A constructive approach to bundles of geometric objects on a differentiable manifold

A constructive approach to bundles of geometric objects of finite rank on a differentiable manifold is proposed, whereby the standard techniques of fiber bundle theory are extensively used. Both the point of view of transition functions (here directly constructed from the jets of local diffeomorphisms of the basis manifold) and that of principal fiber bundles are developed in detail. These, together with the absence of any reference to the current functorial approach, provide a natural clue from the point of view of physical applications. Several examples are discussed. In the last section the functorial approach is also presented in a constructive way, and the Lie derivative of a field of geometric objects is defined.