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).
