 Solution

Starting at $(1,0)$, traverse the right, top, left and bottom sides in that order. For $\mathbf{F}$, the differential expression is $-ydx+xdy$. The four line integral contributions are respectively $1,1,0,0$, so the total is two.

For $\mathbf{G}$, the expression is $2x{y}^{2}dx+2y{x}^{2}dy$. The right side contributes ${\int }_0^{1}2ydy=1$, and the top side contributes ${\int }_1^{0}2xdx=-1$; the other sides contribute zero. Therefore