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  "body": "\n## Summary\n\nThe Christoffel symbols $\\Gamma^k_{ij}$ are the components of an [[affine connection|JX65KSVHSJ]] in a coordinate chart.\nThey record how the covariant derivative of one coordinate vector field along another fails to vanish.\nFor the Levi-Civita connection of a Riemannian metric they are given by an explicit formula in the first derivatives of the metric.\nThey appear in the geodesic equation and in coordinate formulas for curvature.\n\n## Explanations\n\n### L1 human\n\nIn curvilinear coordinates, such as polar coordinates in the plane, the coordinate directions change from point to point.\nEven a constant vector field then has components that vary.\nThe Christoffel symbols are the correction numbers that account for this turning of the coordinate axes.\nIn Cartesian coordinates on the flat plane they are all zero, while in polar coordinates some of them are not, for example $\\Gamma^r_{\\theta\\theta} = -r$.\n",
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