In the original definition by A. Weil, an infinitely near point on a smooth manifold M is introduced as a homomorphisms of the algebra of smooth functions C^infty on M into a local (=Weil) algebra A. This point of view is close to algebraic geometry. On the other hand, there is the idea of (k; r)-velocity based on the concept of jet by C. Ehresmann, that seems to be more appropriate for differential geometry. The author introduces an original concept of A-velocity, that unifies both points of view. Beside the geometry of Weil bundles, some further jet-like functors are studied. The last Chapter 8 is devoted to nonholonomic jets, non-holonomic contact elemets and their applications to the submanifolds of Cartan spaces.
