Let $M$ be differentiable n-manifold. Suppose that at $p \in M$ we are given a basis of tangent space $T_pM$ denoted as $(X^1,\ldots,X^n)$. Can we construct a coordinate representation $(x^1,\ldots,x^n)$, such that $\left. \frac{\partial}{\partial x^i} \right|_{p} = X^i$ for all $i=1,\ldots,n$? Should it be clear that such coordinate chart exists?
Coordinate representation for given tangent space
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differential-geometry
manifolds
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4Take any coordinates around p and use a linear change of basis. – 2012-12-22