Let the force \(\vec{F}=\left(2 x y-z^{3}ight) \hat{i}+x^{2} \hat{j}-\left(3 x z^{2}+1ight) \hat{k}\).

(a) Show that \(\vec{F}\) is conservative.

(b) Find \(\phi(x, y, z)-\phi(0,0,0)\). Note that from (a), since the force is conservative, any path can be taken to evaluate the line integral. Therefore, use simple straight line paths along the same directions as the axes starting from the origin \((0,0,0)\) to \((x, 0,0)\), then from \((x, 0,0)\) to \((x, y, 0)\), and finally up to the final endpoint from \((x, y, 0)\) to \((x, y, z)\).