To show F is solenoidal, show that its divergence is zero. Hence, make the following calculation.
To show F is not conservative, show that its curl is not the zero vector. Hence, make the following calculation.
=
Maple's VectorPotential command gives the vector
as a vector potential.
Recipe 2 in Table 9.7.4 gives
=
The curl of the difference
is
so that the difference C is a gradient. A scalar potential for this gradient is
Indeed, the gradient of is C, as can be verified by the appropriate, but tedious, differentiations.