$(15$ pts$.)$ Let $F(x,y)=($:$x^{2}y-y,-x{y}^2$:$).$

$($a$)$ Compute the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr$ if $C$ is given by

$r(t)=ti,-1t1$

$($b$)$ Find the simple piece$-$wise smooth curve $C_1$ from $(1,0)$ to $(-1,0)$ in the upper half plane $y0$ for which ${\displaystyle \underset{C_1}{\overset{\phantom{}}{}}}F*dr$ is a maximum.

$($A curve is simple if it does not cross itself.$)$

$($c$)$ Find any simple piece$-$wise smooth curve $C_2$ from $(1,0)$ to $(-1,0)$ in the whole $xy-$plane for which ${\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}F*dr$ is greater than ${\displaystyle \underset{}{\overset{\phantom{}}{}}}F*dr$ in $($b$).$