For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your answer.

(a) $\vec F(x,y,z) = 5 x y\,\vec i + 5\,y z\vec j + 5\,x z\,\vec k$
$\vec F(x,y,z)$ is ? a gradient vector field not a gradient vector field

(b) $\vec F(x,y,z) = \langle 4z, 0, 4x\rangle$
$\vec F(x,y,z)$ is ? a gradient vector field not a gradient vector field

(c) $\vec F(x,y,z) = 3 y\,\vec i + 3 x\,\vec j$
$\vec F(x,y,z)$ is ? a gradient vector field not a gradient vector field

(d) $\vec F(x,y,z) = 6 x\,\vec i + 6 y\,\vec j + 6 z\,\vec k$
$\vec F(x,y,z)$ is ? a gradient vector field not a gradient vector field