Hello! I am a bit confused about non-coordinate basis. I understand the way they are defined (I think) and the main purpose is to get on a manifold a coordinate system that is orthonormal at any point on the manifold (right?). So if you have a coordinate basis ##e_\alpha##, you get to a non-coordinate by doing a transformation ##\hat{e_\mu}=a_\mu^\alpha e_\alpha##, such that ##g_{\mu\nu}e_\alpha^\mu e_\beta^\nu = \delta_{\alpha\beta}## (or ##=\eta_{\alpha \beta}##). In polar coordinates in ##R^2##, the way it is done is to divide the angular basis by r i.e. ##\hat{e}_r = \partial_r## and ##\hat{e}_\phi = \frac{1}{r}\partial_\phi##. I am a bit confused why this new basis is non-coordinate. If initially a point had coordinates ##(2,1)##, not it has coordinates ##(2,1/2)##. Why is this not a coordinate basis? You can still identify any point in space by providing 2 numbers, so it seems to be a good coordinate system. Can someone explain this to me? Thank you!(adsbygoogle = window.adsbygoogle || []).push({});

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# I Non-coordinate bases

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