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  "body": "\n## Definition\n\nLet $S \\subseteq \\mathbb{R}^3$ be a surface that has a tangent plane at each point.\nThe surface $S$ is orientable if there is a continuous map $n \\colon S \\to \\mathbb{R}^3$ with the following property. For each $p \\in S$, $n(p)$ is a [[unit normal vector|0BJDQ2HBH1]] of $S$ at $p$.\nSuch a map $n$ is an orientation of $S$, and $S$ together with $n$ is an oriented surface.\n\n## Remarks\n\nIf $S$ is connected and orientable, then it has exactly two orientations, $n$ and $-n$.\nA regular injective [[parametrisation|05788J51WZ]] $r$ gives the orientation $n = (r_u \\times r_v) / \\| r_u \\times r_v \\|$. Swapping $u$ and $v$ gives the opposite orientation.\nSpheres, planes and tori are orientable. The boundary of a bounded region in $\\mathbb{R}^3$ has the outward normal as its standard orientation. The Möbius strip is not orientable.\nIntegrals of vector fields across a surface, such as flux, need an orientation. Reversing the orientation changes their sign. [[Integrals of scalar functions|0DQ8ERJZNS]] do not need one.\n\n## Explanations\n\n### L1 human\n\nAn orientable surface has two sides, an inside and an outside, or a top and a bottom.\nOn a Möbius strip, an ant can walk from one side to the other without crossing an edge. So there is no consistent choice of side.\n",
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