The Covariant Derivative of a Contravariant Vector

If we differentiate a contravariant vector with respect to one of the coordinates, the result will not be a tensor. We use the transformation rule for the Christoffel symbols (which themselves are not a tensor), to find a tensorial rule that corresponds to differentiation. The result of this calculation is covariant differentiation of a contravariant vector. When covariant differentiation takes place, the resulting tensor will be one additional covariant component. #mikethemathematician, #profdabkowski, #mikedabkowski, #tensoranalysis