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.