### Parameter dependent pull-back of closed differential forms and invariant integrals

DOI: http://dx.doi.org/10.12775/TMNA.2005.023

#### Abstract

We prove, given a closed differential $k$-form $\omega$ in an

arbitrary open set $D \subset

{\mathbb R}^n$, and a parameter dependent

smooth map $F(\cdot,\lambda)$

from

an arbitrary open set $G \subset {\mathbb R}^m$ into $D$, that the

derivative

with respect to $\lambda$ of the pull-back

$F(\cdot,\lambda)^{*}\omega$ is exact in $G$. We give applications

to various theorems in topology, dynamics and hydrodynamics.

arbitrary open set $D \subset

{\mathbb R}^n$, and a parameter dependent

smooth map $F(\cdot,\lambda)$

from

an arbitrary open set $G \subset {\mathbb R}^m$ into $D$, that the

derivative

with respect to $\lambda$ of the pull-back

$F(\cdot,\lambda)^{*}\omega$ is exact in $G$. We give applications

to various theorems in topology, dynamics and hydrodynamics.

#### Keywords

Differential forms; invariant integrals; bifurcation; Kelvin theorem; Helmholtz theorem

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