A solenoidal Vector Field satisfies
![\begin{displaymath}
\nabla\cdot{\bf B} = 0
\end{displaymath}](s2_474.gif) |
(1) |
for every Vector
, where
is the Divergence.
If this condition is satisfied, there exists a vector
, known as the
Vector Potential, such that
![\begin{displaymath}
{\bf B}\equiv\nabla\times{\bf A},
\end{displaymath}](s2_478.gif) |
(2) |
where
is the Curl. This follows from the vector identity
![\begin{displaymath}
\nabla\cdot {\bf B} = \nabla\cdot(\nabla \times {\bf A}) = 0.
\end{displaymath}](s2_480.gif) |
(3) |
If
is an Irrotational Field, then
![\begin{displaymath}
{\bf A}\times {\bf r}
\end{displaymath}](s2_481.gif) |
(4) |
is solenoidal. If
and
are irrotational, then
![\begin{displaymath}
{\bf u}\times {\bf v}
\end{displaymath}](s2_484.gif) |
(5) |
is solenoidal. The quantity
![\begin{displaymath}
(\nabla u)\times (\nabla v),
\end{displaymath}](s2_485.gif) |
(6) |
where
is the Gradient, is always solenoidal. For a function
satisfying Laplace's Equation
![\begin{displaymath}
\nabla^2\phi = 0,
\end{displaymath}](s2_487.gif) |
(7) |
it follows that
is solenoidal (and also Irrotational).
See also Beltrami Field, Curl, Divergence, Divergenceless Field, Gradient, Irrotational Field,
Laplace's Equation, Vector Field
References
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
1999-05-26