Famous PDEs, Maxwell's equations
Important systems of linear partial differential equations are the famous Maxwell equations
$$
\frac{\partial \mathbf{E}}{\partial t} = \mathrm{curl}\mathbf{B},
$$
and
$$
-\mathrm{curl} \mathbf{E} = \mathbf{B}
$$
and
$$
\mathrm{div} \mathbf{E} = \mathrm{div}\mathbf{B} = 0.
$$