8. (12 pts) Let $\boldsymbol{F}$ be a vector field given by $\quad \boldsymbol{F}=\left\langle 2 x y z^{-1}, z+x^{2} z^{-1}, y-x^{2} y z^{-2}\right\rangle$. Determine whether the vector field is conservative or not.

Step 1 ：Given the vector field F = <2*x*y/z, z+x^2/z, y-x^2*y/z^2>, we need to determine whether the vector field is conservative or not.

Step 2 ：A vector field F is conservative if its curl is zero. The curl of a vector field F = is given by the cross product of the del operator with the vector field, i.e., curl F = ∇ × F = .

Step 3 ：Let's compute the curl of the given vector field. Here, P = 2*x*y/z, Q = x^2/z + z, and R = -x^2*y/z^2 + y.

Step 4 ：Calculating the curl, we get curl_F = [0, 0, 0].

Step 5 ：Since the curl of the vector field F is zero, the vector field is conservative.

Step 6 ：Final Answer: The vector field F = <2*x*y/z, z+x^2/z, y-x^2*y/z^2> is \(\boxed{conservative}\).