---
tag: 1QE0Z0756D
kind: concept
name: Jet
msc: [58A20]
sources:
  - url: https://en.wikipedia.org/wiki/Jet_(mathematics)
same_as:
  wikidata: Q923480
  wikipedia: Jet (mathematics)
---

## Summary

The $k$-jet of a smooth map at a point records the value and all derivatives up to order $k$ there, and nothing else.
In coordinates, it is the [[Taylor polynomial|0ZZ4RHTR9X]] of degree $k$, but the notion makes sense on [[smooth manifolds|F45VQ47GPV]] without coordinates.
Jets turn differential equations into geometric objects: a differential equation of order $k$ becomes a subset of a space of $k$-jets.
They are the basic tool of the [[jet bundles|HC3VBBCJQP]] and of singularity theory.

## Explanations

### L1 human

Two curves through a point have the same $1$-jet if they pass through the point with the same velocity.
They have the same $2$-jet if, in addition, they have the same acceleration there, and so on.
For example, $\sin x$ and $x$ have the same $2$-jet at $0$, since both have value $0$, slope $1$ and second derivative $0$ there.
They have different $3$-jets, since the third derivative of $\sin$ at $0$ is $-1$ while that of $x$ is $0$.
A jet is a finite list of numbers that describes a map up to a chosen order of contact.