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where ^ is an arbitrary scalar, its curl satisfies condition (1). The result (4) together with Stokes' theorem gives §-Jb-dl-O, i.e., that the line integral of B about a circuit which moves with the fluid is constant. This may also be proven directly. The velocity vector q satisfies (5) as a dynamical condition and the vorticity vector V X q satisfies (1). Thus (4) gives the theorem for the constancy of flux of vorticity and (6), the constancy of circulation.doi:10.1090/qam/29707 fatcat:yvymjyfzsbaxfdjvdc2qg4nlxq