Maxwell's equations

	Differential Form	Integral Form
Gauss’s law for electricity	$\nabla \cdot \vec{\mathbf{E}}=\frac{\rho }{{\epsilon }_0}$ $\rho$ is the charge density	${\iint }_S\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}=\frac{Q_{\text{enc}}}{{\epsilon }_0}$ where $S$ is any closed surface
Gauss’s law for magnetism	$\nabla \cdot \vec{\mathbf{B}}=0$	${\iint }_S\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}=0$ where $S$ is any closed surface
Maxwell–Faraday equation	$\nabla \times \vec{\mathbf{E}}=-\frac{\partial \vec{\mathbf{B}}}{\partial t}$	${\oint }_{\partial S}\vec{\mathbf{E}}\cdot d\vec{\mathbf{\ell }}=-\frac{d}{dt}{\iint }_S\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}$ where $\partial S$ is the boundary curve of a surface $S$
Ampère–Maxwell law	$\nabla \times \vec{\mathbf{B}}={\mu }_0\left(\vec{\mathbf{J}}+{\epsilon }_0\frac{\partial \vec{\mathbf{E}}}{\partial t}\right)$	${\oint }_{\partial S}\vec{\mathbf{B}}\cdot d\vec{\mathbf{\ell }}={\mu }_0\left(I_{\text{enc}}+{\epsilon }_0\frac{d}{dt}{\iint }_S\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}\right)$ where $\partial S$ is the boundary curve of a surface $S$