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  "body": "\n## Summary\n\nA covariant derivative differentiates a vector field in the direction of a tangent vector, and the result is again a tangent vector.\nIt applies more generally to sections of a vector bundle and to tensor fields.\nIt is the operator $\\nabla_X$ that a [[connection|5ZYRRVE8BW]] provides.\nUnlike partial derivatives of components, it does not depend on the choice of coordinates.\nIt also differentiates vector fields along curves, which leads to parallel transport and geodesics.\n\n## Explanations\n\n### L1 human\n\nThe partial derivatives of the components of a vector field change when you change coordinates in a non-linear way.\nThe covariant derivative corrects them by a term that accounts for how the coordinate axes themselves bend.\nFor a car driving on a sphere, the covariant derivative of its velocity is the part of the acceleration that is tangent to the sphere.\nA curve with zero covariant acceleration is as straight as possible on the surface; on a sphere these are the great circles.\n",
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