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Let $M$ be smooth manifold and $x \in M$. $\langle v_1,\dots,v_n \rangle = T_xM$ i want to find chart $(U,x)$ such $v_i = \frac{\partial }{\partial x_i}$.

Ok there is some chart $(W, y)$ and we have that $v_i = \frac{\partial }{\partial x_i} = \frac{\partial y_j}{\partial x_i}\frac{\partial}{\partial y_j}$. But i have no idea how to reconstruct $x_i$ by this data. Probably it's quite simple, but i can't..

qwenty
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    This is a local question, so WLOG you can take $M$ to be $\mathbb{R}^n$ and $x$ to be the origin. Then just perform a linear change of coordinates. – Qiaochu Yuan Aug 11 '15 at 18:26

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Via the chart $(W,y)$, we may think of $M$ as $\mathbb{R}^n$. Then, the desired chart can be the linear endomorphism carrying the standard basis $e_1,\ldots,e_n$ to the given basis $v_1,\ldots,v_n$.

Amitai Yuval
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