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Mathematics
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Definition of Cartesian Coordinate System
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[QUOTE="andrewkirk, post: 6117338, member: 265790"] I don't think there is any such thing as normalised polar coordinates. What there is is a normalised [I]frame[/I] (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 [I]coordinate system[/I], a [I]basis [/I]for the tangent space at a point and a [I]frame[/I] 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. [/QUOTE]
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Definition of Cartesian Coordinate System
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