Equations of Electrostatics and Magnetostatics

In the most general case of distributions of charges and currents, Maxwell's Laws are:
\[\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}\]
 
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
 
\[\mathbf{\nabla} \times \mathbf{E}=- \mu \frac{\partial H}{\partial t}\]
 
\[\mathbf{\nabla} \times \mathbf{H}=- \epsilon \frac{\partial E}{\partial t}+ \mathbf{J}\]
 
In the case of static fields,  
\[\mathbf{E}, \mathbf{H}\]
  are constant, so that  
\[\frac{\partial \mathbf{E}}{\partial t} =\frac{\partial \mathbf{H}}{\partial t}=0\]

Hence
\[\mathbf{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon}\]
 
\[\mathbf{\nabla} \cdot \mathbf{H}=0\]
 
\[\mathbf{\nabla} \times \mathbf{E}=0\]
 
\[\mathbf{\nabla} \times \mathbf{H}= \mathbf{J}\]