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Helmholtz's Theorem

Any Vector Field ${\bf v}$ satisfying

$\displaystyle {[}\nabla\cdot {\bf v}]_\infty$ $\textstyle =$ $\displaystyle 0$ (1)
$\displaystyle {[}\nabla \times {\bf v}]_\infty$ $\textstyle =$ $\displaystyle 0$ (2)

may be written as the sum of an Irrotational part and a Solenoidal part,
{\bf v} = -\nabla \phi +\nabla \times {\bf A},
\end{displaymath} (3)

where for a Vector Field $F$,
$\displaystyle \phi$ $\textstyle =$ $\displaystyle - \int_V {\nabla\cdot {\bf F}\over 4\pi \vert{\bf r}'-{\bf r}\vert}\, d^3{\bf r}'$ (4)
$\displaystyle {\bf A}$ $\textstyle =$ $\displaystyle \int_V {\nabla \times {\bf F}\over 4\pi \vert{\bf r}'-{\bf r}\vert} d^3{\bf r}'.$ (5)

See also Irrotational Field, Solenoidal Field, Vector Field


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.

© 1996-9 Eric W. Weisstein