This is the definition I was given of Lie bracket:
Let be $M$ a differentiable manifold and $v$ and $w$ two vector fields on $M$. The Poisson bracket $[v,w]$ between $v$ and $w$ is a vector field on $M$ defined in this way: $$[v,w]_x (f_x)=v(w(f_x))-w(v(f_x))$$ for every $f_x \in C_x^\infty (M)$ space of germs at the point $x\in M$.
I interprete this definition in this way:
I have that $$[v,w]_x:f_x\in C_x^\infty(M) \mapsto[v,w]_x(f_x)\in \mathbb{R}$$ where $$[v,w]_x(f_x)=v_x(w(f_x))-w_x(v(f_x)).$$ Here I have problems in understanding what is $w(f_x)$. Since $v_x$ is an element of the tangent space at $x$ I have that $w(f_x)\in C_x^\infty$. Now I think that this germ is the equivalence class associated to the function $$y\in U_x \mapsto w_y(f_y).$$
Is my interpretation correct? Thanks for the help!