Consider the flow field in cartesian coordinates $(x, y, z)$ given by

$$\mathbf{u}=\left(-\frac{A y}{x^{2}+y^{2}}, \frac{A x}{x^{2}+y^{2}}, U(z)\right)$$

where $A$ is a constant. Let $\mathcal{D}$ denote the whole of $\mathbb{R}^{3}$ excluding the $z$ axis.

(a) Determine the conditions on $A$ and $U(z)$ for the flow to be both incompressible and irrotational in $\mathcal{D}$.

(b) Calculate the circulation along any closed curve enclosing the $z$ axis.