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

##### 1 Answer

The force field is *not* conservative;

#### Explanation:

If *if and only if*

As stated above, the curl is given by the cross product of the gradient of

We have

The curl of the vector field is then given as:

#abs((veci,vecj,veck),(del/(delx),del/(dely),del/(delz)),(xy,2z-y^2+x,2y-zx))#

We take the cross product as we usually would with vectors, except we'll be taking partial derivatives each time we multiply by a partial differential.

For the

So far, so good.

For the

Remember that if we take the partial of a function with respect to some variable which is not present, the partial derivative is

#0# as we treat all other variables as constants.

For the

This gives a final answer of

*not* conservative