---
tag: RX5MYWE73J
kind: concept
name: Pfaffian system
msc: [58A17, 58A15]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_ideal
---

## Summary

A Pfaffian system is an [[exterior differential system|901CCKQHVY]] generated by [[one-forms|02Z5B2HA3F]].
Equivalently, it is given by a field of linear conditions on tangent vectors, that is, by a distribution, the common kernel of the generating forms.
Its integral manifolds are the submanifolds tangent to this distribution.
The [[Frobenius theorem|5Q4BHBFXJY]] tells exactly when the integral manifolds fill out the manifold as the leaves of a foliation.

## Explanations

### L1 human

A one-form at each point says which directions are allowed: those on which it vanishes.
A Pfaffian system is a list of such restrictions.
For example, the form $dz$ on $\R^3$ allows only horizontal directions, and the horizontal planes are surfaces that obey it everywhere.
The form $dz - y\, dx$ also allows a plane of directions at each point, but these planes twist, and no surface is tangent to them everywhere.
Pfaffian systems describe rolling without slipping, parking a car and many other constrained motions.