Any vector field 
satisfying
![[del ·v]_infty](/images/equations/HelmholtzsTheorem/Inline2.svg) |  |  |
(1)
|
![[del xv]_infty](/images/equations/HelmholtzsTheorem/Inline5.svg) |  |  |
(2)
|
may be written as the sum of an irrotational part and a solenoidal part,
 |
(3)
|
where
 |  |  |
(4)
|
 |  |  |
(5)
|
Helmholtz's Theorem
See also
Irrotational Field, Solenoidal Field, Vector FieldExplore with Wolfram|Alpha
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 TheoremCite this as:
Weisstein, Eric W. "Helmholtz's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzsTheorem.html