Assuming the functions u and v to be differentiable,

$($a$)$ Show that a necessary and sufficient condition that u$($x$,$ y$,$ z$)$ and v$($x$,$ y$,$ z$)$ are related by some function f $($u$,$ v$)=0$ is that $($u$)\backslash$times $($v$)=0$;

$($a$)f(z)=u(x,y)+iv(x_{1}y)$

where $u$ is the velocity potential and $v$ is the stream function vec$(v)=$vec$(grad)u$

Show that vec$(grad)*vec(v)=0$ and vec$(grad)$vec$(V)=0.$