The Laplace equation for φ implies that the integrability condition for ψ is satisfied:

then the incompressibility condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines.

The first Maxwell equation is the integrability condition for the differential

The integrability condition

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup-but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.

Schwarz integrability condition.

The structure equation is the integrability condition for the existence of such a local isomorphism.

There is thus an integrability condition at work, and Cartan's method for realizing integrability conditions was to introduce a differential form.

This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε ( hence a G < sub > δ </ sub > set ) – and gives a very quick proof of one direction of the Lebesgue integrability condition.

In modern geometric terms, the theorem gives necessary and sufficient conditions for the existence of a foliation by maximal integral manifolds each of whose tangent bundles are spanned by a given family of vector fields ( satisfying an integrability condition ) in much the same way as an integral curve may be assigned to a single vector field.

It is a one-form defined on P satisfying an integrability condition known as the Maurer – Cartan equation.

Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain, locally, a group action on P.

An integrability condition is a condition on the α < sub > i </ sub > to guarantee that there will be integral submanifolds of sufficiently high dimension.

) Thus a static spacetime is a stationary spacetime satisfying this additional integrability condition.

A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields an unitary structure ( U ( n ) structure ) on the manifold.

This form is always non-degenerate, with the suitable integrability condition ( of it also being closed and thus a symplectic form ) we get an almost Kähler structure.

A Kähler manifold is an almost Hermitian manifold satisfying an integrability condition.

The modern formulation of this is sometimes called the Liouville-Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability.

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.

In this context, the Frobenius theorem relates integrability to foliation ; to state the theorem, both concepts must be clearly defined.

The integrability follows because the equation defining the curve is a first-order ordinary differential equation, and thus its integrability is guaranteed by the Picard – Lindelöf theorem.

The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius 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 I.

Although such a characterization of " integrability " has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.

* Riemann integrability ; see Riemann integral

* Lebesgue integrability ; see Lebesgue integral

* Darboux integrability ; see Darboux integral

These were " On interval functions and their integrals " I ( 21, 1946 ) and II ( 23, 1948 ); " The efficiency of matrices for Taylor series " ( 22, 1947 ); " The efficiency of matrices for bounded sequences " ( 25, 1950 ); " The efficiency of convergence factors for functions of a continuous real variable " ( 30, 1955 ); " A new description of the Ward integral " ( 35 1960 ); and " The integrability of functions of interval functions " ( 39 1964 ).

This shows that some care must be taken when working with functions which do not have enough regularity ( e. g. compact support, integrability ) since, we know that the desired solution is, while the above integral diverges for all x.

( See the article on integrability conditions for differential systems for a detailed discussion of foliations by maximal integral manifolds.

Ten years after, in, Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of two variables.

" Soliton stability is due to topological constraints, rather than integrability of the field equations.

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space ( called, in this context, flat space ).

This notion of integrability need only be defined locally ; that is, the existence of the vector fields X and Y and their integrability need only be defined on subsets of M.

The soliton solutions are typically obtained by means of the inverse scattering transform and owe their stability to the integrability of the field equations.

Schwarz integrability criterion, which makes them multivalued.

In order for this constraint to be consistent, we require the integrability conditions that for some coefficients c.

It turned out to be impossible in general to construct such a frame, and that there were integrability conditions which needed to be satisfied first.

Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis.

This idea of integrability can be extended to collections of vector fields as well.

The condition was named after John Cheyne and William Stokes, the physicians who first described it in the 19th century.