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)


T=del x(Tr)
=-rx(del T)
S=del ^2(Sr)
=del [partial/(partialr)(rS)]-rdel ^2S.

Following Lamb's 1932 treatise (Lamb 1993), T and S are called toroidal field and poloidal field.

See also

Beltrami Field, Divergence, Irrotational Field, Poloidal Field, Solenoidal Field, Toroidal Field, Vector Field

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Lamb, H. Hydrodynamics, 6th ed. Cambridge, England: Cambridge University Press, 1993.

Referenced on Wolfram|Alpha

Divergenceless Field

Cite this as:

Weisstein, Eric W. "Divergenceless Field." From MathWorld--A Wolfram Web Resource.

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