Christoffel Symbols - Covariant Derivatives of Tensors

Covariant Derivatives of Tensors

The covariant derivative of a vector field is

The covariant derivative of a scalar field is just

and the covariant derivative of a covector field is

The symmetry of the Christoffel symbol now implies

for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor).

The covariant derivative of a type (2,0) tensor field is

that is,

If the tensor field is mixed then its covariant derivative is

and if the tensor field is of type (0,2) then its covariant derivative is