Necessary and Sufficient Conditions
The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Read more about this topic: Integrability Conditions For Differential Systems
Famous quotes containing the words sufficient and/or conditions:
“It is a well-settled principle of the international code that where one nation owes another a liquidated debt which it refuses or neglects to pay the aggrieved party may seize on the property belonging to the other, its citizens or subjects, sufficient to pay the debt without giving just cause of war.”
—Andrew Jackson (1767–1845)
“Under all conditions well-organized violence seems to him the shortest distance between two points.”
—Leon Trotsky (1879–1940)