A divergenceless vector field, also called a solenoidal field, is a vector field for which 
. Therefore, there exists a 
such that 
. Furthermore, 
can be written as
 |  |  |
(1)
|
 |  |  |
(2)
|
where
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  | ![del [partial/(partialr)(rS)]-rdel ^2S.](/images/equations/DivergencelessField/Inline22.svg) |
(6)
|
Following Lamb's 1932 treatise (Lamb 1993), 
and 
are called toroidal field
and poloidal field.
