Chapter 14 - Vector Calculus - 14.3 Conservative Vector Fields - 14.3 Exercises - Page 1084: 17

Not Conservative

For a vector field to be Conservative, $\dfrac{\partial f}{\partial y}=\dfrac{\partial g}{\partial x}$ We have: $f(x,y)=x^3-xy$ and $g(x,y)=\dfrac{x^2}{2}+y$ Thus, $\dfrac{\partial f}{\partial y}=-x$ and $\dfrac{\partial g}{\partial x}=2x$ Therefore, a vector field $F$ is Not Conservative.