# I don't get how the word "jet" is used

I don't quite get how to use the word "jet" in differential topology.

Question. Are jets more like tangent vectors, or are they more like vector fields?

In other words, are $k$-jets elements of the order-$k$ jet bundle, or are they sections of the order-$k$ jet bundle?

Also, what do we call the other thing? For example, if "$k$-jet" means a section of the order-$k$ jet bundle, then what do we call the elements of the order-$k$ jet bundle? And vice versa.

Given smooth manifolds $M$ and $N$ (without boundary), consider pairs $(f, m)$ consisting of a smooth mapping $f:M \to N$ and a point $m \in M$. Fix an integer $k \geq 0$. Two such pairs $(f, m)$ and $(\bar{f}, \bar{m})$ are said to be equivalent if $\bar{m} = m$ and there agree the Taylor polynomials of order $k$ of $f$ and $\bar{f}$, in any local charts centered at $m$ and $f(m)$. (One has to check that this defines an equivalence relation.) A $k$-jet is defined to be an equivalence class $[f, x]$. It is often written with some notation such as $j^{k}f(x)$. The space of $k$-jets of mappings from $M$ to $N$ is written $J^{k}(M, N)$.
The usual usage of $k$-jets therefore refers to an element rather than to a section. However, one can also refer to the $k$-jet of a smooth mapping $f$, meaning $j^{k}f(x)$ for all $x$ in the domain of definition of $f$. However, one has to keep in mind that not every section arises in this way: There are well-defined source and target maps from $J^{k}(M, N)$ to $M$ and $N$, respectively, associating with $[f, x] \in J^{k}(M, N)$ the source, $x \in M$, and the target, $f(x) \in N$. When $N$ has linear structure (for example $N = \mathbb{R}^{n}$), the space $J^{k}(M, N)$ can be given a structure as a vector bundle over $M$. Note, however, that a section $s$ of a such a jet bundle need not have the form $s(x) = j^{k}f(x)$ for some smooth mapping $f$ (in general $s(x) = [f, x]$, but the $f$ depends on the $x$). Such sections are called holonomic. They are characterized by being tangent to a certain canonically defined distribution on $J^{k}(M, N)$.
The basic example is the case of $1$-jets. A $1$-jet of a smooth mapping $f:M \to \mathbb{R}$ carries the information of the values, $f(x)$, of $f$, and the values of its differential, $df(x)$. So a section of the bundle $J^{1}(M, \mathbb{R}) \to M$ is more like a one-form than it is like a vector field. In this sense, jets are more like forms than they are like vector fields (they transform covariantly rather than contravariantly). A choice of local coordinates, $x^{1}, \dots, x^{m}$, on $M$ determines induced local coordinates $(x^{1}, \dots, x^{m}, z, y_{1}, \dots, y_{m})$ on $J^{1}(M, \mathbb{R})$ defined by $x^{i}(j^1f(p)) = x^{i}(p)$, $z(j^{1}f(p)) = f(p)$, and $y_{i}(j^{1}f(p)) = \tfrac{\partial f}{\partial x^{i}}(p)$. The kernel of the one-form $dz - y_{i}dx^{i}$ does not depend on the choice of local coordinates; it is a contact distribution on $J^{1}(M, \mathbb{R})$. A section of $J^{1}(M, \mathbb{R}) \to M$ is holonomic if and only if it is tangent to this contact distribution.