What are the general requirements for defining a coordinate system in R^3?

[mordechai9]

Say we have a vector field defined in R^3. That is, at every point p in R^3, we have the corresponding set (p, v(p)). In representing this field, as far as I can tell, we have a certain list of very general requirements. That seems to be

a.) an origin,
b.) three everywhere non-coplanar curves, call them c1, c2, c3.

Where "everywhere non-coplanar" means that at any given choice of parameters t1, t2, t3, then c1'(t1), c2'(t2), c3'(t3) form a linearly independent set.

The requirement (a.) is necessary in order for us to locate distinct points. The requirement (b.) gives us a way in which to measure the position of the points. I suppose this could work by saying that when you specify a point p = (p1,p2,p3), a three-tuple, then that means the point lies at the intersection of the planes perpendicular to the velocity vectors c1'(p1), c2'(p2), c3'(p3).

Similarly, (b.) also gives us a way to provide a basis at every point, since the basis at point p=(p1,p2,p3) can be written as c1'(p1), c2'(p2), c3'(p3).

Does this seem correct, and if not, what are the general requirements for a coordinate system?

 

[mordechai9]

I'm thinking that maybe my suggestion above doesn't work, so let me rephrase/restate my question.

Say we want to define a vector field in R^3. Then we need a way to consistently define a directional basis at each point, and a position for each point. The common choices are well known -- Cartesian coordinates, spherical coordinates, cylindrical coordinates, and so on. But what are the generic rules or axioms that we must follow in constructing an entirely arbitrary system?

The motivation for this question is the seemingly limitless flexibility in choosing a coordinate system. In practice it seems we always specify coordinate systems by relating them to the standard Cartesian or "natural" coordinates and then going from there. But I see no reason why we should have to reference the Cartesian coordinates in general.