This can be shown mathematically for a flux tube using the induction equation of a perfectly conducting fluid ∂ ∂ B ∂ ∂ t = ∇ ∇ × × ( v × × B ) where B is the magnetic field and v is the velocity field of the fluid. The change in magnetic flux over time through any open surface of the flux tube S enclosed by C with a differential line element d l can be written as d Φ Φ B d t = ∫ ∫ S ∂ ∂ B ∂ ∂ t ⋅ ⋅ d S + ∮ ∮ C B ⋅ ⋅ v × × d l . Using the induction equation gives d Φ Φ B d t = ∫ ∫ S ∇ ∇ × × ( v × × B ) ⋅ ⋅ d S + ∮ ∮ C B ⋅ ⋅ v × × d l which can be rewritten using Stokes' theorem and an elementary vector identity on the first and second term, respectively, to give ∫ ∫ S B ⋅ ⋅ d S = const .