Divergenceless Field

A divergenceless field, also called a Solenoidal Field, is a Field for which $\nabla\cdot {\bf F}\equiv {\bf0}$ . Therefore, there exists a ${\bf G}$ such that ${\bf F}=\nabla\times{\bf G}$ . Furthermore, ${\bf F}$ can be written as

$\begin{displaymath} {\bf F}=\nabla\times(T{\bf r})+\nabla^2(S{\bf r}) \equiv {\bf T}+{\bf S}, \end{displaymath}$

(1)

where

$\displaystyle {\bf T}$	$\textstyle \equiv$	$\displaystyle \nabla\times(T{\bf r}) = -{\bf r}\times(\nabla T)$	(2)
$\displaystyle {\bf S}$	$\textstyle \equiv$	$\displaystyle \nabla^2(S{\bf r}) = \nabla\left[{{\partial \over \partial r} (rS)}\right]- {\bf r}\nabla^2 S.$	(3)

Following Lamb, ${\bf T}$ and ${\bf S}$ are called Toroidal Field and Poloidal Field.