7. a) The Burgers vortex is defined in cylindrical coordinates \((r, \theta, z)\) through the velocity field \(\mathbf{u} = u_r \hat{\mathbf{e}}_r + u_\theta \hat{\mathbf{e}}_\theta + u_z \hat{\mathbf{e}}_z\), whose components are
\(u_r = -\frac{1}{2} \alpha r\), \(u_\theta = \frac{\Gamma}{2\pi r} (1 - e^{-\alpha r^2/4\nu})\), \(u_z = \alpha z\),
where \(\alpha\) and \(\Gamma\) are constants. Calculate the convective derivative of this velocity field and determine the vorticity field \(\mathbf{\omega} = \nabla \times \mathbf{u}\). Is it an incompressible flow? Hint: Check in the literature the form that the operators curl, divergence and convective derivative take in cylindrical coordinates (see for example the appendices of the books by Currie, Acheson or Davidson).