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Helmholtz's Theorem

Any vector field v satisfying

[del ·v]_infty=0
(1)
[del xv]_infty=0
(2)

may be written as the sum of an irrotational part and a solenoidal part,

v=-del phi+del xA,
(3)

where

phi=int_V(del ·v)/(4pi|r^'-r|)d^3r^'
(4)
A=int_V(del xv)/(4pi|r^'-r|)d^3r^'.
(5)

See also

Irrotational Field, Solenoidal Field, Vector Field

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References

Arfken, G. "Helmholtz's Theorem." §1.15 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 78-84, 1985.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1084, 2000.

Referenced on Wolfram|Alpha

Helmholtz's Theorem

Cite this as:

Weisstein, Eric W. "Helmholtz's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzsTheorem.html

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