---
tag: AME1ZE5HBX
kind: definition
name: Exterior differential system
msc: [58A15]
defines: 901CCKQHVY
uses: [F45VQ47GPV, 0E3G0GRPZ6, 0ZHSTAG8ZY, 0GRAATMGP1, 096Z9MP2ME, 6RW68JGBHC]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_ideal
---

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] and let $\Omega(M) = \bigoplus_{k \ge 0} \Omega^k(M)$ be the algebra of smooth [[differential forms|0E3G0GRPZ6]] with the [[wedge product|0GRAATMGP1]].
An *exterior differential system* on $M$ is a subset $\mathcal{I} \subseteq \Omega(M)$ with these properties.

1. $\mathcal{I}$ is a two-sided [[ideal|096Z9MP2ME]]: it is closed under addition, and $\alpha \wedge \varphi \in \mathcal{I}$ for all $\alpha \in \Omega(M)$ and $\varphi \in \mathcal{I}$.
2. $\mathcal{I}$ is graded: if $\varphi \in \mathcal{I}$, then each homogeneous component of $\varphi$ is in $\mathcal{I}$.
3. $\mathcal{I}$ is closed under the [[exterior derivative|0ZHSTAG8ZY]]: $d\varphi \in \mathcal{I}$ for all $\varphi \in \mathcal{I}$.

An ideal with these properties is also called a differential ideal.
An *integral manifold* of $\mathcal{I}$ is an [[immersion|6RW68JGBHC]] $\iota \colon N \to M$ with $\iota^* \varphi = 0$ for all $\varphi \in \mathcal{I}$.

## Remarks

Usually $\mathcal{I}$ is given by finitely many generators $\varphi_1, \dots, \varphi_r$.
The differential ideal they generate is the algebraic ideal generated by the $\varphi_a$ and the $d\varphi_a$.
Adding the $d\varphi_a$ changes no integral manifold, because $\iota^* d\varphi_a = d(\iota^* \varphi_a) = 0$ whenever $\iota^* \varphi_a = 0$.

Example: on $M = \R^3$ with coordinates $(x, u, p)$, let $\mathcal{I}$ be generated by $\theta = du - p\, dx$ and $d\theta = dx \wedge dp$.
For every smooth function $f$, the curve $x \mapsto (x, f(x), f'(x))$ is an integral manifold: $\theta$ pulls back to $(f' - f')\, dx = 0$, and every $2$-form pulls back to $0$ on a curve.

Non-example: the algebraic ideal generated by $\theta$ alone fails clause 3.
Every element of it has the form $\alpha \wedge \theta$, so every $2$-form in it is zero after wedging with $\theta$.
But $\theta \wedge d\theta = du \wedge dx \wedge dp \ne 0$, so $d\theta$ is not in that ideal.