I am reading Huybrechts book Complex Geometry.
On page 259, below Def.6.1.3: the author talks about families of diffeomorphism and then he gives a power series expansion and uses it to define a vector field.
If necessary, we may assume the manifold is compact.
Q1: How did we get the power series expansion?
Q2: Is $\frac{dF_t}{dt}|_{t=0} = \sum_i F_i \frac{\partial}{\partial x_i }$ just a notation? or is it a formula?
Could someone help understand this part, please?
Suppose M is $R^n$, $F_t(x)$ is a smooth function on $(-1,1)\times M$, $F: (t,x)\mapsto F_t(x)$ so it can be expanded with respect to $t$ in a neighbor of 0, $F_t(x)=x+t\cdot\frac{dF}{dt}(x)|_{t=0}+o(t)$. (Note $F_0=id$, so $F_0(x)=x$). And$\frac{dF}{dt}(x)|_{t=0}=(\frac{dF^1}{dt}, \frac{dF^2}{dt}, \dots, \frac{dF^n}{dt})=\sum{\frac{dF^i}{dt}\frac{\partial}{\partial{x^i}}}$. For a general manifold just do that in a coordinate chart.