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:
“Take therefore no thought for the morrow: for the morrow shall take thought for the things of itself. Sufficient unto the day is the evil thereof.”
—Bible: New Testament Jesus, in Matthew, 6:34.
From the Sermon on the Mount.
“The depth and strength of a human character are defined by its moral reserves. People reveal themselves completely only when they are thrown out of the customary conditions of their life, for only then do they have to fall back on their reserves.”
—Leon Trotsky (1879–1940)