criteria for existence of antidervatives

Let X be a normed spaceMathworldPlanetmath, Y a Banach spaceMathworldPlanetmath, UX a connected open set, f:UL(X;Y) a continuous functionMathworldPlanetmathPlanetmath, where L(X;Y) is the space of continuous linear operators. In this article a path is a curve that has bounded variationMathworldPlanetmath. The following theorems give necessary and sufficient conditions for f to have an antiderivatives.

Theorem 1.

The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    f has an antiderivative on U,

  2. 2.

    for any γ closed path in U γf=0,

  3. 3.

    for any γ, δ paths in U that have the same starting and endpoints γf=δf.

The next theorem states criteria for the existence of local antiderivatives.

Theorem 2.

The following conditions are equivalent:

  1. 1.

    f has an antiderivative locally,

  2. 2.

    for γ, δ homotopic closed paths in U γf=δf,

  3. 3.

    if γ is a triangular path such that its convex hull is in U, then γf=0.

With the stronger assumptionPlanetmathPlanetmath that f is differenciable we can obtain a more easily applicable condition. We introduce the canonical isometric isomorphism


where L2(X;Y) is the space of bilinear operators from X to Y. If F is an antiderivative of f, then π1,1(Df(x))=D2F(x) and by Clairaut’s theorem the second derivative is symmetricPlanetmathPlanetmath. The following theorems assert that the reverse is also true.

Theorem 3.

If f is differentiableMathworldPlanetmathPlanetmath, then it has an antiderivative locally if and only if π1,1(Df(x)) is symmetric for all xU.

Combining these three theorems immediately gives the following.

Corollary 1.

If U is simply connected and f is differentiable, then it has an antiderivative on U if and only if π1,1(Df(x)) is symmetric for all xU.

Title criteria for existence of antidervatives
Canonical name CriteriaForExistenceOfAntidervatives
Date of creation 2013-03-22 19:14:00
Last modified on 2013-03-22 19:14:00
Owner scineram (4030)
Last modified by scineram (4030)
Numerical id 9
Author scineram (4030)
Entry type Theorem
Classification msc 46G05