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  "body": "\n## Summary\n\nThe germ of a [[smooth function|0V33BNKZ9S]] at a point $p$ of a [[smooth manifold|F45VQ47GPV]] is the function seen only on arbitrarily small neighbourhoods of $p$.\nIt is the special case of a [[germ|VZXF7RNFKS]] for smooth real functions.\nThe germs at $p$ form a [[local ring|MZ02BKPHJA]], and this ring carries all local information of smooth calculus at $p$.\nTangent vectors, jets and the classification of singularities are all built on germs.\n\n## Explanations\n\n### L1 human\n\nTwo smooth functions have the same germ at $p$ when they agree near $p$, however far apart they are elsewhere.\nThink of a germ as what a function looks like under a microscope pointed at $p$.\nFor example, on $\\R$ the function $f(x) = x$ and a smooth function $g$ that equals $x$ on $(-1, 1)$ and equals $0$ for $\\abs{x} \u003e 2$ have the same germ at $0$.\nGerms can be added and multiplied, just like functions.\nA germ is invertible exactly when its value at $p$ is not zero.\n",
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