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We prove, given a closed differential omega-form to in an arbitrary open set D subset of R-n, and a parameter dependent smooth map F((.),lambda) from an arbitrary open set G subset of R-m into D, that the derivative with respect to lambda of the pull-back F((.),lambda)* omega is exact in G. We give applications to various theorems in topology, dynamics and hydrodynamics.

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