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

Oct 16, 2016

Not conservative

#### Explanation:

For a field to be a conservative field, a necessary condition is that its curl is zero. So:

$\nabla \times \vec{F} = \det \left(\begin{matrix}\hat{i} & \hat{j} & \hat{k} \\ {\partial}_{x} & {\partial}_{y} & {\partial}_{z} \\ x y + z & x y - x & x z\end{matrix}\right)$

$= \hat{i} \left(0\right) - \hat{j} \left(z - 1\right) + \hat{k} \left(y - x\right) \ne 0$

Not conservative