# A force field is described by <F_x,F_y,F_z> = < xy , 2z-y^2 +x, 2y -z > . Is this force field conservative?

Aug 1, 2016

Not conservative

#### Explanation:

the conclusive test is the existence of a potential function $f$ such that $\vec{F} = \nabla f$. you could try and reverse-engineer a potential function from the partials but...

.....a necessary (though insufficient) condition is that the curl of the force field is zero because, if $f$ indeed exists, then $\nabla \times \vec{F} = \nabla \times \nabla f = 0$ as curl ( grad ) =0.

here

$\nabla \times \vec{F} = | \left(\hat{x} , \hat{y} , \hat{z}\right) , \left({\partial}_{x} , {\partial}_{y} , {\partial}_{z}\right) , \left(x y , 2 z - {y}^{2} + x , 2 y - z\right) |$

$= \hat{x} \left(2 - 2\right) - \hat{y} \left(0 - 0\right) + \hat{z} \left(1 - x\right) = \left(\begin{matrix}0 \\ 0 \\ 1 - x\end{matrix}\right) \ne 0$

So this is not conservative.