= Solution
Let $\phi_s$ be the <local flow> of the smooth <vector field> $X$, with $\phi_0=\mathrm{id}$ and $\frac{d}{ds}\phi_s(p)=X_{\phi_s(p)}$. The <flow definition of the Lie derivative of a tensor field> is
$$
\boxed{(\mathcal L_XT)_p=\left.\frac{d}{ds}\right|_{s=0}(\phi_s^*T)_p.}
$$
For a <tensor field> of type $(r,s)$, the <mixed tensor pullback> transports the tensor at $\phi_s(p)$ back to $p$: it applies $(d\phi_s)_p^{-1}$ to every contravariant factor and $(d\phi_s)_p^*$ 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.
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