Prolongation of a section to a jet bundle

Remarks

In adapted coordinates $(x^i, u^\alpha, u^\alpha_I)$, if $s$ has components $s^\alpha(x)$, then $$j^k s(x) = \Big(x^i,\ s^\alpha(x),\ \frac{\partial^{\abs{I}} s^\alpha}{\partial x^I}(x)\Big)_{1 \le \abs{I} \le k}.$$ If $\sigma = j^k s$, then $s = \pi_{k,0} \circ \sigma$, so $s$ is determined by $\sigma$.

Example: for $\pi \colon \R \times \R \to \R$ and $s(x) = x^2$, the first prolongation is $j^1 s(x) = (x, x^2, 2x)$.

Non-example: the section $\sigma(x) = (x, 0, 1)$ of $J^1\pi \to \R$ for the same bundle is not holonomic. If $\sigma = j^1 s$, then $s = \pi_{1,0} \circ \sigma = 0$, so $j^1 s(x) = (x, 0, 0) \ne \sigma(x)$.

A partial differential equation of order $k$ for sections of $\pi$ can be given as a subset $\mathcal{R} \subseteq J^k\pi$. A local section $s$ is a solution exactly when $j^k s$ takes its values in $\mathcal{R}$.