Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 1 / a

Let ${\varphi }_s$ be the local flow of the smooth vector field $X$, with ${\varphi }_0=\mathrm{id}$ and $\frac{d}{ds}{\varphi }_s(p)=X_{{\varphi }_s(p)}$. The flow definition of the Lie derivative of a tensor field isFor a tensor field of type $(r,s)$, the mixed tensor pullback transports the tensor at ${\varphi }_s(p)$ back to $p$: it applies $(d{\varphi }_s{)}_p^{-1}$ to every contravariant factor and $(d{\varphi }_s{)}_p^{\ast }$ to every covariant factor. Thus every difference quotient belongs to the same tensor space at $p$. This definition gives another tensor field of the same type and uses no choice of affine connection. The tensor Lie derivative is defined wherever the local flow exists, including points where $X$ vanishes.