A vector F is solenoidal if ∇ ⋅ vector F = 0. To show that ∇f × ∇g is solenoidal. First we show that ∇f × ∇g can be expressed as a curl of a vector. We can show this using the identity.
∇ × (f∇g) = ∇f × ∇g + f∇ + ∇g
= ∇f × ∇g (∵ curl glad is zero)
Operating divergence on this and using the identity
∇ × ∇(f ∇ g) = 0 (∵ div curl of any vector is zero)
This gives ∇.(∇f × ∇g) = 0
Hence, ∇f × ∇g is solenoidal and hence proved.
