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integrabil

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Integrabil might refer to
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example, including Lax pairs of integrable systems. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system.
* Given a collection of differential 1-forms
*
*
*
*
*
* α
*
* i
*
*
* ,
* i
* =
* 1
* ,
* 2
* ,
* …
* ,
* k
*
*
*
* {\displaystyle \textstyle \alpha _{i},i=1,2,\dots ,k}
* on an
*
*
*
*
* n
*
*
*
* {\displaystyle \textstyle n}
* -dimensional manifold
*
*
*
* M
*
*
* {\displaystyle M}
* , an integral manifold is a submanifold whose tangent space at every point
*
*
*
*
* p
* ∈
* M
*
*
*
* {\displaystyle \textstyle p\in M}
* is annihilated by each
*
*
*
*
*
* α
*
* i
*
*
*
*
*
* {\displaystyle \textstyle \alpha _{i}}
* .
* A maximal integral manifold is a submanifold*
*
*
* i
* :
* N
* ⊂
* M
*
*
* {\displaystyle i:N\subset M}
* such that the kernel of the restriction map on forms
*
*
*
*
*
* i
*
* ∗
*
*
* :
*
* Ω
*
* p
*
*
* 1
*
*
* (
* M
* )
* →
*
* Ω
*
* p
*
*
* 1
*
*
* (
* N
* )
*
*
* {\displaystyle i^{*}:\Omega _{p}^{1}(M)\rightarrow \Omega _{p}^{1}(N)}
* is spanned by the
*
*
*
*
*
* α
*
* i
*
*
*
*
*
* {\displaystyle \textstyle \alpha _{i}}
* at every point
*
*
*
* p
*
*
* {\displaystyle p}
* of
*
*
*
* N
*
*
* {\displaystyle N}
* . If in addition the
*
*
*
*
*
* α
*
* i
*
*
*
*
*
* {\displaystyle \textstyle \alpha _{i}}
* are linearly independent, then
*
*
*
* N
*
*
* {\displaystyle N}
* is (
*
*
*
* n
* −
* k
*
*
* {\displaystyle n-k}
* )-dimensional. Note that
*
*
*
*
* ...
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