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
where
Following Lamb's 1932 treatise (Lamb 1993), 
and 
are called toroidal field
and poloidal field.
See also
Beltrami Field, Divergence, Irrotational Field, Solenoidal Field, Toroidal Field, Vector Field
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References
Lamb, H. Hydrodynamics, 6th ed. Cambridge, England: Cambridge University Press, 1993.
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Cite this as:
Weisstein, Eric W. "Divergenceless Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DivergencelessField.html