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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $k \\ge 0$.\nA *smooth vector bundle* of rank $k$ over $M$ is a smooth [[fiber bundle|CX4M8K61AB]] $\\pi \\colon E \\to M$ with fiber $\\R^k$ and the following structure.\nEach fiber $E_p = \\pi^{-1}(p)$ is a real [[vector space|000AYPA9AJ]], and the local trivializations $\\pi^{-1}(U) \\to U \\times \\R^k$ can be chosen linear on each fiber.\nA *smooth section* of $E$ is a [[smooth|0V33BNKZ9S]] map $s \\colon M \\to E$ such that\n$$\\pi \\circ s = \\id_M,$$\nthat is, $s(p) \\in E_p$ for every $p \\in M$.\nThe set of smooth sections is written $\\Gamma(E)$.\n\n## Remarks\n\nThe zero section $p \\mapsto 0_p$ is a section of every vector bundle.\nFor the trivial bundle $E = M \\times \\R^k$, the sections are the maps $p \\mapsto (p, u(p))$ with $u \\colon M \\to \\R^k$ smooth.\nA smooth vector field on $M$ is a section of the [[tangent bundle|WY6K3P4YCG]] $TM$.\n\nNon-example: on $E = S^1 \\times \\R$, the map $s(p) = (-p, 0)$ is smooth, but $\\pi(s(p)) = -p \\neq p$, so it is not a section.\n\nSections are added and multiplied by smooth functions pointwise, so $\\Gamma(E)$ is a module over $C^\\infty(M)$.\nThe same definition, without linearity, gives sections of any fiber bundle; continuous sections are defined with continuous maps.\n\n## Explanations\n\n### L1 human\n\nA section of a vector bundle is a smooth choice of one vector in the fiber over each point of the base.\n",
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