Coordinate tensor Lie derivative ($({\mathcal{L}}_{X}T{)}^a{}_b=X^c{\partial }_c{T}^a{}_b-T^c{}_b{\partial }_c{X}^a+T^a{}_c{\partial }_b{X}^c$)

For every contravariant slot of a tensor field, the tensor Lie derivative contributes a negative derivative of its generating vector field; every covariant slot contributes a positive one. The remaining term differentiates the tensor's components along the generator. This follows from the flow definition of the Lie derivative of a tensor field and its product and contraction rules, without choosing a affine connection.