$\mathbf{F}(x,y)=2xy\mathbf{i}+x^{2}\mathbf{j}$ is a conservative vector field, since $\mathbf{F}$ is the gradient of the function $f(x,y)=x^{2}y$. That is, $$\nabla\! \ f(x,y)=f_{x}(x,y)\mathbf{i}+f_{y}(x,y)\mathbf{j}=2xy\mathbf{i}+x^{2}\mathbf{j}=\mathbf{F}(x,y)$$

The function $f(x,y)=x^{2}y$ is the potential function for $\mathbf{F}$.