A steady two-dimensional velocity field is given by

$$\mathbf{u}(x, y)=(\alpha x-\beta y, \beta x-\alpha y), \quad \alpha>0, \quad \beta>0$$

(i) Calculate the vorticity of the flow.

(ii) Verify that $\mathbf{u}$ is a possible flow field for an incompressible fluid, and calculate the stream function.

(iii) Show that the streamlines are bounded if and only if $\alpha<\beta$.

(iv) What are the streamlines in the case $\alpha=\beta ?$