Question

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

Answer 1

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= |(hat x, hat y, hat z),(del_x, del_y, del_z),(xy, 2z-y^2+x, 2y-z)|`

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

So this is not conservative.