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  "body": "\n## Summary\n\nThe tangent bundle of a smooth manifold collects all tangent vectors at all points into one space.\nA point of the tangent bundle is a pair: a point of the manifold and a velocity vector at that point.\nThe tangent bundle is itself a smooth manifold of twice the dimension, and it is the basic example of a vector bundle.\nVector fields are its sections, and the derivative of a smooth map is a map between tangent bundles.\n\n## Explanations\n\n### L1 human\n\nAt each point of a sphere there is a tangent plane, the flat plane that just touches the sphere there.\nThe tangent bundle puts all these planes together, one for each point.\nTo specify a point of it, you give a place on the sphere and an arrow lying in the tangent plane at that place.\nFor the plane $\\R^2$ itself, the tangent bundle is just $\\R^2 \\times \\R^2$: a position and an arrow.\n",
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