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Let $v,w$ be vector fields on a smooth manifold $M$ (i.e. $v : M \rightarrow TM = \lbrace (p, v_p) : p \in M, v_p \in T_p M \rbrace$). The Lie brackets of $v,w$ are defined as $$ [v,w](f)|_p = v_p(w_p(f)) - w_p(v_p(f)),$$ for every $f \in C^{\infty}(U,\mathbb R)$, with $U$ a neighborhood of $p$.

But $v_p, w_p \in T_p M$, therefore $w_p(f), v_p(f) \in \mathbb R$ by definition, and therefore $v_p(w_p(f)) = w_p(v_p(f)) = 0$. So $[v,w](f)|_p \equiv 0,$ which doesn't make sense...

What am I getting wrong?

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Your formula is incorrect: the right side should have $v_p(w(f))$ instead of $v_p(w_p(f))$ where $w(f)$ is the function whose value at $p$ is $w_p(f)$, and similarly for the second term.

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