for a simply closed loop ∂ ∂ Σ Σ with the smooth oriented surface Σ Σ in it. If the open domain D is simply connected open space (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a C 1 vector field v which curl is zero, i.e., ∇ ∇ × × v = 0 ) has the path independence by the Stokes' theorem, so the following statement is made; In a simply connected open region, any C 1 vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vice versa. The equality of the path independence and conservative vector fields is shown here.