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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $C^\\infty(M)$ be the algebra of [[smooth functions|0V33BNKZ9S]] $M \\to \\R$.\nFor $p \\in M$, a *tangent vector* at $p$ is a linear map $v \\colon C^\\infty(M) \\to \\R$ such that $v(fg) = f(p)\\, v(g) + g(p)\\, v(f)$ for all $f, g \\in C^\\infty(M)$; that is, a derivation at $p$.\nThe tangent vectors at $p$ form a real vector space $T_pM$, the *tangent space* at $p$.\nThe *tangent bundle* of $M$ is the disjoint union\n$$TM = \\bigsqcup_{p \\in M} T_pM = \\set{(p, v) : p \\in M, \\ v \\in T_pM},$$\nwith the projection $\\pi \\colon TM \\to M$, $\\pi(p, v) = p$.\nLet $(U, (x^1, \\ldots, x^n))$ be a chart of $M$.\n$TM$ has a unique smooth structure in which each such chart gives a chart on $\\pi^{-1}(U)$, namely\n$$(p, v) \\mapsto \\big(x^1(p), \\ldots, x^n(p), v^1, \\ldots, v^n\\big), \\qquad v = \\sum_{i=1}^n v^i \\left.\\frac{\\partial}{\\partial x^i}\\right|_p.$$\n\n## Remarks\n\nEach $T_pM$ has dimension $n$, with basis $\\partial/\\partial x^1|_p, \\ldots, \\partial/\\partial x^n|_p$ for any chart around $p$, so $TM$ is a smooth manifold of dimension $2n$.\nFor $M = \\R^n$ a single chart covers everything, and $T\\R^n = \\R^n \\times \\R^n$.\nFor the circle $S^1$, the vector field $\\partial/\\partial\\theta$ is nowhere zero, and $TS^1$ is diffeomorphic to $S^1 \\times \\R$.\n\nThe tangent bundle of $S^2$ is not isomorphic, as a vector bundle, to the product $S^2 \\times \\R^2$.\nBy the hairy ball theorem every tangent vector field on $S^2$ vanishes somewhere, while $S^2 \\times \\R^2$ has nowhere zero sections.\n\nOther common definitions of $T_pM$ use equivalence classes of curves through $p$, or vectors in coordinates that change by the Jacobian matrix; they give isomorphic spaces.\n\n## Explanations\n\n### L1 human\n\nThe tangent bundle is the set of all pairs made of a point of the manifold and a tangent vector at that point.\n",
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