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:
$$
\pdv{E_x}{x} + \pdv{E_y}{x} +
\pdv{E_z}{x} =
\frac{\rho}{\vacuumPermittivity}
\\

\pdv{B_x}{x} +
\pdv{B_y}{x} +
\pdv{B_z}{x} =
0
\\

\pdv{E_z}{y} - \pdv{E_y}{z} = -\pdv{B_x}{t} \\
\pdv{E_x}{z} - \pdv{E_z}{x} = -\pdv{B_y}{t} \\
\pdv{E_y}{x} - \pdv{E_x}{y} = -\pdv{B_z}{t} \\

\pdv{B_z}{y} - \pdv{B_y}{z} = \vacuumPermeability \left(J_x + \vacuumPermittivity \pdv{E_x}{t} \right) \\
\pdv{B_x}{z} - \pdv{B_z}{x} = \vacuumPermeability \left(J_y + \vacuumPermittivity \pdv{E_y}{t} \right) \\
\pdv{B_y}{x} - \pdv{B_x}{y} = \vacuumPermeability \left(J_z + \vacuumPermittivity \pdv{E_z}{t} \right) \\
$$
{id=equation-maxwells-equation-explicit}


Discussion on "Explicit scalar form of the Maxwell's equations" by Ciro Santilli (@cirosantilli, 34)