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  "body": "\n## Summary\n\nThe tension field of a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] is a vector field along the map.\nIt is the trace of the covariant derivative of the differential of the map.\nIt generalizes the Laplacian of a function and the acceleration of a curve.\nMinus the tension field is the gradient of the [[energy|E6HDE1DFXD]], so a map is [[harmonic|KZJYG9DS3M]] exactly when its tension field vanishes.\nThe [[harmonic map heat flow|MD99PYBCE6]] moves a map in the direction of its tension field.\n\n## Explanations\n\n### L1 human\n\nThe tension field tells, at each point, in which direction the map is pulled if one wants to lower its energy.\nFor a curve, it is the acceleration as seen from inside the target, the part of the bending that is not forced by the curvature of the target.\nFor a real function on a domain, it is the Laplacian, which compares the value at a point with the average of nearby values.\nFor example, the map $t \\mapsto (\\cos t, \\sin t)$ into the plane has tension $(-\\cos t, -\\sin t)$, pointing inward, but the same circle seen as a map into the unit circle has tension $0$.\n",
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