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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold, let $X$ be a smooth vector field on $M$, and let $\\varphi_t$ be the local [[flow|06738GGWWK]] of $X$.\nLet $T$ be a smooth [[tensor field|WX302T93TW]] on $M$, and let $\\varphi_t^* T$ be its pullback by the local diffeomorphism $\\varphi_t$.\nThe *Lie derivative* of $T$ along $X$ is the tensor field $\\mathcal{L}_X T$ given at each $p \\in M$ by\n$$(\\mathcal{L}_X T)_p = \\frac{d}{dt}\\Big|_{t = 0} (\\varphi_t^* T)_p.$$\n\n## Remarks\n\nFor a smooth function $f$, $\\mathcal{L}_X f = X f$, the derivative of $f$ in the direction of $X$.\nFor a vector field $Y$, $\\mathcal{L}_X Y = [X, Y]$, the [[Lie bracket|68FGQEBD6J]].\nFor a differential form $\\omega$, Cartan's formula gives $\\mathcal{L}_X \\omega = d(\\iota_X \\omega) + \\iota_X d\\omega$.\nExample: on $\\R^2$ let $X = -y \\, \\partial_x + x \\, \\partial_y$ and $f = x^2 + y^2$; then $\\mathcal{L}_X f = -y \\cdot 2x + x \\cdot 2y = 0$, as $f$ is invariant under the rotations $\\varphi_t$.\nNon-example of tensoriality: $\\mathcal{L}_X Y$ is not $C^\\infty(M)$-linear in $X$, since $\\mathcal{L}_{fX} Y = f [X, Y] - (Y f) X$.\nSo $(\\mathcal{L}_X T)_p$ depends on $X$ near $p$, not only on $X_p$; this is the difference with a covariant derivative.\nThe Lie derivative is a derivation: $\\mathcal{L}_X (S \\otimes T) = \\mathcal{L}_X S \\otimes T + S \\otimes \\mathcal{L}_X T$, and it commutes with contractions.\n",
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