I Definition of Cartesian Coordinate System

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

Science Advisor
Homework Helper
Insights Author
Gold Member
3,715
1,344
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

Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2018 Award
15,294
5,447
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

Science Advisor
Gold Member
4,138
1,726
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.
 

Want to reply to this thread?

"Definition of Cartesian Coordinate System" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top