Chapter 14 - Vector Calculus - 14.3 Conservative Vector Fields - 14.3 Exercises - Page 1084: 14

Yes; Conservative

For a vector field to be Conservative, $\dfrac{\partial f}{\partial y}=\dfrac{\partial g}{\partial x}$ We have: $f(x,y)=2x^3+xy^2$ and $g(x,y)=2y^3+x^2y$ Thus, $\dfrac{\partial f}{\partial y}=2xy$ and $\dfrac{\partial g}{\partial x}=2xy$ So, $\dfrac{\partial f}{\partial y} = \dfrac{\partial g}{\partial x}$ Therefore, a vector field $F$ is Conservative.