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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

{\bf F}=\nabla\times(T{\bf r})+\nabla^2(S{\bf r}) \equiv {\bf T}+{\bf S},
\end{displaymath} (1)

$\displaystyle {\bf T}$ $\textstyle \equiv$ $\displaystyle \nabla\times(T{\bf r}) = -{\bf r}\times(\nabla T)$ (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.

See also Beltrami Field, Irrotational Field, Poloidal Field, Solenoidal Field, Toroidal Field

© 1996-9 Eric W. Weisstein