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∂Ex+∂x∂Ey+∂x∂Ez=ε0ρ∂x∂Bx+∂x∂By+∂x∂Bz=0∂y∂Ez−∂z∂Ey=−∂t∂Bx∂z∂Ex−∂x∂Ez=−∂t∂By∂x∂Ey−∂y∂Ex=−∂t∂Bz∂y∂Bz−∂z∂By=μ0(Jx+ε0∂t∂Ex)∂z∂Bx−∂x∂Bz=μ0(Jy+ε0∂t∂Ey)∂x∂By−∂y∂Bx=μ0(Jz+ε0∂t∂Ez)