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  "body": "\n## Summary\n\nA partition of unity is a family of nonnegative functions on a space that add up to $1$ at every point.\nUsually each function is required to live inside a chosen open set.\nIt is the standard tool for gluing local constructions into global ones.\nOn a smooth manifold one builds an object in each coordinate chart, multiplies it by the matching function, and adds the results.\nSmooth partitions of unity exist on every smooth manifold, which is why Riemannian metrics and many other structures exist on all of them.\n\n## Explanations\n\n### L1 human\n\nImagine several lamps lighting a long hall, each lamp covering only part of it.\nYou want the total light to be exactly $1$ everywhere.\nSo you dim each lamp near the edges of its region, in such a way that the overlaps add up correctly.\nThe brightness profiles of the lamps form a partition of unity.\nOn the line, two smooth bumps $\\rho_1$ and $\\rho_2 = 1 - \\rho_1$, where $\\rho_1$ is $1$ far to the left and $0$ far to the right, give a simple example.\n",
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