$(1)$ The stream function for a flow field is given by the equation $(x,y)=x^3-3x{y}^2$ where $$ is in $f\frac{t^2}{s}$ and $x$ and $y$ are in ft $.($a$)$ calculate the volumetric dilatation rate in $\frac{1}{s}.$ Is this flow incompressible? $($b$)$ calculate the vorticity in $z$ direction $($normal to page$)$ in $\frac{1}{s}.$ Is this flow irrotational? $($c$)$ calculate the angular deformation ${}_{xy}^{}$ in $\frac{1}{s},($d$)$ determine the pressure difference $(p_1-p_2)$ between points $1$ at $)=1ft,y=(2ft$ and $2$ at $)=4ft,y=(3ft.$ Elevation changes can be neglected, and the fluid is water with specific weight of $62\mathrm{.}4l\frac{b}{f}{t}^3.$ The acceleration of gravity is $32\mathrm{.}2f\frac{t}{s^2}.[$Answers: $($a$)0,$ incompressible, $($b$)0,$ irrotational, $($c$)-12x[$ft$2/$s$],($d$)36\mathrm{.}3$ psi$]$