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:
“Generally speaking, the political news, whether domestic or foreign, might be written today for the next ten years with sufficient accuracy. Most revolutions in society have not power to interest, still less alarm us; but tell me that our rivers are drying up, or the genus pine dying out in the country, and I might attend.”
—Henry David Thoreau (1817–1862)
“The worst of my actions or conditions seem not so ugly unto me as I find it both ugly and base not to dare to avouch for them.”
—Michel de Montaigne (1533–1592)