Coordinate basis vs local frame?

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SUMMARY

The discussion clarifies the distinction between local frames and coordinate bases in the context of vector bundles and smooth manifolds. A local frame refers to a basis of local sections applicable to any vector bundle, while a coordinate frame specifically pertains to the tangent bundle of a smooth manifold, represented by the basis of vector fields \{\frac{\partial}{\partial x^i}\}_i. The notation e_\alpha is used for basis elements in local frames, contrasting with the partial derivative notation \partial_\alpha for coordinate bases, highlighting that not all bases qualify as coordinate bases.

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The wikipedia article on connection forms refers to a local frame. What is the relationship between local frames and coordinate bases? Are they the same thing? Is one a subset of the other?

The connection form article uses general notation e_\alpha for the basis elements instead of the partial derivative notation \partial_\alpha typically used for coordinate bases. Is it because not all bases are coordinate bases?
 
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It is my understanding that 'local frame' is defined for any vector bundle (a basis of local sections), while the term 'coordinate frame' is reserved for the special case where the vector bundle is the tangent bundle of a smooth manifold, and the local frame is the usual basis of vector fields \{\frac{\partial}{\partial x^i}\}_i.
 
To my own amazement, I actually get it. Thanks, Landau.
 

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