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Let $f$ be a smooth function, $f\colon\mathbb{R}^2\to \mathbb{R}$.

What is $\left[\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right]$ ?

I want to say it's $0$ since $\frac{\partial f^2}{\partial x\partial y}=\frac{\partial f^2}{\partial y\partial x}$ but I am unsure about why for the two functions : $\frac{\partial f}{\partial x}$ ,$\frac{\partial f}{\partial y}$ the composition is the same...

Can someone please help with this ? [I didn't know what tag to give this post, hope it's ok]

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    Yeah, sorry, I implicitly assumed (and didn't notice the $f$, and even copied it into the first draft of my answer!) that the OP was talking about Lie bracket of vector fields.2012-04-23

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Your first answer is correct for the reason that you said. There is no composition of functions involved in Lie bracketing vector fields, only composition of derivations (= composition of operators). It may help to look at some nontrivial Lie brackets, e.g., compute $ \left[\frac{\partial}{\partial x}, x\frac{\partial}{\partial y} \right]. $

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    Actually, I am thw one with the mistake. I didn't understand the question and it's interpreted like you said! thank you2012-04-23