the initial values of all fields (E, H, B and D) everywhere (in the entire volume considered) is specified; For all times (of consideration), the component of either the electric field E or the magnetic field H tangential to the boundary surface ( n ^ × E {\displaystyle {\hat {n}}\times \mathbf {E} } or n ^ × H {\displaystyle {\hat {n}}\times \mathbf {H} } , where n ^ {\displaystyle {\hat {n}}} is the normal vector at a point on the boundary surface) is specified. Note that this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution, when the source distribution is outside of the volume specified in the initial condition.

Source: Wikipedia — Electromagnetism uniqueness theorem (CC BY-SA 4.0)