Jet

Summary

The $k$-jet of a smooth map at a point records the value and all derivatives up to order $k$ there, and nothing else. In coordinates, it is the Taylor polynomial of degree $k$, but the notion makes sense on smooth manifolds without coordinates. Jets turn differential equations into geometric objects: a differential equation of order $k$ becomes a subset of a space of $k$-jets. They are the basic tool of the jet bundles and of singularity theory.