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9.4: The Covariant Derivative
https://phys.libretexts.org/Bookshelves/Relativity/Special_Relativity_(Crowell)/09%3A_Flux/9.04%3A_The_Covariant_Derivative
The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, i.e., linearly via the Jacobian matrix of the coordinate transformation.
5.7: The Covariant Derivative
https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/05%3A_Curvature/5.07%3A_The_Covariant_Derivative
The covariant derivative is the derivative that under a general coordinate transformation transforms covariantly, that is, linearly via the Jacobian matrix of the coordinate transformation.

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