Chapter 7.5, Problem 13P

To determine

To prove: If $F_x+G_y$ has the same sign throughout $D$, then there is no closed trajectory of the system $\frac{dx}{dt}=F\left(x,y\right),\frac{dy}{dt}=G\left(x,y\right)$ lying entirely in $D$, where the functions $F\text{and}G$ have continuous first partial derivatives in a simply connected domain $D$ of the $xy-$ plane. Show that the line integral is zero for this curve.