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  "body": "\n## Summary\n\nA Pfaffian system is an [[exterior differential system|901CCKQHVY]] generated by [[one-forms|02Z5B2HA3F]].\nEquivalently, it is given by a field of linear conditions on tangent vectors, that is, by a distribution, the common kernel of the generating forms.\nIts integral manifolds are the submanifolds tangent to this distribution.\nThe [[Frobenius theorem|5Q4BHBFXJY]] tells exactly when the integral manifolds fill out the manifold as the leaves of a foliation.\n\n## Explanations\n\n### L1 human\n\nA one-form at each point says which directions are allowed: those on which it vanishes.\nA Pfaffian system is a list of such restrictions.\nFor example, the form $dz$ on $\\R^3$ allows only horizontal directions, and the horizontal planes are surfaces that obey it everywhere.\nThe 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.\nPfaffian systems describe rolling without slipping, parking a car and many other constrained motions.\n",
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