dated 2010-02-16 may be checked for validity and style.
Given a charge density ρ(t, xi) and 3‑current density Ji(t, xi) at time t and location xi in a space with three spatial dimensions where i ∈ {1, 2, 3}, the Maxwell equations of electrodynamics including improvements initiated by Heaviside and Hertz are
| c εijk ∇j Bk − ∂t Ei = Ji , ∇i Ei = ρ , |
| c εijk ∇j Ek + ∂t Bi = 0 , ∇i Bi = 0 , |
∇i here referring to ∂/∂xi and
∂t referring to ∂/∂t,
c being the limiting velocity of electrodynamics
and εijk
the totally antisymmetric Levi-Civitá symbol.
In Minkowski space-time, the Maxwell equations take the form
| c gμρ ∂μ Fρσ = Jσ , |
| ∈μνρσ ∂μ Fρσ = 0 , |
from the first equation of which follows conservation of charge
∂σ Jσ = 0 ,
and from the second equation, the Bianchi identity
| ∂μ Fρσ + ∂ρ Fσμ + ∂σ Fμρ = 0 |
http://www.maths.tcd.ie/~nhb/y/maxwell.php
dated
2010-02-16 may be checked for
validity and
style.