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The curl measures local rotation, with components (∂Fz/∂y − ∂Fy/∂z, ∂Fx/∂z − ∂Fz/∂x, ∂Fy/∂x − ∂Fx/∂y). A zero curl means the field is conservative and derives from a scalar potential.
Curl
∇ × F = (∂Fz/∂y − ∂Fy/∂z, ∂Fx/∂z − ∂Fz/∂x, ∂Fy/∂x − ∂Fx/∂y)
The curl is (3, −3, 4), of magnitude √34 ≈ 5.8309519.
It means line integrals depend only on the endpoints, so the field is conservative and work around any closed loop is zero.