The notion of jet space or jet bundle is a generalization of the notion of tangent spaces and tangent bundles, respectively. While a tangent vector is an equivalence class of germs of curves with order- tangency at a given point in the target, jet spaces are equivalence classes of germs of smooth maps with respect to (finite) order- tangency at some point in the target.
Jet bundles were first introduced by Charles Ehresmann.
wikipedia: jet, jet bundle
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