Consider a two$-$dimensional flow in the region defined by $x0$ and $y0.$

The $x-$component of the velocity is given by $v_x=A{x}^{},$ where both A and $$ are constant.

$($a$)$ Assuming that the fluid is incompressible $($i$.$e$.,$ grad$*v=0)$ and that $v_y=0$ for $y=0$ and all $x,$ i$.$e$.,v_y(x,0)=0,$ find $v_y(x,y).$

$($b$)$ For what values of $$ is the flow also irrotational $($i$.$e$.,$ grad$v=0)?$