The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_C F·d s = 0. 2. For any two oriented simple curves C_1 and C_2 with the same endpoints, integral_(C_1) F·d s = integral_(C_2) F·d s. 3. There exists a scalar potential function f such that F = del f, where del is the gradient. 4. If D is simply connected, then curl del xF = 0.

curl | gradient | line integral | Poincaré's theorem | potential function | simply connected | vector field