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  "body": "\n## Summary\n\nThe differential of a smooth map between manifolds is its best linear approximation at a point.\nIt sends tangent vectors at a point of the source to tangent vectors at the image point in the target.\nIn local coordinates it is given by the Jacobian matrix of partial derivatives.\nIt extends the derivative of calculus to curved spaces, and it obeys the chain rule.\n\n## Explanations\n\n### L1 human\n\nImagine a smooth map that wraps a line around a circle.\nIf a point moves along the line with some velocity, its image moves along the circle with some velocity.\nThe differential is the rule that turns the first velocity into the second.\nFor $t \\mapsto (\\cos t, \\sin t)$, the velocity $1$ at time $t$ becomes the tangent arrow $(-\\sin t, \\cos t)$.\n",
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