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
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA:
Academic Press, pp. 1084, 1980.
© 19969 Eric W. Weisstein
19990526