 Lie derivative at a zero of its generator

The tensor Lie derivative need not vanish at a point where its generating vector field vanishes. For $X=x{\partial }_x$, the local flow fixes zero but its differential rescales tangent vectors; ${\mathcal{L}}_{X}dx=dx$ even there. Straightening coordinates exist only at nonzero generator points. The flow definition remains valid at zeros and includes derivatives of the generator acting on tensor slots.