\chapter{Jet spaces}


Tangent spaces are a useful tool for setting up a notion of a derivative on a manifold independently of chosen coordinates (the pushforward). It would be useful to generalize this idea to support the notion of higher order derivatives; this is particularly important in modelling differential varieties (submanifolds representing space of solutions to some differential equation).

We do this by \emph{jet spaces}; these are spaces that keep track of functions up to their Taylor series truncated to $n$ terms, or more essentially, a space that captures the first $n$ derivatives.

\begin{definition}[Jet space]
	\[J^{n}\]
\end{definition}


Similar to tangent spaces, we often want to represent all Jet spaces as sets for the image of some functions, hence we analogously consider objects called Jet bundles.

\begin{definition}[Jet bundle]
\end{definition}