Iterative calculus on tangent oors.

Tangent fibrations generate a "multi-oored tower", while raising from one of its oors to the next one, one practically reiterates the previously performed actions. In this way, the "tower" admits a ladder-shaped structure. Raising to the first oors sufaces for iteratively performing the subsequent steps. The paper mainly studies the tangent functor. We describe the structure of multiple vector bundle which naturally appears on the oors, tangent maps, sector-forms, the lift of vector elds to upper oors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant di erentiation. Finally, we will point out within the oors special subbundles - the osculating bundles, which play an essential role in classical theories. [ABSTRACT FROM AUTHOR]