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##\displaystyle{[\mathcal{L}_{\bf{X}}Y]_{x}:=\lim_{t\rightarrow 0} \frac{[{\bf{Y}}_{\phi_{t}x}-\phi_{t*}{\bf{Y}}_{x}]}{t}}##

Now, I understand that ##{\bf{Y}}_{\phi_{t}x}## is the tangent vector of the vector field ##\bf{Y}## at the point ##\phi_{t}x##, where the point ##\phi_{t}x## is obtained by starting at point ##x## at time ##0## and traversing along the orbit of ##x## to time ##t##.

But I don't understand how to interpret ##\phi_{t*}{\bf{Y}}_{x}##. Given the map ##\phi_{t}## which maps points ##x## in ##M^{n}## to points ##\phi_{t}(x)## in ##M^{n}## along the orbit of ##x## parameterised by time ##t##, we can define the differential ##\phi_{t*}## that maps the tangent vector of the vector field ##Y## at ##x## to some tangent vector at the point ##\phi_{t}x##. Now,

*there is only one tangent vector of the vector field ##\bf{Y}## at the point ##\phi_{t}x##, and this tangent vector is the vector ##{\bf{Y}}_{\phi_{t}x}##*. This seems to suggest that ##{\bf{Y}}_{\phi_{t}x}## and ##\phi_{t*}{\bf{Y}}_{x}##.

What am I missing?