Coordinate basis vs local frame?

[pellman]

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?

 

[Landau]

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.

 

[pellman]

To my own amazement, I actually get it. Thanks, Landau.