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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]], let $F \\colon M \\to N$ be a [[smooth|0V33BNKZ9S]] map, and let $p \\in M$.\nTangent vectors are taken as derivations, as in the [[tangent bundle|WY6K3P4YCG]].\nThe *differential* of $F$ at $p$ is the linear map\n$$dF_p \\colon T_pM \\to T_{F(p)}N, \\qquad (dF_p\\, v)(f) = v(f \\circ F)$$\nfor $v \\in T_pM$ and $f \\in C^\\infty(N)$.\nThe maps $dF_p$ for all $p$ together form the smooth map $dF \\colon TM \\to TN$, $(p, v) \\mapsto (F(p), dF_p v)$.\n\n## Remarks\n\nIn charts $(x^1, \\ldots, x^m)$ near $p$ and $(y^1, \\ldots, y^n)$ near $F(p)$, with $F^j = y^j \\circ F$,\n$$dF_p\\left(\\left.\\frac{\\partial}{\\partial x^i}\\right|_p\\right) = \\sum_{j=1}^{n} \\frac{\\partial F^j}{\\partial x^i}(p) \\left.\\frac{\\partial}{\\partial y^j}\\right|_{F(p)},$$\nso the matrix of $dF_p$ is the [[Jacobian matrix|047NDTEPCV]] of $F$ in these coordinates.\n\nExample: for $F \\colon \\R \\to \\R^2$, $F(t) = (\\cos t, \\sin t)$, one gets $dF_t(d/dt) = -\\sin t \\, \\partial_x + \\cos t \\, \\partial_y$.\nCheck: for $f \\in C^\\infty(\\R^2)$, $(d/dt)(f(\\cos t, \\sin t)) = -\\sin t \\, f_x + \\cos t \\, f_y$ by the chain rule.\n\nThe formula $v \\mapsto v(f \\circ F)$ needs $F$ to be smooth: if $F$ is only continuous, $f \\circ F$ need not be smooth and $v(f \\circ F)$ is not defined.\n\nThe chain rule reads $d(G \\circ F)_p = dG_{F(p)} \\circ dF_p$.\nOther notations are $F_*$, $DF$, $TF$ and $F'$.\n\n## Explanations\n\n### L1 human\n\nThe differential of a smooth map at a point is the linear map that sends each velocity vector at the point to the velocity vector of the image curve.\n",
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