Definition of Cartesian Coordinate System

In summary: Thus, in summary, a Cartesian coordinate system is an affine coordinate system on a Euclidean space with orthonormal basis vectors. This means that for every point in the space, the derived sets of basis vectors are orthonormal, allowing for a unique representation of points in n dimensions.
  • #1
kent davidge
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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##.
 
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  • #2
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.
 
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  • #3
andrewkirk said:
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.
 
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  • #4
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.
 
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Related to Definition of Cartesian Coordinate System

What is a Cartesian Coordinate System?

A Cartesian Coordinate System is a mathematical system used to locate points in a two-dimensional or three-dimensional space. It is named after the French mathematician and philosopher, René Descartes, who developed the concept in the 17th century.

How does a Cartesian Coordinate System work?

A Cartesian Coordinate System uses two or three perpendicular lines, known as axes, to create a grid on which points can be plotted. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. In three-dimensional space, a third axis, known as the z-axis, is added. Points are located by their distance from the origin, which is where the axes intersect.

What are the benefits of using a Cartesian Coordinate System?

A Cartesian Coordinate System allows for precise and accurate representation of points in a two-dimensional or three-dimensional space. It is a fundamental concept in mathematics and is used in many fields, including physics, engineering, and computer graphics.

What is the difference between a Cartesian Coordinate System and a polar coordinate system?

A Cartesian Coordinate System uses two axes, while a polar coordinate system uses a single axis and a distance from the origin. In a Cartesian system, points are located based on their x and y coordinates, while in a polar system, points are located based on a distance from the origin and an angle.

How is a Cartesian Coordinate System used in real life?

A Cartesian Coordinate System is used in many practical applications, such as mapping and navigation systems, graphing and analyzing data, and creating computer graphics. It is also used in geometry to calculate distances and angles between points in a two-dimensional or three-dimensional space.

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