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Tangent fibrations generate a “multi-floored tower”, while raising from one of its floors 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 floors suffices 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 floors, tangent maps, sector-forms, the lift of vector fields to upper floors. Further, we show how tangent groups of Lie groups lead to gauge theory, and explain in this context the meaning of covariant differentiation. Finally, we will point out within the floors special subbundles - the osculating bundles, which play an essential role in classical theories.

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]

We present an infinitesimal interpretation of the control theory, particularly of the part concerning dynamic systems. We use the original concept of a bundle connection, which lies in the idea of fibre transportation along a path on the base manifold. The control of a process leads also to the transportation of fibres, and the control strategy, i.e. the choice of a suitable system control in order to optimize the process corresponds to the choice of a path on the base manifold. The triple of crucial terms of control, aim-control-strategy, translates in the terms of connections as fibre-connection-curve. Such a scheme is quite convincing, but it also works well in dynamic systems analysis. [ABSTRACT FROM AUTHOR]