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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$.\nA smooth [[atlas|J7P32JQTVE]] $\\set{(U_i, \\varphi_i)}$ of $M$ is *oriented* if each transition map $\\varphi_j \\circ \\varphi_i^{-1}$ has a positive [[Jacobian determinant|0ZCSMYHTBP]] at every point of $\\varphi_i(U_i \\cap U_j)$.\n$M$ is *orientable* if it has an oriented smooth atlas.\n\n## Remarks\n\nThe circle $S^1$ is orientable.\nThe stereographic charts $\\varphi_N(x, y) = x/(1 - y)$ and $\\varphi_S(x, y) = x/(1 + y)$ have transition map $t \\mapsto 1/t$, whose derivative $-1/t^2$ is negative; replacing $\\varphi_S$ by $-\\varphi_S$ gives transition map $t \\mapsto -1/t$ with derivative $1/t^2 \u003e 0$.\n\nThe open [[Möbius strip|04AJE0M271]] is not orientable.\nFollowing charts around the central circle forces a transition map with negative Jacobian determinant somewhere, so no atlas meets the condition.\nThe real projective plane and the Klein bottle are not orientable either, since they contain an open Möbius strip.\n\nEquivalently, $M$ is orientable if and only if it has a nowhere vanishing differential form of degree $n$.\nAn orientation of $M$ is a choice of [[orientation|0QJJAQ6ZYZ]] of each tangent space that is given by the charts of one oriented atlas.\nA connected orientable manifold has exactly two orientations.\nFor a [[topological manifold|QQQ9B1DK4P]], orientability is defined with [[singular homology|M0Q5CFJ7ZH]].\nEach local group $H_n(M, M \\setminus \\set{x}; \\Z)$ is infinite cyclic.\nAn orientation is a choice of generator at each point $x$ that is locally constant in $x$.\nFor smooth manifolds the two notions agree.\n",
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