The Maxwell Equations

See The Integral, The Differential.

$$\begin{array}{rlrl}\nabla \cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0}& & \text{(Gauss’s law)}\\ \nabla \cdot \mathbf{B}& =0& & \text{(Gauss’s law for magnetism)}\\ \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}& & \text{(Faraday’s law)}\\ \nabla \times \mathbf{B}& ={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}& & \text{(Ampeˋre–Maxwell law)}\end{array}$$ $$\begin{array}{rl}{\oint }_{\partial V}\mathbf{E}\cdot d\mathbf{A}& =\frac{1}{{\epsilon }_0}{\int }_V\rho dV\\ {\oint }_{\partial V}\mathbf{B}\cdot d\mathbf{A}& =0\\ {\oint }_{\partial S}\mathbf{E}\cdot d\mathbf{\ell }& =-\frac{d}{dt}{\iint }_S\mathbf{B}\cdot d\mathbf{A}\\ {\oint }_{\partial S}\mathbf{B}\cdot d\mathbf{\ell }& ={\mu }_0{\iint }_S\mathbf{J}\cdot d\mathbf{A}+{\mu }_0{\epsilon }_0\frac{d}{dt}{\iint }_S\mathbf{E}\cdot d\mathbf{A}\end{array}$$ $$\begin{array}{rl}\nabla \cdot \mathbf{D}& ={\rho }_{\mathrm{free}}\\ \nabla \cdot \mathbf{B}& =0\\ \nabla \times \mathbf{E}& =-\frac{\partial \mathbf{B}}{\partial t}\\ \nabla \times \mathbf{H}& ={\mathbf{J}}_{\mathrm{free}}+\frac{\partial \mathbf{D}}{\partial t}\end{array}$$ $$\nabla \cdot \mathbf{J}+\frac{\partial \rho }{\partial t}=0$$