What does "curl E = const." on Ω say about E on ∂Ω? Assume I have a simply connected domain Ω and a twice differentiable vector field E for which I know that "∇×E = const." (1) and "∇E = 0" (2) on Ω - I am interested in solving a BC Problem on ∏ = (Ʃ ⊃ Ω)\Ω, the remainder of Ʃ less Ω. (1) and (2) imply certain restrictions on the BC on ∏. Question: Which are the restrictions equivalent to (1) and (2)? By Stokes' theorem, ∫dr·E = ∫dA·const. along the boundary of ∂Ω, but that alone can't possibly be equivalent, can it? I might pick an E which satisfies a certain curve integral value along ∂Ω and which can't satisfy (1) and (2), I assume. Context: A conductor is forming a loop the hole in which is pierced by a changing magnetic field - how this can be re-formulated into BCs on the conductor's domain? Can it, at all?