Let $M$ be a smooth manifold, and $V,W$ smooth vector fields on $M$. If $\theta:\mathcal{D}\to M$ denotes the flow of $V$, where $\mathcal{D}$ is an open subset of $\mathbb{R}\times M$. The Lie Derivative of $W$ along $V$ at $p$ is given by the following
$$
(\mathcal{L}_V W)_p = \dfrac{d}{dt}\biggr|_{t=0} d(\theta_{-t}(p))(W_{\theta_{t}(p)}) = \lim_{t\to 0}\dfrac{d(\theta_{-t}(p))(W_{\theta_t(p)})-W_p}{t}
$$
The tangent space at $p$ is a vector space, so $T_pM$ is endowed with a smooth structure. If we define $X(t) = d(\theta_{-t}(p))(W_{\theta_t(p)})$, this is a curve $X:(-\varepsilon,+\varepsilon)\to T_pM$ starting at $W_p$, since $X(0)=W_p$. The left hand side becomes
$$
(\mathcal{L}_V W)_p = \dfrac{d}{dt}\biggr\vert_{t=0}X(t) = X'(0)\in T_{W_p}(T_pM)
$$
While the right hand side looks like some kind of Frechet derivative on $T_pM$. My question is, what is the highlighted part trying to say? This is from Lee's Introduction to Smooth Manifolds.
Best Answer
Highlighted part explains why can you switch order of derivative and $d\theta$. Next it argues that this is required result.
\begin{align} \frac d {ds}\bigg|_{s=0} d(\theta_{-t_0}) &\circ d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))}) \\ &= \lim_{s\to 0} \frac{ d(\theta_{-t_0}) \circ d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- d(\theta_{-t_0})(W_{\theta_{t_0}(p)})}{s} \\ &= \lim_{s\to 0} d(\theta_{-t_0}) \left(\frac{d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- W_{\theta_{t_0}(p)}}{s}\right) \\ &= d(\theta_{-t_0}) \left(\lim_{s\to 0}\frac{d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- W_{\theta_{t_0}(p)}}{s}\right) \\ &= d(\theta_{-t_0})( (\mathcal{L}_V W)_{\theta_{t_0}(p)}) \\ \end{align}
In second equality we used linearity of $d \theta_{-t_0}$. Next independence $d \theta_{-t_0}$ of $s$ and continuity of linear function on finite dimensional space (not emphasized by author) allows third equality. Final equality is definition of Lie derivative, where second vector field is $p \mapsto W_{\theta_{t_0}(p)}$. Since $\theta_{t_0}$ is locally a diffeomorphism, it is the same as $ (\mathcal{L}_V W)_{\theta_{t_0}(p)}$.