Consider the vector feld vec$(F)$ in the figure and the closed circular path $C$ oriented counter$-$clockwise.

$($a$)$ Is ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$ positive, negative, or zero? $?$

$($b$)$ True or False: vec$(F)=$gradf for some function $f.$ Hint: use your answer to part $($a$).?$

$($c$)$ Which of the following formulas best fits vec$(F)?$

A$.$ vec$(F)=$vec$(y)-$xvec$(j)$

B$.$ vec$(F)=\frac{y}{(x^2+y^2{)}^2}$vec$(i)-\frac{x}{(x^2+y^2{)}^2}$vec$(j)$

C$.$ vec$(F)=$xvec$(i)+$yvec$(j)$

D$.$ vec$(F)=\frac{x}{(x^2+y^2{)}^2}$vec$(i)+\frac{y}{(x^2+y^2{)}^2}$vec$(j)$

$($Click on graph to enlarge$)$