Suppose we have a manifold ##M## and at ##p \in M## we have a basis for the tangent space of vectors ##X_i##. Since ##M## is a manifold, there exists a local chart ##(U,\phi)## about ##p##. Now the question is, given such a chart , how can we construct a new chart in a such that ##X_i = \left. \frac{\partial}{\partial x^\mu} \right|_{p}##.(adsbygoogle = window.adsbygoogle || []).push({});

I know there is a theorem that says given commuting vector fields we can find a chart such that these vector fields locally is the coordinate basis of that chart.

However, I want to prove in a transparent manner the less general statement above; that we can construct a coordinate system such that the vector ##X_i## _in the tangent space at p_ is the coordinate basis ##\left. \tfrac{\partial}{\partial x^\mu} \right|_{p}## just at p.

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# Constructing a chart with coord. basis equal to given basis at one pt.

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