Show that for a 2D flow or plane confined to the \((x, y)\) plane only the \(z\)-component of the vorticity is non-zero. This component, \(\omega_{z}\), is simply abbreviated as \(\omega\) and treated like a scalar. Also verify by direct substitution that

\[-\omega=abla^{2} \psi\]

for 2D flow, where \(abla^{2}\) is the Laplacian operator with only \(x\) and \(y\) terms included. Here \(\psi\) is the streamfunction for 2D flows.

Also derive the corresponding relations for axisymmetric flows.