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:
“Cleverness is serviceable for everything, sufficient for nothing.”
—Henri-Frédéric Amiel (1821–1881)
“There is no society known where a more or less developed criminality is not found under different forms. No people exists whose morality is not daily infringed upon. We must therefore call crime necessary and declare that it cannot be non-existent, that the fundamental conditions of social organization, as they are understood, logically imply it.”
—Emile Durkheim (1858–1917)