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  "body": "\n## Summary\n\nThe shape operator of a surface in $\\R^3$ at a point is a linear map of the tangent plane to itself.\nIt tells how the unit normal turns as one moves on the surface.\nIt is minus the differential of the Gauss map, and it is also called the Weingarten map.\nIts eigenvalues are the principal curvatures, its determinant is the Gaussian curvature, and half its trace is the mean curvature.\n\n## Explanations\n\n### L1 human\n\nImagine a tiny arrow standing straight out of a surface at each point, like a hair.\nWalk a little in some direction, and watch how the hair tilts.\nThe shape operator turns your step into the change of tilt of the hair.\nOn a flat floor the hair never tilts, and on a ball it tilts in the direction you walk, at the same rate in every direction.\n",
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