For numerical algorithms and to get a more low level understanding of the equations, we can expand all terms to the simpler and more explicit form:

$∂x∂E_{x} +∂x∂E_{y} +∂x∂E_{z} =ε_{0}ρ ∂x∂B_{x} +∂x∂B_{y} +∂x∂B_{z} =0∂y∂E_{z} −∂z∂E_{y} =−∂t∂B_{x} ∂z∂E_{x} −∂x∂E_{z} =−∂t∂B_{y} ∂x∂E_{y} −∂y∂E_{x} =−∂t∂B_{z} ∂y∂B_{z} −∂z∂B_{y} =μ_{0}(J_{x}+ε_{0}∂t∂E_{x} )∂z∂B_{x} −∂x∂B_{z} =μ_{0}(J_{y}+ε_{0}∂t∂E_{y} )∂x∂B_{y} −∂y∂B_{x} =μ_{0}(J_{z}+ε_{0}∂t∂E_{z} )$

As seen from explicit scalar form of the Maxwell's equations, this expands to 8 equations, so the question arises if the system is over-determined because it only has 6 functions to be determined.

As explained on the Wikipedia page however, this is not the case, because if the first two equations hold for the initial condition, then the othe six equations imply that they also hold for all time, so they can be essentially omitted.

It is also worth noting that the first two equations don't involve time derivatives. Therefore, they can be seen as spacial constraints.

TODO: the electric field and magnetic field can be expressed in terms of the electric potential and magnetic vector potential. So then we only need 4 variables?

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