Cartan–Kähler theorem

Cartan–Kähler theorem In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals {stile di visualizzazione I} . It is named for Élie Cartan and Erich Kähler.

Contenuti 1 Meaning 2 Dichiarazione 3 Proof and assumptions 4 Riferimenti 5 External links Meaning It is not true that merely having {displaystyle dI} contained in {stile di visualizzazione I} is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.

Dichiarazione Let {stile di visualizzazione (M,io)} be a real analytic EDS. Supponi che {displaystyle Psubseteq M} is a connected, {stile di visualizzazione k} -dimensionale, real analytic, regular integral manifold of {stile di visualizzazione I} insieme a {stile di visualizzazione r(P)geq 0} (cioè., the tangent spaces {stile di visualizzazione T_{p}P} sono "extendable" to higher dimensional integral elements).

Inoltre, assume there is a real analytic submanifold {displaystyle Rsubseteq M} of codimension {stile di visualizzazione r(P)} contenente {stile di visualizzazione P} e tale che {stile di visualizzazione T_{p}Rcap H(T_{p}P)} has dimension {displaystyle k+1} per tutti {displaystyle pin P} .

Then there exists a (localmente) unique connected, {stile di visualizzazione (k+1)} -dimensionale, real analytic integral manifold {displaystyle Xsubseteq M} di {stile di visualizzazione I} che soddisfa {displaystyle Psubseteq Xsubseteq R} .

Proof and assumptions The Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary.

References Jean Dieudonné, Eléments d'analyse, vol. 4, (1977) Chapt. XVIII.13 R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffith, Exterior Differential Systems, Casa editrice Springer, New York, 1991. External links Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Enciclopedia della matematica, EMS Press R. Bryant, "Nine Lectures on Exterior Differential Systems", 1999 e. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich Categories: Partial differential equationsTheorems in analysis