Observe that the given vector field is a gradient field:
Let [tex]f(x,y,z)=\nabla g(x,y,z)[/tex], so that
[tex]\dfrac{\partial g}{\partial x} = x y^2 z^2[/tex]
[tex]\dfrac{\partial g}{\partial y} = x^2 y z^2[/tex]
[tex]\dfrac{\partial g}{\partial z} = x^2 y^2 z[/tex]
Integrating the first equation with respect to [tex]x[/tex], we get
[tex]g(x,y,z) = \dfrac12 x^2 y^2 z^2 + h(y,z)[/tex]
Differentiating this with respect to [tex]y[/tex] gives
[tex]\dfrac{\partial g}{\partial y} = x^2 y z^2 + \dfrac{\partial h}{\partial y} = x^2 y z^2 \\\\ \implies \dfrac{\partial h}{\partial y} = 0 \implies h(y,z) = i(z)[/tex]
Now differentiating [tex]g[/tex] with respect to [tex]z[/tex] gives
[tex]\dfrac{\partial g}{\partial z} = x^2 y^2 z + \dfrac{di}{dz} = x^2 y^2 z \\\\ \implies \dfrac{di}{dz} = 0 \implies i(z) = C[/tex]
Putting everything together, we find a scalar potential function whose gradient is [tex]f[/tex],
[tex]f(x,y,z) = \nabla \left(\dfrac12 x^2 y^2 z^2 + C\right)[/tex]
It follows that the curl of [tex]f[/tex] is 0 (i.e. the zero vector).
