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A vaguely defined yet very popular notion which may mean one of the following:
- For functions - existence of the integral in some sense (Riemann integrability, Lebesgue integrability, improper integrals etc.);
- For geometric structures and partial differential equations - conditions guaranteeing existence of solutions (Frobenius integrability condition
- For differential equations (both ordinary and partial) and dynamical systems:
- a possibility to find solution in a given class of functions (Darbouxian integrability, Liouville integrability, integrability in quadratures etc.) or just in some closed form;
- existence of one or more first integrals, functions which remain constant along solutions;
- preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
- complete integrability for Hamiltonian systems means existence of the maximal possible number of first integrals in involution.
Follow the links for more details.
Integrability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability&oldid=25868