= Coordinate tensor Lie derivative
{title2=$(\mathcal L_XT)^a{}_b=X^c\partial_cT^a{}_b-T^c{}_b\partial_cX^a+T^a{}_c\partial_bX^c$}
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>.
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