567 ! APÉNDICE IV. LAS ECUACIONES DE MAXWELL EN UN SOLO SISTEMA AXIOMATICO Por John Auping con base en dos artículos de José Heras Gómez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Ó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`L/!ab^Pa^b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Ó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ÓN DE LAS ECUACIONES DE MAXWELL DE LA ECUACIÓN DE CONTINUIDAD ! G.;7H2+.3!I24)7!*2!;)76<246@,!+),46.,2;2!),!)*!)3:246.!-76;6+),36.,2*8!532,;.!<)4-.7)38!:)7.! )3-)!42+6,.!)3!+59!*27?.J!G.7!)3.8!.:-2+.3!:.7!)*!42+6,.!)3:2462*K-)+:.72*8!),!L.7+2!-),3.762*J! 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