Question

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

Answer 1

Non-conservative

Explanation:

The definitive evidence of a conservative field `vec F(x,y,z)` is the existence of potential function `f(x,y,z)` which defines the potential at every point and makes the transitions between points in the field path-independent.

And because `vec F = \nabla f` and thus `nabla times vec F = nabla times nabla f = 0` by definition as curl grad is zero....

.....then a necessary but insufficient condition is that the curl of the field be zero. (IOW the curl could be zero but the field still be non -conservative, but if the field is conservative the curl has to be zero)

We test that here:

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

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

Non-conservative