criteria for existence of antidervatives

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

Theorem 1.

The following conditions are equivalent:

1. $f$ has an antiderivative on $U$,

2. for any $\gamma$ closed path in $U$ ${\int }_{\gamma }f=0$,

3. for any $\gamma$, $\delta$ paths in $U$ that have the same starting and endpoints ${\int }_{\gamma }f={\int }_{\delta }f$.

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

Theorem 2.

The following conditions are equivalent:

1. $f$ has an antiderivative locally,

2. for $\gamma$, $\delta$ homotopic closed paths in $U$ ${\int }_{\gamma }f={\int }_{\delta }f$,

3. if $\gamma$ is a triangular path such that its convex hull is in $U$, then ${\int }_{\gamma }f=0$.

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

$${\pi }_{1,1}:L(X;L(X;Y))\to L_2(X;Y),u\mapsto ((x_1,x_2)\mapsto u(x_1)(x_2))$$

where $L_2(X;Y)$ is the space of bilinear operators from $X$ to $Y$. If $F$ is an antiderivative of $f$, then ${\pi }_{1,1}(Df(x))=D^{2}F(x)$ and by Clairaut’s theorem the second derivative is symmetric. The following theorems assert that the reverse is also true.

Theorem 3.

If $f$ is differentiable, then it has an antiderivative locally if and only if ${\pi }_{\mathrm{1}\mathrm{,}\mathrm{1}}\mathit{}\mathrm{(}D\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ is symmetric for all $x\mathrm{\in }U$.

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 ${\pi }_{\mathrm{1}\mathrm{,}\mathrm{1}}\mathit{}\mathrm{(}D\mathit{}f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{)}$ is symmetric for all $x\mathrm{\in }U$.

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