We're looking for [tex]f(x,y,z)[/tex] such that [tex]\nabla f(x,y,z)=\vec F(x,y,z)[/tex], which requires
[tex]\dfrac{\partial f}{\partial x}=xyz^2[/tex]
[tex]\dfrac{\partial f}{\partial y}=x^9yz^2[/tex]
[tex]\dfrac{\partial f}{\partial z}=x^9y^2z[/tex]
Integrating both sides of the first PDE wrt [tex]x[/tex] gives
[tex]f(x,y,z)=\dfrac12x^2yz^2+g(y,z)[/tex]
Differenting this wrt [tex]y[/tex] gives
[tex]\dfrac{\partial f}{\partial y}=x^9yz^2=\dfrac12x^2z^2+\dfrac{\partial g}{\partial y}[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=\dfrac12x^2z^2(2x^7y-1)[/tex]
but we're assuming [tex]g(y,z)[/tex] is a function that doesn't depend on [tex]x[/tex], which is contradicted by this result, and so there is no such [tex]f[/tex] and [tex]\vec F[/tex] is not conservative.
