Consider the steady flow

$$u_{x}=\sin x \cos y, \quad u_{y}=-\cos x \sin y, \quad u_{z}=0$$

where $(x, y, z)$ are Cartesian coordinates. Show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and determine the streamfunction. Calculate the vorticity and verify that the vorticity equation is satisfied in the absence of viscosity. Sketch the streamlines in the region $0<x<2 \pi, 0<y<2 \pi$.