Prolongation of a section to a jet bundle

Summary

Let $s$ be a section of a fiber bundle. Its $k$-th prolongation is the section $j^k s$ of the jet bundle that sends each point $p$ to the $k$-jet of $s$ at $p$. It lifts a section to a section that also records all derivatives up to order $k$. A section of the jet bundle that arises in this way is called holonomic. Prolongations let one write a differential equation as the condition that $j^k s$ takes values in a given subset of the jet bundle.