Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time.
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Another example is for a foliation, which requires a reduction of the tangent bundle to a block matrix subgroup, together with an integrability condition so that the Frobenius theorem applies.
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*Since step functions are integrable and the integrability and the value of a Riemann integral are compatible with uniform limits, the regulated integral is a special case of the Riemann integral.
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Integrability means that there is a change of variables ( action-angle variables ) such that the evolution equation in the new variables is equivalent to a linear flow at constant speed.
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The proof of the classical result that for " n " = 3 the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by integrability argument.
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Imposing integrability conditions on a particular " G "-structure ( for instance, with the case of a symplectic form ) can be dealt with via the process of prolongation.
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With the extra integrability condition that it is closed ( i . e ., it is a symplectic form ), we get an "'almost K�hler structure " '.
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The study of non-linear equations was enhanced by this discovery, opening up the door to analytical work on the integrability of the KdV equation and the equations of the KP hierarchy.
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The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist.
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There one starts with an integrable system ( so " decoupled " into harmonic oscillators ), and then perturbs away from the integrability . talk ) 12 : 40, 26 March 2015 ( UTC)
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