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  "body": "\n## Summary\n\nA tensor field on a [[smooth manifold|F45VQ47GPV]] assigns to each point a tensor built from the tangent space and its dual.\nThe assignment varies smoothly with the point.\nVector fields, differential forms, Riemannian metrics and curvature tensors are all tensor fields.\nTensor fields are the basic objects of differential geometry, because their meaning does not depend on the choice of coordinates.\n\n## Explanations\n\n### L1 human\n\nAt each point of a curved space there is a flat space of directions, the tangent space.\nA tensor at a point is a multilinear gadget that eats some directions and some covectors and returns a number.\nA tensor field places one such gadget at every point, smoothly.\nFor example, the dot product $dx \\otimes dx + dy \\otimes dy$ at every point of the plane is a tensor field that measures lengths of arrows.\nA wind map that puts an arrow at each point is a simpler tensor field, of type $(1,0)$.\n",
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