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

Jean Mawhin

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.

Keywords


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

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