Definition of coordinate system

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In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to ##\mathbb R^m##, can we say that the polar coordinates on ##\mathbb R^2## are a coordinate system?

I was thinking about this and the answer sounds to be a no, because the polar coordinates are not everywhere bijective to the cartesian coordinates, which we know, is a coordinate system that spans ##\mathbb R^2##.
 

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  • #2
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Your definition on a manifold is called a chart (##f(U)##) or a coordinate chart ##(U,f)##.
The usual coordinates, are the Cartesian coordinates.
Polar, or cylindrical coordinates are also coordinates, even though not Cartesian.

In general, coordinates are any system which allows to uniquely specify a point in some space.
 
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  • #3
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In general, coordinates are any system which allows to uniquely specify a point in some space.
But do the polar coordinates uniquely specify a point in ##\mathbb R^2##? I think there's a issue when ##r = 0##.
 
  • #4
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But do the polar coordinates uniquely specify a point in ##\mathbb R^2##? I think there's a issue when ##r = 0##.
Yes, the origin has to be assigned separately by a definition. ##r=0## is o.k. but it has no angle, but we can simply require ##0:=(0,0)## and have a unique system again. ##(0,\varphi)## with ##\varphi > 0## will then be undefined. But this is more of a debate for logicians (or linguists), and I'm neither.
 
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Yes, coordinate systems can be locally- or globally- defined. In the Polar case, they are defined only locally. EDIT: Most coord systems are locally, otherwise the manifold is isomorphic to the space where it is embedded, i.e., local homeos become global ones.
 
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