Divergenceless Field
A divergenceless vector field, also called a solenoidal field, is a vector field for which del ·F=0. Therefore, there exists a G such that F=del xG. Furthermore, F can be written as
F = del x(Tr)+del ^2(Sr)
(1)
= T+S,
(2)
where
T = del x(Tr)
(3)
= -rx(del T)
(4)
S = del ^2(Sr)
(5)
![画像:del [partial/(partialr)(rS)]-rdel ^2S.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/DivergencelessField/Inline22.svg){kind=link}
Following Lamb's 1932 treatise (Lamb 1993), T and S are called toroidal field and poloidal field.
See also
Beltrami Field, Divergence, Irrotational Field, Solenoidal Field, Toroidal Field, Vector FieldExplore with Wolfram|Alpha
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References
Lamb, H. Hydrodynamics, 6th ed. Cambridge, England: Cambridge University Press, 1993.Referenced on Wolfram|Alpha
Divergenceless FieldCite this as:
Weisstein, Eric W. "Divergenceless Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DivergencelessField.html