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Suppose I have a local nbd $U$ in a manifold centered around $p$ and a chart on it. The chart is given by $X=(x_1,x_2,\cdots,x_n):U\rightarrow V\subset\mathbb{R}^n$. I consider the vector fields $\frac{\partial}{\partial x_i}$, ($i=1,\cdots,n$) which are given by the curve $c_i:(-\epsilon,\epsilon):\rightarrow U$, $c_i(t)=X^{-1}(0,0,\cdots,t,\cdots,0)$ ($t$ at the $i$-th coordinate).

I want to ask whether $\bigg[\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\bigg]$ is always $0$ ?

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Hint: Compute the action of your $\bigg[\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\bigg]$ on an arbitrary smooth function $f$ – Yuri Vyatkin Mar 1 '13 at 11:03

This is Schwarz's theorem on symmetric second derivatives.

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I knew it as Young's theorem.. – Berci Mar 1 '13 at 11:17

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