Maxwell's equations

Differential form

$\boldsymbol{\nabla} \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$ ,

$\boldsymbol{\nabla} \cdot \mathbf{B} = 0$ ,

$\boldsymbol{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ ,

$\boldsymbol{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}$ .

$\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {Q_S}{\epsilon_0}$ ,

$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {d \Phi_{B,S}}{dt}$ ,

$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \epsilon_0 \frac {d \Phi_{E,S}}{dt}$ .

Name	Differential form	Integral form
Gauss's law:	$\boldsymbol{\nabla} \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$	$\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {Q_S}{\epsilon_0}$
Gauss' law for magnetism (absence of magnetic monopoles):	$\boldsymbol{\nabla} \cdot \mathbf{B} = 0$	$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$
Faraday's law of induction:	$\boldsymbol{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$	$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {d \Phi_{B,S}}{dt}$
Ampère's Law (with Maxwell's correction):	$\boldsymbol{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}$	$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \epsilon_0 \frac {d \Phi_{E,S}}{dt}$

on to boundary conditions for E and B