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integrabilit
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| Integrabilit might refer to |
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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 * * * * * ... |