A solenoidal vector field satisfies
 |
(1) |
for every vector 
, where 
is the divergence.
If this condition is satisfied, there exists a vector 
, known as the vector potential,
such that
 |
(2) |
where 
is the curl. This follows from the vector identity
 |
(3) |
If 
is an irrotational field, then
 |
(4) |
is solenoidal. If 
and 
are irrotational, then
 |
(5) |
is solenoidal. The quantity
 |
(6) |
where 
is the gradient, is always solenoidal. For a function
satisfying Laplace's
equation
 |
(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
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1084, 2000.
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Cite this as:
Weisstein, Eric W. "Solenoidal Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SolenoidalField.html