Problem $3$:

The velocity components for a particular two$-$dimensional flow field are defined as follows:

$u=-\frac{y}{x^2+y^2},$;$,v=\frac{x}{x^2+y^2}$

$($a$)$ Determine the circulation $()$ for this flow field around the following contour by integrating the velocity around it:

$-1x+1,y=-1$

$-1y+1,x=+1$

$-1x+1,y=+1$

$-1y+1,x=-1$

$($b$)$ Calculate the vorticity vector for the given flow field.

$($c$)$ Apply Stoke theorem, check if Stoke theorem is valid or not?

$($d$)$ Calculate the divergrence of the velocity vector for the given flow field.

$(10\%)$