Definition of Cartesian Coordinate System

[kent davidge]

I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that

- the basis vectors are the same $\forall x \in R^n$
- the basis vectors are orthonormal at each $x \in R^n$

So for instance, normalized polar coordinates do not constitute a Cartesian coordinate System, because despite being orthonormal they will change with $x \in R^n$.

 

[andrewkirk]

I don't think there is any such thing as normalised polar coordinates. What there is is a normalised frame (set of two vector fields) that gives a basis of two vectors for the tangent space at each point. It is important to understand the difference between a coordinate system, a basis for the tangent space at a point and a frame that maps each point to a basis for the tangent space at that point.

Unlike the frame that is derived from a polar coordinate system in the natural way, there is no coordinate system from which the frame of normalised polar vectors can be derived. Schutz calls this a non-coordinate basis (I'd call it a 'frame') and he uses the normalised polar basis (frame) as his prime example of such a thing. See section 5.5 of his 'A First Course in General Relativity'.

I think it follows that we can say that a Cartesian coordinate system for $\mathbb R^n$ is one for which the derived sets of basis vectors are everywhere orthonormal.

 

[Orodruin]

I think it follows that we can say that a Cartesian coordinate system for RnRn\mathbb R^n is one for which the derived sets of basis vectors are everywhere orthonormal.

This is correct. The definition of a Cartesian coordinate system is that it is an affine coordinate system on a Euclidean space sich that the basis vectors are orthonormal. It is easy to show that this is equivalent to the above statement.

 

[WWGD]

Cartesian coordinates have the property that the point with x=a, y=b has the coordinate representation (a,b). This comes from the Ortho projection into the axes. And, of course, this generalizes to n dimensions.