I don't get the following theorem from my lecture: $M$ smooth manifold
Let $T \in \mathcal{T}^{r,s}(M)$ a tensor field, $\Phi$ the global flow of a complete vector field $X$ and $\varphi_t(p) := \Phi(t,p)$.
Then $\dfrac{\partial}{\partial t}|_{t=0} \varphi^{*}_t T = \mathcal{L}_X$ where $\mathcal{L}_X$ is the Lie-derivative.
What I don't get is, how do you take the partial derivative towards $t$ from this function?
I mean even if we take the case that $T=f \in C^{\infty}(M)$, then $t \mapsto \varphi^{*}_t f=f \circ \varphi_t$ is a function-valued function.
There are two ways I think about this. One is to work in coordinates and simply take derivatives of all component functions. Another is to apply the tensor to an arbitrary element of its domain. (Of course these are equivalent.)
I'm not sure I understand the objection to the fact that $t \mapsto \phi_t^{\ast}f$ is a function-valued function. For instance, you can take the derivative of $f(x,t) = xt$ with respect to $t$, at a specified value $t = t_0$. The result is a function of $x$. Maybe the issue is realizing that the derivative is taken at $t = 0$.