\(\mathbf{F}(x,y)=2xy\mathbf{i}+x^{2}\mathbf{j}\) 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 \(\mathbf{F}\).