I am struggling a lot with the concept of Lie derivative. I am studying it just in $\mathbb{R}^n$ not in a general manifold context. I have that its definition is:
$$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)|_{t=0}$$
where $v,w$ are vector fields on an open $U \subset R^n$, $g_v^t$ is the "phase flow" of $v$ and where the lower star index denotes the "push-forward map".
Could you please provide an example of its calculation? Maybe working in $\mathbb{R}^2$ and choosing simple vector fields. For instance, let $v$ be the vector field given by $v_{(x,y)} = x (d/dx) + y (d/dy)$ and $w_{(x,y)} = (d/dx) + (d/dy)$. What does $[v,w]$ equal to here?
To compute this, we will use the fact that $[\partial_x,\partial_y]= 0$ and if that $X,Y$ are vector fields and $f$ a function, then : $$[X,fY] = f[X,Y] + X(f) Y$$
Let $v = x\partial_x + y\partial_y$ and $w = \partial_x + \partial_y$. Then : \begin{align} [v,w] &= [x\partial_x+y\partial_y,\partial_x+ \partial_y] \\ &= [x\partial_x,\partial_x] + [x\partial_x,\partial_y] + [y\partial_y,\partial_x] + [y\partial_y,\partial_y] \\ &= -\partial_x +0 + 0 -\partial_y \\ &= -w \end{align}
The Lie derivative
If we want to compute $\mathcal L_v w$ directly from the definition, we can. The flow of $v$ can be computed easily. It is given by : $$\operatorname{Fl}_v^t(x,y)= (e^tx,e^ty)$$
Its derivative is (in the $(\partial_x,\partial_y)$ basis) : $$T_{(x,y)}\operatorname{Fl}_v^t = \begin{pmatrix}e^t&0\\0&e^t\end{pmatrix}$$
Therefore, the push-forward is : \begin{align} (\operatorname {Fl}_v^t)_* w(x,y) &= T_{ \operatorname{Fl}_v^{-t}(x,y)}\operatorname{Fl}_v^t \circ w \circ \operatorname{Fl}_v^{-t}(x,y) \\ &= T_{ \operatorname{Fl}_v^{-t}(x,y)}\operatorname{Fl}_v^t (\partial_x + \partial_y) \\ &= e^t w \end{align}
Therefore, the Lie derivative is : $$\mathcal L_v w = \partial_t(\operatorname {Fl}_v^{-t})_* w|_{t=0} = -w$$