Coordinate basis vs local frame?

In summary, there is a difference between local frames and coordinate bases, with the former being a more general concept that applies to any vector bundle and the latter being specific to the tangent bundle of a smooth manifold. The connection form article uses notation that can apply to both types of frames, while the term 'coordinate frame' is reserved for the special case of a tangent bundle.
  • #1
pellman
684
5
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 [tex]e_\alpha[/tex] for the basis elements instead of the partial derivative notation [tex]\partial_\alpha[/tex] typically used for coordinate bases. Is it because not all bases are coordinate bases?
 
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  • #2
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 [itex]\{\frac{\partial}{\partial x^i}\}_i[/itex].
 
  • #3
To my own amazement, I actually get it. Thanks, Landau.
 

What is the difference between coordinate basis and local frame?

Coordinate basis refers to a set of vectors used to describe the position of a point in a space, while local frame refers to a set of vectors used to describe the orientation of an object in a space.

How are coordinate basis and local frame related?

Coordinate basis and local frame are related in that they both use vectors to describe the position and orientation of objects in a space. However, they serve different purposes and are not interchangeable.

When should I use coordinate basis or local frame?

Coordinate basis is typically used for describing the position of a point in space, while local frame is used for describing the orientation of an object in space. The choice depends on the specific needs of the problem being solved.

What are some examples of coordinate basis and local frame in real life?

Examples of coordinate basis include latitude and longitude coordinates on a map, or Cartesian coordinates in a 3D space. Examples of local frame include the orientation of a car on a road or the orientation of an airplane in flight.

Can coordinate basis and local frame be used together?

Yes, coordinate basis and local frame can be used together to fully describe the position and orientation of an object in a space. For example, in robotics, both coordinate basis and local frame are used to program the movement of a robot arm.

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