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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?

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# What does curl E = const. on Ω say about E on ∂Ω?

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