I am quite confused as to what is the difference between a frame and a co-ordinate system. The wikipedia page was not very helpful for me. I would be very happy if someone could give me a non-rigorous idea about what exactly the difference is. My background involves basic differential geometry. I have also done some very basic differential topology and am aware of manifolds and some topological properties associated with them.
I'll try and elaborate a bit about the context. The Frenet frame fields are useful in that they express the rate of change of the unit vectors constituting the frame in terms of the vectors themselves. Also the fact that it is a "moving frame" is supposedly useful. I am thinking, this means analysis of the various properties associated with the curve is easier as the frame moves along with the observer.Though I couldn't explicitly point out what these properties are. Am I right??
Why exactly is this frame so useful??Also doesn't the notion of a frame reject path independence?
I apologise if my question seems a bit vague. I do hope its enough to convey what I am looking for.
I am not much exposed to physics. But I would welcome answers involving examples from physics if it helps matters.
P.S.: If this can be tied in with notions about the coordinate basis of vector space, then all the better.I am looking for something all encompassing. I hope that doesnt come across as foolish.
Best Answer
A coordinate system is a homeomorphism from a neighborhood of $M$ to $\mathbb{R}^n$. This gives you coordinates and induces coordinate vector fields $\frac{\partial}{\partial x_i}$. These form a basis in each tangent space.
If you specify a metric on $M$, you will see that unless your manifold has no curvature, the basis in each tangent space induced by coordinates is not orthonormal. By performing Gram-Schmidt in each tangent space to the coordinate fields, you can create a "frame field" - $n$ orthonormal vector fields which form an orthonormal basis of each tangent space in this neighborhood. These fields cannot be integrated to coordinates (again because of curvature).
The short answer to your question: coordinate systems are charts to $\mathbb{R}^n$ that give coordinate functions locally on a manifold, while frames are vector fields that give orthonormal bases for each tangent space in a neighborhood of the manifold.
The Frenet frame is an orthonormal basis of the pullback of the tangent bundle $T\mathbb{R}^3$ via a curve in $\mathbb{R}^3$. It encodes information about the embedding of the curve in $\mathbb{R}^3$. (A frame field on a vector bundle with a bundle metric is a little bit more general than a frame field on the tangent bundle on a manifold, but the basic idea is still the same.)