Helmholtz's Theorem

Any Vector Field ${\bf v}$ satisfying

$${[}\nabla\cdot {\bf v}]_\infty = 0$$ (1)

$${[}\nabla \times {\bf v}]_\infty = 0$$ (2)

may be written as the sum of an Irrotational part and a Solenoidal part,

$${\bf v} = -\nabla \phi +\nabla \times {\bf A},$$ (3)

where for a Vector Field $F$,

$$\phi = - \int_V {\nabla\cdot {\bf F}\over 4\pi \vert{\bf r}'-{\bf r}\vert}\, d^3{\bf r}'$$ (4)

$${\bf A} = \int_V {\nabla \times {\bf F}\over 4\pi \vert{\bf r}'-{\bf r}\vert} d^3{\bf r}'.$$ (5)

See also Irrotational Field, Solenoidal Field, Vector Field

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, 5th ed. San Diego, CA: Academic Press, pp. 1084, 1980.