Invariance of electric charges covariance of 557 write the maxwell equations themselves in an explicitly covariant form. the equations are (11.135) car c of f. and the 4-current尸these take on the covariant form in terms se equa- are the (11.141 similarly, the homogeneous maxwell equations 11.136) 1 ab can be written in terms of the dual field-strength tensor as (11.142) 1.137) in terms of fo rather thanso, these homogeneous equations are the four a- equations (11.143) where a, b, y are any three of the integers 0, 1, 2, 3. with the definitions of j (11.128), a" (11.132), and b (11.136), together with the wave equations (11.133) or the maxwell equations (11.141) and (11.142), the covariance of the equations of electromagnetism is established. to complete the discussion, we put the lorentz force and rate of change of energy equations .138) seful sym- (11.125) and (11.126) in manifestly covariant form, (11.144) dr c the covariant description of the conservation laws of a combined system of elec- tromagnetic fields and charged particles and a covariant solution for the fields of a moving charge are deferred to chapter 12, where a lagrangian formulation is presented 139) ace tial and ant for the macroscopic maxwell equations it is necessary to distinguish two field-strength tensors, fw (e, b) and g- (d, h), where fus is given by (11.137) and gas is obtained from (11.137) by substituting e-, d and b- h, the covariant form of the maxwell equations is then (11.145) it is clear that with the fields (e, b) and (d, h) transforming as antisymmetric second-rank tensors the polarization p and the negative magnetization -m form a similar second-rank tensor. with these quantities given meaning as averages of atomic properties in the rest frame of the medium, the electrodynam ics of macroscopic matter in motion is specified.