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  "body": "\n## Statement\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]], let $X$ be a smooth [[vector field|0299VCW1TH]] on $M$ and let $\\omega$ be a smooth [[differential form|0E3G0GRPZ6]] on $M$.\nLet $\\mathcal{L}_X$ be the [[Lie derivative|PJWK0PKN6Q]] along $X$, and let $d$ be the [[exterior derivative|0ZHSTAG8ZY]].\nLet $\\iota_X$ be the interior product, given on a $k$-form by $(\\iota_X \\omega)(Y_1, \\dots, Y_{k-1}) = \\omega(X, Y_1, \\dots, Y_{k-1})$.\nThen\n$$\\mathcal{L}_X \\omega = d(\\iota_X \\omega) + \\iota_X(d\\omega).$$\n\n## Remarks\n\nThe idea of the proof: both sides are derivations of degree $0$ of the algebra of forms, and both commute with $d$.\nOn a function $f$ both sides give $X f$, since $\\iota_X f = 0$ and $\\iota_X df = Xf$.\nOn an exact $1$-form $df$ both sides give $d(Xf)$.\nLocally every form is a sum of products $f\\, dg_1 \\wedge \\dots \\wedge dg_k$, so the two sides agree everywhere.\n\nExample: on $\\R^2$, let $X = -y\\, \\partial_x + x\\, \\partial_y$, the infinitesimal rotation, and $\\omega = dx \\wedge dy$.\nThen $d\\omega = 0$ and $\\iota_X \\omega = -y\\, dy - x\\, dx$, so $d(\\iota_X \\omega) = 0$.\nHence $\\mathcal{L}_X \\omega = 0$: rotations preserve area.\n\nThe formula has two quick consequences.\nFirst, $\\mathcal{L}_X$ commutes with $d$, since $d \\mathcal{L}_X = d \\iota_X d = \\mathcal{L}_X d$ by $d^2 = 0$.\nSecond, if $\\omega$ is closed, then $\\mathcal{L}_X \\omega = d(\\iota_X \\omega)$ is exact.\n\n## Explanations\n\n### L1 human\n\nThe Lie derivative measures how a form changes when it is dragged along the flow of a vector field.\nCartan's formula computes this change from two simple operations: plugging the vector field into the form, and taking the exterior derivative.\nIt turns a question about flows into a short algebraic computation.\n\n### L2 human\n\nThe second consequence above is the heart of the homotopy invariance of de Rham cohomology.\nIntegrating $\\mathcal{L}_X$ along a flow shows that the pullbacks of a closed form by homotopic maps differ by an exact form.\nThis gives the Poincaré lemma for star-shaped domains.\nIn symplectic geometry the formula shows that a vector field $X$ preserves a symplectic form $\\omega$ exactly when $\\iota_X \\omega$ is closed, which leads to Hamiltonian vector fields.\n",
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