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  "body": "\n## Summary\n\nLet $s$ be a section of a fiber bundle.\nIts $k$-th prolongation is the section $j^k s$ of the [[jet bundle|HC3VBBCJQP]] that sends each point $p$ to the [[$k$-jet|1QE0Z0756D]] of $s$ at $p$.\nIt lifts a section to a section that also records all derivatives up to order $k$.\nA section of the jet bundle that arises in this way is called holonomic.\nProlongations let one write a differential equation as the condition that $j^k s$ takes values in a given subset of the jet bundle.\n\n## Explanations\n\n### L1 human\n\nTake a function $u = s(x)$ of one variable.\nIts first prolongation is the curve $x \\mapsto (x, s(x), s'(x))$ in the space with coordinates $(x, u, u_x)$.\nFor $s(x) = x^2$ this curve is $x \\mapsto (x, x^2, 2x)$.\nNot every curve $x \\mapsto (x, u(x), p(x))$ in that space is a prolongation: the third coordinate must be the derivative of the second.\nThat is the whole point of prolongation: it ties the slope coordinate to the actual slope.\n",
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