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Alfven arbitrary assumed basis boundary conditions bounded charge closed curve closed surface compatibility conditions conservation equations consider constant curl cylindrical defined denote density determined differential equations dimensional displacement current disturbance domain electric field electromagnetic equations entropy example existence theorems exterior flow velocity fluid dynamics fluid magnetics flux formula function gas dynamics given grad harmonic vectors homogeneous homology class homology group inhomogeneous initial values inner product integral irrotational Lagrange multiplier Lagrange multiplier rule linear magnetic field magneto-hydrodynamics mass mathematical Maxwell Maxwell's equations momentum equation normal boundary values normal component NOTES ON MAGNETO-HYDRODYNAMICS obtain Ohm's law open arc open surface oscillations particle perfect conductor plasma potential pre-Maxwell equations pre-Maxwell system prescribed pressure Prom quantities relation scalar Section simple waves solenoidal solenoidal vector sound speed specified surface gradient tangential component theory torus Transverse Waves vanish variables well-posed problem wire zero