## Solenoidal Field

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).