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  "body": "\n## Statement\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nLet $D \\subset TM$ be a smooth subbundle of rank $k$ of the [[tangent bundle|WY6K3P4YCG]], called a distribution.\nCall $D$ *involutive* if for all smooth vector fields $X, Y$ with values in $D$, the [[Lie bracket|68FGQEBD6J]] $[X, Y]$ also has values in $D$.\nCall $D$ *integrable* if every point of $M$ lies in an integral manifold of $D$.\nAn integral manifold is an immersed submanifold $N$ with $T_qN = D_q$ for all $q \\in N$.\nThen the following are equivalent.\n\n1. $D$ is involutive.\n2. $D$ is integrable.\n3. Every point of $M$ has a chart with coordinates $x^1, \\dots, x^n$ in which $D$ is spanned by $\\partial_1, \\dots, \\partial_k$.\n\n## Remarks\n\nCondition 3 implies condition 2, since the slices $x^{k+1} = c_{k+1}, \\dots, x^n = c_n$ are integral manifolds.\nCondition 2 implies condition 1, since vector fields tangent to a submanifold have a bracket tangent to it.\nThe main step is that condition 1 implies condition 3.\nOne picks local vector fields spanning $D$ that pairwise commute, which involutivity allows after a change of basis, and then uses their commuting flows as coordinates.\nExample: on $\\R^3$ the distribution spanned by $X = \\partial_x$ and $Y = \\partial_y + x \\, \\partial_z$ is the kernel of $dz - x \\, dy$.\nHere $[X, Y] = \\partial_z$, which is not in $D$, so no surface is tangent to $D$; this is the standard contact structure.\nIn the global form of the theorem, an involutive distribution splits $M$ into maximal connected integral manifolds, the leaves of a foliation.\n\n## Explanations\n\n### L1 human\n\nAt every point of a space, choose a plane of allowed directions.\nCan the space be sliced into surfaces that are tangent to these planes everywhere, like the pages of a book?\nThe Frobenius theorem says this happens exactly when moving back and forth along allowed directions never produces a forbidden direction, a condition tested by the Lie bracket.\n\n### L2 human\n\nThe theorem turns a question about solving a system of first-order PDEs into an algebraic check on brackets.\nFor $k = 1$ every line field is involutive, and the theorem reduces to the existence of integral curves of a vector field.\nA dual form uses differential forms: $D = \\ker \\theta^1 \\cap \\dots \\cap \\ker \\theta^{n-k}$ is involutive if and only if each $d\\theta^i$ lies in the ideal generated by the $\\theta^j$.\nThe theorem is used to integrate Lie subalgebras to Lie subgroups, to show that flat connections are locally trivial, and as the foundation of the theory of foliations.\n",
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