$4-67$ A two$-$dimensional fluid element of dimensions $dx$

and $dy$ translates and distorts as shown in Fig. P$4-67$ during

the infinitesimal time period $dt=t_2-t_1.$ The velocity com$-$

ponents at point $P$ at the initial time are $u$ and $v$ in the $x-$ and

$y-$directions, respectively. Show that the magnitude of the rate

of rotation $($angular velocity$)$ about point $P$ in the $xy-$plane is

${}_z=\frac{1}{2}(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{x}}-\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}})$

Fluid element

Fluid element

at time $t_2$

at time $t_1$

Please answer concisey$>{?}_{?}$