Green’s theorem in the plane

Note that Green s theorem in the plane expressed as... 

While the Poincare-Bendixson theorem yields the existence of limit cycles, it is often important to know when limit cycles do not exist. For two-dimensional systems, a result in this direction which complements the Poincare-Bendixson theorem is called the Dulac criterion. Its proof is a direct application of the classical Green s theorem in the plane (after an assumption that the theorem is false) and will not be given here a good reference is [ALGM]. 

This statement is called Green s theorem in the plane. A proof may be found in Spiegel (1971). We conclude immediately that if dA (x, y) is an exact differential, and therefore... 

---