Solution
= Solution
On the same domain, differentiate the Cartesian components:
$$
\nabla\cdot\boldsymbol u
=-\frac{2axy}{(x^2+y^2)^2}
+\frac{2axy}{(x^2+y^2)^2}+0.
$$
Thus
$$
\boxed{\nabla\cdot\boldsymbol u=0\qquad(x^2+y^2>0).}
$$
This is also an instance of <divergence of a curl is zero>, since the potential is smooth off the axis. The classical field is undefined on the axis, so that identity must not be applied there without specifying an extension.