the first two Maxwell's equations in tensor form


dμFνλ+dνFλμ+dλFμν=0
given that Fμν=dμAνdνAμ=dAνdxμdAμdxν
assuming μ=0 and ν=j=(1,2,3) in the first equation, we get
d0Fjλ+djFλ0+dλF0j=0

now let’s write λ=(0,k) where k = (1,2,3)
d0Fj0+d0Fjk+djF00+djFk0+d0F0j+dkF0j=0

d0(Ej)+d0(Bk)+dj(Ek)+d0(+Ej)+dk(+Ej)=0

d0(Bi)+dj(Ek)+dk(+Ej)=0

d0(Bi)+(×E)i=0

similarly when ν=0 and λ=0, we get two more equations
d0(Bj)+(×E)j=0

d0(Bk)+(×E)k=0

in general we can write
d0B+×E=0

1cdBdt+×E=0

when d=(d0,di)=(1cddt,)
now, assuming μ=i and ν=j
diFjλ+djFλi+dλFij=0

now, for λ=(0,k) where k=(1,2,3) as before
diFj0+diFjk+djF0i+djFki+d0Fij+dkFij=0

diFj0+djF0i+d0Fij+diFjk+djFki+dkFij=0

di(Ej)+dj(+Ei)+d0(Bk)+di(Bi)+dj(Bj)+dk(Bk)=0

di(Bi)+dj(Bj)+dk(Bk)=0

B=0

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