Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / a / ii / Solution

At a point where $X\ne 0$, the flow-box theorem supplies a transverse hypersurface with coordinates $x^i$. Flow each point on it for parameter $t$ and keep the $x^i$ constant along the flow. In the resulting coordinates,Because the coordinate basis is transported by this flow, the Lie derivative of any tensor is obtained by differentiating its coordinate components: