We let $F,G :\mathbb R^3\rightarrow\mathbb R^3$ be vector fields given by: $$F(x,y,z)=(x^2,0,y^2)$$ $$G(x,y,z)=\left(\frac{1}{1+\|(x,y,z)\|^2}x,\frac{1}{1+\|(x,y,z)\|^2}y,\frac{1}{1+\|(x,y,z)\|^2}z \right)$$ and we let $\gamma_1,\gamma_2:\mathbb R^3\to\mathbb R^3$ be two paths from $(0, 0, 0)$ to $(1, 1, 1)$ defined as: $$\gamma_1(t)=(t,t^2,t^3), \gamma_2(t)=\left(\sin\left(\frac{\pi t}{2}\right), 1-\cos\left(\frac{\pi t}{2}\right),t\right)$$
Then I have to prove that $F$ is not a conservative vector field and find out if there exist an open subset $O \subset \mathbb R^3$ such that the restriction $F : O \to\mathbb R^3$ is conservative? And I have to find out whether $G$ is conservative or not? I think I can prove that F is not a conservative vector field by that: $\frac{\partial F_2}{\partial z}=\frac{\partial F_3}{\partial y} \Leftrightarrow \frac{\partial }{\partial z}(0)=\frac{\partial }{\partial y}(y^2 ) \Leftrightarrow 0\neq 2y$ therefore we have proof that the curl is not zero, so $F$ is path-dependent and thereby not conservative. But how do I find whether if there exists these interval $O$ and how do I deal with differentiating the norms in $G$ so I can show whether $G$ is conservative or not? Hope anyone can help me?