Answer
Conservative
Work Step by Step
A vector field is said to be conservative when $ curl \space F=\nabla \times F=0$
Now, $$ curl F= \nabla \times F\\=(\dfrac{\partial z}{\partial y}-\dfrac{\partial y}{\partial z})i+(\dfrac{\partial x}{\partial z}-\dfrac{\partial z}{\partial x})j+(\dfrac{\partial y}{\partial x}-\dfrac{\partial x}{\partial y}) \space k $$
From the given equation, we have
$$ curl \space F=-3yz \times (x^2+y^2+z^2)^{-5/2}+3yz \times
(x^2+y^2+z^2)^{-5/2}--3xz \times
(x^2+y^2+z^2)^{-5/2}+3xz \times
(x^2+y^2+z^2)^{-5/2}-3xy \times
(x^2+y^2+z^2)^{-5/2}+3xy \times (x^2+y^2+z^2)^{-5/2}\\=0$$
Thus, the given $ F $ field is conservative.
