Question

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

Answer 1

NO

Explanation:

if the field `vec F` is conservative, then there exists a potential function, `f`, such that `vec F = - nabla f`

and as `nabla times nabla f = 0`, curl of gradient, it follows that if `f` exists then `nabla times vec F = 0` also

so we can test the curl of the vector field to see if it is indeed zero, as follows

`nabla times vec F = det[(hat x ,hat y ,hat z) , (del_x, del_y, del_z),(xy + z,xy-x,2y-zx)]`

`= hat x(2-0) - hat y (-z-1) + hat z (y-x)`

`= [(2), (z+1), (y - x)] ne 0`

This is not conservative. It is necessary that the `nabla times vec F = 0` for `vec F` to be conservative