Show by expanding and collecting real and imaginary terms that \(f=z^{6}\) (where \(z\) is the complex number \(z=x+i y\) ) leads to a valid velocity potential (the real part of \(f\) ) and a corresponding stream function (the negative of the imaginary part of \(f\) ) of an irrotational and incompressible flow. Then show that the real and imaginary parts of \(d f / d z\) yield \(-u\) and \(v\), respectively.