Answer
$F(x, y)=(1, 1)$
Work Step by Step
Let us consider a vector field $F=(f, g)$ such that
$\dfrac{\partial f}{\partial x}+\dfrac{\partial g}{\partial y}=0$
This gives: $\dfrac{\partial f}{\partial x}=0=\dfrac{\partial g}{\partial y}$
In order to achieve the above result, we must have: $F(x, y)=(1, 1)$
